Main:z from

Percentage Accurate: 91.8% → 99.3%

Time: 25.4s

Alternatives: 25

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{x + 1}\\ t_4 := \sqrt{1 + t}\\ t_5 := t\_4 + \sqrt{t}\\ \mathbf{if}\;z \leq 1.52 \cdot 10^{+29}:\\ \;\;\;\;\frac{\left(\left(\frac{1}{\sqrt{z} + t\_1} + t\_2\right) + t\_3\right) - \left(\sqrt{y} + \sqrt{x}\right)}{t\_5} \cdot \left(\frac{-1}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) - \left(t\_3 + \left(t\_2 + t\_1\right)\right)} + t\_5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{\sinh^{-1} \left(-\sqrt{y}\right)} + e^{\sinh^{-1} \left(-\sqrt{x}\right)}\right) - \left(\sqrt{z} - t\_1\right)\right) - \left(\sqrt{t} - t\_4\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (sqrt (+ 1.0 t)))
        (t_5 (+ t_4 (sqrt t))))
   (if (<= z 1.52e+29)
     (*
      (/
       (- (+ (+ (/ 1.0 (+ (sqrt z) t_1)) t_2) t_3) (+ (sqrt y) (sqrt x)))
       t_5)
      (+
       (/ -1.0 (- (+ (+ (sqrt y) (sqrt z)) (sqrt x)) (+ t_3 (+ t_2 t_1))))
       t_5))
     (-
      (-
       (+ (exp (asinh (- (sqrt y)))) (exp (asinh (- (sqrt x)))))
       (- (sqrt z) t_1))
      (- (sqrt t) t_4)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((x + 1.0));
	double t_4 = sqrt((1.0 + t));
	double t_5 = t_4 + sqrt(t);
	double tmp;
	if (z <= 1.52e+29) {
		tmp = (((((1.0 / (sqrt(z) + t_1)) + t_2) + t_3) - (sqrt(y) + sqrt(x))) / t_5) * ((-1.0 / (((sqrt(y) + sqrt(z)) + sqrt(x)) - (t_3 + (t_2 + t_1)))) + t_5);
	} else {
		tmp = ((exp(asinh(-sqrt(y))) + exp(asinh(-sqrt(x)))) - (sqrt(z) - t_1)) - (sqrt(t) - t_4);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = math.sqrt((y + 1.0))
	t_3 = math.sqrt((x + 1.0))
	t_4 = math.sqrt((1.0 + t))
	t_5 = t_4 + math.sqrt(t)
	tmp = 0
	if z <= 1.52e+29:
		tmp = (((((1.0 / (math.sqrt(z) + t_1)) + t_2) + t_3) - (math.sqrt(y) + math.sqrt(x))) / t_5) * ((-1.0 / (((math.sqrt(y) + math.sqrt(z)) + math.sqrt(x)) - (t_3 + (t_2 + t_1)))) + t_5)
	else:
		tmp = ((math.exp(math.asinh(-math.sqrt(y))) + math.exp(math.asinh(-math.sqrt(x)))) - (math.sqrt(z) - t_1)) - (math.sqrt(t) - t_4)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = sqrt(Float64(1.0 + t))
	t_5 = Float64(t_4 + sqrt(t))
	tmp = 0.0
	if (z <= 1.52e+29)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(z) + t_1)) + t_2) + t_3) - Float64(sqrt(y) + sqrt(x))) / t_5) * Float64(Float64(-1.0 / Float64(Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x)) - Float64(t_3 + Float64(t_2 + t_1)))) + t_5));
	else
		tmp = Float64(Float64(Float64(exp(asinh(Float64(-sqrt(y)))) + exp(asinh(Float64(-sqrt(x))))) - Float64(sqrt(z) - t_1)) - Float64(sqrt(t) - t_4));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	t_2 = sqrt((y + 1.0));
	t_3 = sqrt((x + 1.0));
	t_4 = sqrt((1.0 + t));
	t_5 = t_4 + sqrt(t);
	tmp = 0.0;
	if (z <= 1.52e+29)
		tmp = (((((1.0 / (sqrt(z) + t_1)) + t_2) + t_3) - (sqrt(y) + sqrt(x))) / t_5) * ((-1.0 / (((sqrt(y) + sqrt(z)) + sqrt(x)) - (t_3 + (t_2 + t_1)))) + t_5);
	else
		tmp = ((exp(asinh(-sqrt(y))) + exp(asinh(-sqrt(x)))) - (sqrt(z) - t_1)) - (sqrt(t) - t_4);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.52e+29], N[(N[(N[(N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] * N[(N[(-1.0 / N[(N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(t$95$3 + N[(t$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Exp[N[ArcSinh[(-N[Sqrt[y], $MachinePrecision])], $MachinePrecision]], $MachinePrecision] + N[Exp[N[ArcSinh[(-N[Sqrt[x], $MachinePrecision])], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.52e29

   1. Initial program 95.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 93.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(z + 1\right) - z, \mathsf{fma}\left(\sqrt{y + 1} - \sqrt{y}, \left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{1 + x} - \sqrt{x}\right), e^{\sinh^{-1} \left(-\sqrt{x}\right) \cdot 2}\right), \left(\sqrt{z} + \sqrt{z + 1}\right) \cdot \left({\left(e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}^{3} + {\left(e^{\sinh^{-1} \left(-\sqrt{x}\right)}\right)}^{3}\right)\right)}{\left(\sqrt{z} + \sqrt{z + 1}\right) \cdot \mathsf{fma}\left(\sqrt{y + 1} - \sqrt{y}, \left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{1 + x} - \sqrt{x}\right), e^{\sinh^{-1} \left(-\sqrt{x}\right) \cdot 2}\right)}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      2. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\color{blue}{\sqrt{1 + z}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{\color{blue}{1 + z}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{\color{blue}{z + 1}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{\color{blue}{z + 1}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \sqrt{\color{blue}{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \sqrt{\color{blue}{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \sqrt{\color{blue}{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      15. rem-square-sqrt N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      16. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      17. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      18. lower-+.f64 97.7

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      20. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{\color{blue}{z + 1}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      21. lift-+.f64 97.7

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{\color{blue}{z + 1}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   8. Applied rewrites 97.7%

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\left(\sqrt{t} + \left(\sqrt{1 + t} + \frac{1}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right)\right) \cdot \left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)}{\sqrt{t} + \sqrt{1 + t}}} \]

   11. Applied rewrites 35.3%

       \[\leadsto \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{t}\right) + \frac{1}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)}\right) \cdot \frac{\left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)}{\sqrt{1 + t} + \sqrt{t}}} \]

   if 1.52e29 < z

   1. Initial program 84.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(z + 1\right) - z, \mathsf{fma}\left(\sqrt{y + 1} - \sqrt{y}, \left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{1 + x} - \sqrt{x}\right), e^{\sinh^{-1} \left(-\sqrt{x}\right) \cdot 2}\right), \left(\sqrt{z} + \sqrt{z + 1}\right) \cdot \left({\left(e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}^{3} + {\left(e^{\sinh^{-1} \left(-\sqrt{x}\right)}\right)}^{3}\right)\right)}{\left(\sqrt{z} + \sqrt{z + 1}\right) \cdot \mathsf{fma}\left(\sqrt{y + 1} - \sqrt{y}, \left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{1 + x} - \sqrt{x}\right), e^{\sinh^{-1} \left(-\sqrt{x}\right) \cdot 2}\right)}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.52 \cdot 10^{+29}:\\ \;\;\;\;\frac{\left(\left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)}{\sqrt{1 + t} + \sqrt{t}} \cdot \left(\frac{-1}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) - \left(\sqrt{x + 1} + \left(\sqrt{y + 1} + \sqrt{z + 1}\right)\right)} + \left(\sqrt{1 + t} + \sqrt{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{\sinh^{-1} \left(-\sqrt{y}\right)} + e^{\sinh^{-1} \left(-\sqrt{x}\right)}\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := e^{\sinh^{-1} \left(-\sqrt{x}\right)}\\ t_3 := \sqrt{1 + t} + \sqrt{t}\\ \frac{\mathsf{fma}\left(\left(1 + t\right) - t, \frac{-1}{\left(\sqrt{z} - t\_1\right) - \left(t\_2 - \left(\sqrt{y} - \sqrt{y + 1}\right)\right)}, t\_3\right)}{\frac{1}{\left(e^{\sinh^{-1} \left(-\sqrt{y}\right)} + t\_2\right) + \frac{\left(z + 1\right) - z}{\sqrt{z} + t\_1}} \cdot t\_3} \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0)))
        (t_2 (exp (asinh (- (sqrt x)))))
        (t_3 (+ (sqrt (+ 1.0 t)) (sqrt t))))
   (/
    (fma
     (- (+ 1.0 t) t)
     (/ -1.0 (- (- (sqrt z) t_1) (- t_2 (- (sqrt y) (sqrt (+ y 1.0))))))
     t_3)
    (*
     (/
      1.0
      (+
       (+ (exp (asinh (- (sqrt y)))) t_2)
       (/ (- (+ z 1.0) z) (+ (sqrt z) t_1))))
     t_3))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = exp(asinh(-sqrt(x)));
	double t_3 = sqrt((1.0 + t)) + sqrt(t);
	return fma(((1.0 + t) - t), (-1.0 / ((sqrt(z) - t_1) - (t_2 - (sqrt(y) - sqrt((y + 1.0)))))), t_3) / ((1.0 / ((exp(asinh(-sqrt(y))) + t_2) + (((z + 1.0) - z) / (sqrt(z) + t_1)))) * t_3);
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = exp(asinh(Float64(-sqrt(x))))
	t_3 = Float64(sqrt(Float64(1.0 + t)) + sqrt(t))
	return Float64(fma(Float64(Float64(1.0 + t) - t), Float64(-1.0 / Float64(Float64(sqrt(z) - t_1) - Float64(t_2 - Float64(sqrt(y) - sqrt(Float64(y + 1.0)))))), t_3) / Float64(Float64(1.0 / Float64(Float64(exp(asinh(Float64(-sqrt(y)))) + t_2) + Float64(Float64(Float64(z + 1.0) - z) / Float64(sqrt(z) + t_1)))) * t_3))
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[ArcSinh[(-N[Sqrt[x], $MachinePrecision])], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(1.0 + t), $MachinePrecision] - t), $MachinePrecision] * N[(-1.0 / N[(N[(N[Sqrt[z], $MachinePrecision] - t$95$1), $MachinePrecision] - N[(t$95$2 - N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] / N[(N[(1.0 / N[(N[(N[Exp[N[ArcSinh[(-N[Sqrt[y], $MachinePrecision])], $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision] + N[(N[(N[(z + 1.0), $MachinePrecision] - z), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 90.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

4. Applied rewrites 89.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(z + 1\right) - z, \mathsf{fma}\left(\sqrt{y + 1} - \sqrt{y}, \left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{1 + x} - \sqrt{x}\right), e^{\sinh^{-1} \left(-\sqrt{x}\right) \cdot 2}\right), \left(\sqrt{z} + \sqrt{z + 1}\right) \cdot \left({\left(e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}^{3} + {\left(e^{\sinh^{-1} \left(-\sqrt{x}\right)}\right)}^{3}\right)\right)}{\left(\sqrt{z} + \sqrt{z + 1}\right) \cdot \mathsf{fma}\left(\sqrt{y + 1} - \sqrt{y}, \left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{1 + x} - \sqrt{x}\right), e^{\sinh^{-1} \left(-\sqrt{x}\right) \cdot 2}\right)}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

6. Applied rewrites 94.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}}} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   2. flip-- N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\color{blue}{\sqrt{1 + z}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{\color{blue}{1 + z}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{\color{blue}{z + 1}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{\color{blue}{z + 1}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \sqrt{\color{blue}{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   10. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \sqrt{\color{blue}{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \sqrt{\color{blue}{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   12. rem-square-sqrt N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   13. lift-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   14. lift-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   15. rem-square-sqrt N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   16. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   17. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   18. lower-+.f64 94.9

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   19. lift-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   20. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{\color{blue}{z + 1}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   21. lift-+.f64 94.9

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{\color{blue}{z + 1}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

8. Applied rewrites 94.9%

   \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]
9. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \color{blue}{e^{\sinh^{-1} \left(-\sqrt{y}\right)}}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   2. sinh-+-cosh-rev N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \color{blue}{\left(\cosh \sinh^{-1} \left(-\sqrt{y}\right) + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   3. lift-asinh.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\cosh \color{blue}{\sinh^{-1} \left(-\sqrt{y}\right)} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   4. lift-neg.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\cosh \sinh^{-1} \color{blue}{\left(\mathsf{neg}\left(\sqrt{y}\right)\right)} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   5. neg-mul-1 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\cosh \sinh^{-1} \color{blue}{\left(-1 \cdot \sqrt{y}\right)} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   6. cosh-asinh-rev N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\color{blue}{\sqrt{\left(-1 \cdot \sqrt{y}\right) \cdot \left(-1 \cdot \sqrt{y}\right) + 1}} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   7. neg-mul-1 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\sqrt{\color{blue}{\left(\mathsf{neg}\left(\sqrt{y}\right)\right)} \cdot \left(-1 \cdot \sqrt{y}\right) + 1} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   8. lift-neg.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\sqrt{\color{blue}{\left(-\sqrt{y}\right)} \cdot \left(-1 \cdot \sqrt{y}\right) + 1} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   9. neg-mul-1 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\sqrt{\left(-\sqrt{y}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{y}\right)\right)} + 1} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   10. lift-neg.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\sqrt{\left(-\sqrt{y}\right) \cdot \color{blue}{\left(-\sqrt{y}\right)} + 1} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   11. lift-neg.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\sqrt{\color{blue}{\left(\mathsf{neg}\left(\sqrt{y}\right)\right)} \cdot \left(-\sqrt{y}\right) + 1} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   12. lift-neg.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\sqrt{\left(\mathsf{neg}\left(\sqrt{y}\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\sqrt{y}\right)\right)} + 1} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   13. sqr-neg-rev N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\sqrt{\color{blue}{\sqrt{y} \cdot \sqrt{y}} + 1} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   14. lift-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\sqrt{\color{blue}{\sqrt{y}} \cdot \sqrt{y} + 1} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   15. lift-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\sqrt{\sqrt{y} \cdot \color{blue}{\sqrt{y}} + 1} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   16. rem-square-sqrt N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\sqrt{\color{blue}{y} + 1} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\sqrt{\color{blue}{y + 1}} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   18. lift-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\color{blue}{\sqrt{y + 1}} + \sinh \sinh^{-1} \left(-\sqrt{y}\right)\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   19. lift-asinh.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \left(\sqrt{y + 1} + \sinh \color{blue}{\sinh^{-1} \left(-\sqrt{y}\right)}\right)\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

10. Applied rewrites 92.8%

    \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

11. Final simplification 92.8%

    \[\leadsto \frac{\mathsf{fma}\left(\left(1 + t\right) - t, \frac{-1}{\left(\sqrt{z} - \sqrt{z + 1}\right) - \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} - \left(\sqrt{y} - \sqrt{y + 1}\right)\right)}, \sqrt{1 + t} + \sqrt{t}\right)}{\frac{1}{\left(e^{\sinh^{-1} \left(-\sqrt{y}\right)} + e^{\sinh^{-1} \left(-\sqrt{x}\right)}\right) + \frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}} \cdot \left(\sqrt{1 + t} + \sqrt{t}\right)} \]
12. Add Preprocessing

Alternative 3: 90.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{z + 1}\\ t_4 := \sqrt{x + 1}\\ t_5 := \sqrt{1 + t}\\ t_6 := \left(\left(\left(t\_4 - \sqrt{x}\right) - \left(\sqrt{y} - t\_2\right)\right) - \left(\sqrt{z} - t\_3\right)\right) - \left(\sqrt{t} - t\_5\right)\\ \mathbf{if}\;t\_6 \leq 1:\\ \;\;\;\;1 - \left(t\_1 - \left(t\_2 + t\_3\right)\right)\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;\left(t\_4 + t\_2\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{elif}\;t\_6 \leq 3:\\ \;\;\;\;\left(\left(t\_2 + 1\right) + t\_3\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(3 + t\_5\right) - \left(t\_1 + \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (+ (sqrt y) (sqrt z)) (sqrt x)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (sqrt (+ x 1.0)))
        (t_5 (sqrt (+ 1.0 t)))
        (t_6
         (-
          (- (- (- t_4 (sqrt x)) (- (sqrt y) t_2)) (- (sqrt z) t_3))
          (- (sqrt t) t_5))))
   (if (<= t_6 1.0)
     (- 1.0 (- t_1 (+ t_2 t_3)))
     (if (<= t_6 2.0)
       (- (+ t_4 t_2) (+ (sqrt y) (sqrt x)))
       (if (<= t_6 3.0)
         (- (+ (+ t_2 1.0) t_3) t_1)
         (- (+ 3.0 t_5) (+ t_1 (sqrt t))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (sqrt(y) + sqrt(z)) + sqrt(x);
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((z + 1.0));
	double t_4 = sqrt((x + 1.0));
	double t_5 = sqrt((1.0 + t));
	double t_6 = (((t_4 - sqrt(x)) - (sqrt(y) - t_2)) - (sqrt(z) - t_3)) - (sqrt(t) - t_5);
	double tmp;
	if (t_6 <= 1.0) {
		tmp = 1.0 - (t_1 - (t_2 + t_3));
	} else if (t_6 <= 2.0) {
		tmp = (t_4 + t_2) - (sqrt(y) + sqrt(x));
	} else if (t_6 <= 3.0) {
		tmp = ((t_2 + 1.0) + t_3) - t_1;
	} else {
		tmp = (3.0 + t_5) - (t_1 + sqrt(t));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = (sqrt(y) + sqrt(z)) + sqrt(x)
    t_2 = sqrt((y + 1.0d0))
    t_3 = sqrt((z + 1.0d0))
    t_4 = sqrt((x + 1.0d0))
    t_5 = sqrt((1.0d0 + t))
    t_6 = (((t_4 - sqrt(x)) - (sqrt(y) - t_2)) - (sqrt(z) - t_3)) - (sqrt(t) - t_5)
    if (t_6 <= 1.0d0) then
        tmp = 1.0d0 - (t_1 - (t_2 + t_3))
    else if (t_6 <= 2.0d0) then
        tmp = (t_4 + t_2) - (sqrt(y) + sqrt(x))
    else if (t_6 <= 3.0d0) then
        tmp = ((t_2 + 1.0d0) + t_3) - t_1
    else
        tmp = (3.0d0 + t_5) - (t_1 + sqrt(t))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = (Math.sqrt(y) + Math.sqrt(z)) + Math.sqrt(x);
	double t_2 = Math.sqrt((y + 1.0));
	double t_3 = Math.sqrt((z + 1.0));
	double t_4 = Math.sqrt((x + 1.0));
	double t_5 = Math.sqrt((1.0 + t));
	double t_6 = (((t_4 - Math.sqrt(x)) - (Math.sqrt(y) - t_2)) - (Math.sqrt(z) - t_3)) - (Math.sqrt(t) - t_5);
	double tmp;
	if (t_6 <= 1.0) {
		tmp = 1.0 - (t_1 - (t_2 + t_3));
	} else if (t_6 <= 2.0) {
		tmp = (t_4 + t_2) - (Math.sqrt(y) + Math.sqrt(x));
	} else if (t_6 <= 3.0) {
		tmp = ((t_2 + 1.0) + t_3) - t_1;
	} else {
		tmp = (3.0 + t_5) - (t_1 + Math.sqrt(t));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (math.sqrt(y) + math.sqrt(z)) + math.sqrt(x)
	t_2 = math.sqrt((y + 1.0))
	t_3 = math.sqrt((z + 1.0))
	t_4 = math.sqrt((x + 1.0))
	t_5 = math.sqrt((1.0 + t))
	t_6 = (((t_4 - math.sqrt(x)) - (math.sqrt(y) - t_2)) - (math.sqrt(z) - t_3)) - (math.sqrt(t) - t_5)
	tmp = 0
	if t_6 <= 1.0:
		tmp = 1.0 - (t_1 - (t_2 + t_3))
	elif t_6 <= 2.0:
		tmp = (t_4 + t_2) - (math.sqrt(y) + math.sqrt(x))
	elif t_6 <= 3.0:
		tmp = ((t_2 + 1.0) + t_3) - t_1
	else:
		tmp = (3.0 + t_5) - (t_1 + math.sqrt(t))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = sqrt(Float64(x + 1.0))
	t_5 = sqrt(Float64(1.0 + t))
	t_6 = Float64(Float64(Float64(Float64(t_4 - sqrt(x)) - Float64(sqrt(y) - t_2)) - Float64(sqrt(z) - t_3)) - Float64(sqrt(t) - t_5))
	tmp = 0.0
	if (t_6 <= 1.0)
		tmp = Float64(1.0 - Float64(t_1 - Float64(t_2 + t_3)));
	elseif (t_6 <= 2.0)
		tmp = Float64(Float64(t_4 + t_2) - Float64(sqrt(y) + sqrt(x)));
	elseif (t_6 <= 3.0)
		tmp = Float64(Float64(Float64(t_2 + 1.0) + t_3) - t_1);
	else
		tmp = Float64(Float64(3.0 + t_5) - Float64(t_1 + sqrt(t)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = (sqrt(y) + sqrt(z)) + sqrt(x);
	t_2 = sqrt((y + 1.0));
	t_3 = sqrt((z + 1.0));
	t_4 = sqrt((x + 1.0));
	t_5 = sqrt((1.0 + t));
	t_6 = (((t_4 - sqrt(x)) - (sqrt(y) - t_2)) - (sqrt(z) - t_3)) - (sqrt(t) - t_5);
	tmp = 0.0;
	if (t_6 <= 1.0)
		tmp = 1.0 - (t_1 - (t_2 + t_3));
	elseif (t_6 <= 2.0)
		tmp = (t_4 + t_2) - (sqrt(y) + sqrt(x));
	elseif (t_6 <= 3.0)
		tmp = ((t_2 + 1.0) + t_3) - t_1;
	else
		tmp = (3.0 + t_5) - (t_1 + sqrt(t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 1.0], N[(1.0 - N[(t$95$1 - N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 2.0], N[(N[(t$95$4 + t$95$2), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 3.0], N[(N[(N[(t$95$2 + 1.0), $MachinePrecision] + t$95$3), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(3.0 + t$95$5), $MachinePrecision] - N[(t$95$1 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 73.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 4.3

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 4.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 97.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 6.3

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 21.2%

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

   if 2 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 26.2

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 26.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 24.5%

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

   if 3 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 95.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 86.7%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.3%

      \[\leadsto 2 + \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \left(3 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 74.4%

       \[\leadsto \left(3 + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \color{blue}{\sqrt{t}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 1:\\ \;\;\;\;1 - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) - \left(\sqrt{y + 1} + \sqrt{z + 1}\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 2:\\ \;\;\;\;\left(\sqrt{x + 1} + \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 3:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + 1\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{z + 1}\\ t_4 := \sqrt{x + 1}\\ t_5 := \sqrt{1 + t}\\ t_6 := \left(\left(\left(t\_4 - \sqrt{x}\right) - \left(\sqrt{y} - t\_2\right)\right) - \left(\sqrt{z} - t\_3\right)\right) - \left(\sqrt{t} - t\_5\right)\\ t_7 := 1 - \left(t\_1 - \left(t\_2 + t\_3\right)\right)\\ \mathbf{if}\;t\_6 \leq 1:\\ \;\;\;\;t\_7\\ \mathbf{elif}\;t\_6 \leq 2:\\ \;\;\;\;\left(t\_4 + t\_2\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{elif}\;t\_6 \leq 3:\\ \;\;\;\;t\_7\\ \mathbf{else}:\\ \;\;\;\;\left(3 + t\_5\right) - \left(t\_1 + \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (+ (sqrt y) (sqrt z)) (sqrt x)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (sqrt (+ x 1.0)))
        (t_5 (sqrt (+ 1.0 t)))
        (t_6
         (-
          (- (- (- t_4 (sqrt x)) (- (sqrt y) t_2)) (- (sqrt z) t_3))
          (- (sqrt t) t_5)))
        (t_7 (- 1.0 (- t_1 (+ t_2 t_3)))))
   (if (<= t_6 1.0)
     t_7
     (if (<= t_6 2.0)
       (- (+ t_4 t_2) (+ (sqrt y) (sqrt x)))
       (if (<= t_6 3.0) t_7 (- (+ 3.0 t_5) (+ t_1 (sqrt t))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (sqrt(y) + sqrt(z)) + sqrt(x);
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((z + 1.0));
	double t_4 = sqrt((x + 1.0));
	double t_5 = sqrt((1.0 + t));
	double t_6 = (((t_4 - sqrt(x)) - (sqrt(y) - t_2)) - (sqrt(z) - t_3)) - (sqrt(t) - t_5);
	double t_7 = 1.0 - (t_1 - (t_2 + t_3));
	double tmp;
	if (t_6 <= 1.0) {
		tmp = t_7;
	} else if (t_6 <= 2.0) {
		tmp = (t_4 + t_2) - (sqrt(y) + sqrt(x));
	} else if (t_6 <= 3.0) {
		tmp = t_7;
	} else {
		tmp = (3.0 + t_5) - (t_1 + sqrt(t));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = (sqrt(y) + sqrt(z)) + sqrt(x)
    t_2 = sqrt((y + 1.0d0))
    t_3 = sqrt((z + 1.0d0))
    t_4 = sqrt((x + 1.0d0))
    t_5 = sqrt((1.0d0 + t))
    t_6 = (((t_4 - sqrt(x)) - (sqrt(y) - t_2)) - (sqrt(z) - t_3)) - (sqrt(t) - t_5)
    t_7 = 1.0d0 - (t_1 - (t_2 + t_3))
    if (t_6 <= 1.0d0) then
        tmp = t_7
    else if (t_6 <= 2.0d0) then
        tmp = (t_4 + t_2) - (sqrt(y) + sqrt(x))
    else if (t_6 <= 3.0d0) then
        tmp = t_7
    else
        tmp = (3.0d0 + t_5) - (t_1 + sqrt(t))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = (Math.sqrt(y) + Math.sqrt(z)) + Math.sqrt(x);
	double t_2 = Math.sqrt((y + 1.0));
	double t_3 = Math.sqrt((z + 1.0));
	double t_4 = Math.sqrt((x + 1.0));
	double t_5 = Math.sqrt((1.0 + t));
	double t_6 = (((t_4 - Math.sqrt(x)) - (Math.sqrt(y) - t_2)) - (Math.sqrt(z) - t_3)) - (Math.sqrt(t) - t_5);
	double t_7 = 1.0 - (t_1 - (t_2 + t_3));
	double tmp;
	if (t_6 <= 1.0) {
		tmp = t_7;
	} else if (t_6 <= 2.0) {
		tmp = (t_4 + t_2) - (Math.sqrt(y) + Math.sqrt(x));
	} else if (t_6 <= 3.0) {
		tmp = t_7;
	} else {
		tmp = (3.0 + t_5) - (t_1 + Math.sqrt(t));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (math.sqrt(y) + math.sqrt(z)) + math.sqrt(x)
	t_2 = math.sqrt((y + 1.0))
	t_3 = math.sqrt((z + 1.0))
	t_4 = math.sqrt((x + 1.0))
	t_5 = math.sqrt((1.0 + t))
	t_6 = (((t_4 - math.sqrt(x)) - (math.sqrt(y) - t_2)) - (math.sqrt(z) - t_3)) - (math.sqrt(t) - t_5)
	t_7 = 1.0 - (t_1 - (t_2 + t_3))
	tmp = 0
	if t_6 <= 1.0:
		tmp = t_7
	elif t_6 <= 2.0:
		tmp = (t_4 + t_2) - (math.sqrt(y) + math.sqrt(x))
	elif t_6 <= 3.0:
		tmp = t_7
	else:
		tmp = (3.0 + t_5) - (t_1 + math.sqrt(t))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = sqrt(Float64(x + 1.0))
	t_5 = sqrt(Float64(1.0 + t))
	t_6 = Float64(Float64(Float64(Float64(t_4 - sqrt(x)) - Float64(sqrt(y) - t_2)) - Float64(sqrt(z) - t_3)) - Float64(sqrt(t) - t_5))
	t_7 = Float64(1.0 - Float64(t_1 - Float64(t_2 + t_3)))
	tmp = 0.0
	if (t_6 <= 1.0)
		tmp = t_7;
	elseif (t_6 <= 2.0)
		tmp = Float64(Float64(t_4 + t_2) - Float64(sqrt(y) + sqrt(x)));
	elseif (t_6 <= 3.0)
		tmp = t_7;
	else
		tmp = Float64(Float64(3.0 + t_5) - Float64(t_1 + sqrt(t)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = (sqrt(y) + sqrt(z)) + sqrt(x);
	t_2 = sqrt((y + 1.0));
	t_3 = sqrt((z + 1.0));
	t_4 = sqrt((x + 1.0));
	t_5 = sqrt((1.0 + t));
	t_6 = (((t_4 - sqrt(x)) - (sqrt(y) - t_2)) - (sqrt(z) - t_3)) - (sqrt(t) - t_5);
	t_7 = 1.0 - (t_1 - (t_2 + t_3));
	tmp = 0.0;
	if (t_6 <= 1.0)
		tmp = t_7;
	elseif (t_6 <= 2.0)
		tmp = (t_4 + t_2) - (sqrt(y) + sqrt(x));
	elseif (t_6 <= 3.0)
		tmp = t_7;
	else
		tmp = (3.0 + t_5) - (t_1 + sqrt(t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(1.0 - N[(t$95$1 - N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, 1.0], t$95$7, If[LessEqual[t$95$6, 2.0], N[(N[(t$95$4 + t$95$2), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 3.0], t$95$7, N[(N[(3.0 + t$95$5), $MachinePrecision] - N[(t$95$1 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1 or 2 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   1. Initial program 85.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 16.0

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 16.0%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.4%

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 97.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 6.3

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 21.2%

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

   if 3 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 95.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 86.7%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.3%

      \[\leadsto 2 + \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \left(3 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 74.4%

       \[\leadsto \left(3 + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \color{blue}{\sqrt{t}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 1:\\ \;\;\;\;1 - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) - \left(\sqrt{y + 1} + \sqrt{z + 1}\right)\right)\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 2:\\ \;\;\;\;\left(\sqrt{x + 1} + \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 3:\\ \;\;\;\;1 - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) - \left(\sqrt{y + 1} + \sqrt{z + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + t}\\ t_4 := \sqrt{y + 1}\\ t_5 := \left(\left(\left(t\_2 - \sqrt{x}\right) - \left(\sqrt{y} - t\_4\right)\right) - \left(\sqrt{z} - t\_1\right)\right) - \left(\sqrt{t} - t\_3\right)\\ t_6 := \sqrt{y} + \sqrt{z}\\ t_7 := \left(t\_6 + \sqrt{x}\right) + \sqrt{t}\\ \mathbf{if}\;t\_5 \leq 1:\\ \;\;\;\;\left(\sqrt{t} - t\_7\right) + 1\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\left(t\_2 + t\_4\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{elif}\;t\_5 \leq 3:\\ \;\;\;\;\left(\left(t\_1 - \sqrt{x}\right) - t\_6\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\left(3 + t\_3\right) - t\_7\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (+ 1.0 t)))
        (t_4 (sqrt (+ y 1.0)))
        (t_5
         (-
          (- (- (- t_2 (sqrt x)) (- (sqrt y) t_4)) (- (sqrt z) t_1))
          (- (sqrt t) t_3)))
        (t_6 (+ (sqrt y) (sqrt z)))
        (t_7 (+ (+ t_6 (sqrt x)) (sqrt t))))
   (if (<= t_5 1.0)
     (+ (- (sqrt t) t_7) 1.0)
     (if (<= t_5 2.0)
       (- (+ t_2 t_4) (+ (sqrt y) (sqrt x)))
       (if (<= t_5 3.0)
         (+ (- (- t_1 (sqrt x)) t_6) 2.0)
         (- (+ 3.0 t_3) t_7))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((1.0 + t));
	double t_4 = sqrt((y + 1.0));
	double t_5 = (((t_2 - sqrt(x)) - (sqrt(y) - t_4)) - (sqrt(z) - t_1)) - (sqrt(t) - t_3);
	double t_6 = sqrt(y) + sqrt(z);
	double t_7 = (t_6 + sqrt(x)) + sqrt(t);
	double tmp;
	if (t_5 <= 1.0) {
		tmp = (sqrt(t) - t_7) + 1.0;
	} else if (t_5 <= 2.0) {
		tmp = (t_2 + t_4) - (sqrt(y) + sqrt(x));
	} else if (t_5 <= 3.0) {
		tmp = ((t_1 - sqrt(x)) - t_6) + 2.0;
	} else {
		tmp = (3.0 + t_3) - t_7;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((z + 1.0d0))
    t_2 = sqrt((x + 1.0d0))
    t_3 = sqrt((1.0d0 + t))
    t_4 = sqrt((y + 1.0d0))
    t_5 = (((t_2 - sqrt(x)) - (sqrt(y) - t_4)) - (sqrt(z) - t_1)) - (sqrt(t) - t_3)
    t_6 = sqrt(y) + sqrt(z)
    t_7 = (t_6 + sqrt(x)) + sqrt(t)
    if (t_5 <= 1.0d0) then
        tmp = (sqrt(t) - t_7) + 1.0d0
    else if (t_5 <= 2.0d0) then
        tmp = (t_2 + t_4) - (sqrt(y) + sqrt(x))
    else if (t_5 <= 3.0d0) then
        tmp = ((t_1 - sqrt(x)) - t_6) + 2.0d0
    else
        tmp = (3.0d0 + t_3) - t_7
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((z + 1.0));
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = Math.sqrt((1.0 + t));
	double t_4 = Math.sqrt((y + 1.0));
	double t_5 = (((t_2 - Math.sqrt(x)) - (Math.sqrt(y) - t_4)) - (Math.sqrt(z) - t_1)) - (Math.sqrt(t) - t_3);
	double t_6 = Math.sqrt(y) + Math.sqrt(z);
	double t_7 = (t_6 + Math.sqrt(x)) + Math.sqrt(t);
	double tmp;
	if (t_5 <= 1.0) {
		tmp = (Math.sqrt(t) - t_7) + 1.0;
	} else if (t_5 <= 2.0) {
		tmp = (t_2 + t_4) - (Math.sqrt(y) + Math.sqrt(x));
	} else if (t_5 <= 3.0) {
		tmp = ((t_1 - Math.sqrt(x)) - t_6) + 2.0;
	} else {
		tmp = (3.0 + t_3) - t_7;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((z + 1.0))
	t_2 = math.sqrt((x + 1.0))
	t_3 = math.sqrt((1.0 + t))
	t_4 = math.sqrt((y + 1.0))
	t_5 = (((t_2 - math.sqrt(x)) - (math.sqrt(y) - t_4)) - (math.sqrt(z) - t_1)) - (math.sqrt(t) - t_3)
	t_6 = math.sqrt(y) + math.sqrt(z)
	t_7 = (t_6 + math.sqrt(x)) + math.sqrt(t)
	tmp = 0
	if t_5 <= 1.0:
		tmp = (math.sqrt(t) - t_7) + 1.0
	elif t_5 <= 2.0:
		tmp = (t_2 + t_4) - (math.sqrt(y) + math.sqrt(x))
	elif t_5 <= 3.0:
		tmp = ((t_1 - math.sqrt(x)) - t_6) + 2.0
	else:
		tmp = (3.0 + t_3) - t_7
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(1.0 + t))
	t_4 = sqrt(Float64(y + 1.0))
	t_5 = Float64(Float64(Float64(Float64(t_2 - sqrt(x)) - Float64(sqrt(y) - t_4)) - Float64(sqrt(z) - t_1)) - Float64(sqrt(t) - t_3))
	t_6 = Float64(sqrt(y) + sqrt(z))
	t_7 = Float64(Float64(t_6 + sqrt(x)) + sqrt(t))
	tmp = 0.0
	if (t_5 <= 1.0)
		tmp = Float64(Float64(sqrt(t) - t_7) + 1.0);
	elseif (t_5 <= 2.0)
		tmp = Float64(Float64(t_2 + t_4) - Float64(sqrt(y) + sqrt(x)));
	elseif (t_5 <= 3.0)
		tmp = Float64(Float64(Float64(t_1 - sqrt(x)) - t_6) + 2.0);
	else
		tmp = Float64(Float64(3.0 + t_3) - t_7);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + 1.0));
	t_2 = sqrt((x + 1.0));
	t_3 = sqrt((1.0 + t));
	t_4 = sqrt((y + 1.0));
	t_5 = (((t_2 - sqrt(x)) - (sqrt(y) - t_4)) - (sqrt(z) - t_1)) - (sqrt(t) - t_3);
	t_6 = sqrt(y) + sqrt(z);
	t_7 = (t_6 + sqrt(x)) + sqrt(t);
	tmp = 0.0;
	if (t_5 <= 1.0)
		tmp = (sqrt(t) - t_7) + 1.0;
	elseif (t_5 <= 2.0)
		tmp = (t_2 + t_4) - (sqrt(y) + sqrt(x));
	elseif (t_5 <= 3.0)
		tmp = ((t_1 - sqrt(x)) - t_6) + 2.0;
	else
		tmp = (3.0 + t_3) - t_7;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$6 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 1.0], N[(N[(N[Sqrt[t], $MachinePrecision] - t$95$7), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(N[(t$95$2 + t$95$4), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 3.0], N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision] + 2.0), $MachinePrecision], N[(N[(3.0 + t$95$3), $MachinePrecision] - t$95$7), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 73.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 53.9%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto 1 + \left(\sqrt{t} - \left(\color{blue}{\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} + \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 18.5%

      \[\leadsto 1 + \left(\sqrt{t} - \left(\color{blue}{\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} + \sqrt{t}\right)\right) \]

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 97.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 6.3

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 21.2%

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

   if 2 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 18.2%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 16.5%

      \[\leadsto 2 + \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 25.3%

       \[\leadsto 2 + \left(\left(\sqrt{1 + z} - \sqrt{x}\right) - \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) \]

   if 3 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 95.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 86.7%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.3%

      \[\leadsto 2 + \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \left(3 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 74.4%

       \[\leadsto \left(3 + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \color{blue}{\sqrt{t}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 1:\\ \;\;\;\;\left(\sqrt{t} - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right) + 1\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 2:\\ \;\;\;\;\left(\sqrt{x + 1} + \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 3:\\ \;\;\;\;\left(\left(\sqrt{z + 1} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{1 + t}\\ t_4 := \sqrt{z + 1}\\ t_5 := \sqrt{z} - t\_4\\ t_6 := \sqrt{y} - t\_2\\ t_7 := \left(\left(t\_1 - \sqrt{x}\right) - t\_6\right) - t\_5\\ t_8 := \sqrt{t} - t\_3\\ \mathbf{if}\;t\_7 \leq 0:\\ \;\;\;\;\left(\left(\frac{0.5}{\sqrt{x}} - t\_6\right) - t\_5\right) - t\_8\\ \mathbf{elif}\;t\_7 \leq 1.0002:\\ \;\;\;\;\left(\left(\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right) \cdot 0.5 + t\_1\right) - \sqrt{x}\right) - t\_8\\ \mathbf{elif}\;t\_7 \leq 2.002:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{z}} - \left(\left(\sqrt{y} + \sqrt{x}\right) - \left(t\_1 + t\_2\right)\right)\right) - t\_8\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{t\_3 + \sqrt{t}} + t\_4\right) + t\_2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (+ 1.0 t)))
        (t_4 (sqrt (+ z 1.0)))
        (t_5 (- (sqrt z) t_4))
        (t_6 (- (sqrt y) t_2))
        (t_7 (- (- (- t_1 (sqrt x)) t_6) t_5))
        (t_8 (- (sqrt t) t_3)))
   (if (<= t_7 0.0)
     (- (- (- (/ 0.5 (sqrt x)) t_6) t_5) t_8)
     (if (<= t_7 1.0002)
       (-
        (- (+ (* (+ (sqrt (/ 1.0 z)) (sqrt (/ 1.0 y))) 0.5) t_1) (sqrt x))
        t_8)
       (if (<= t_7 2.002)
         (- (- (/ 0.5 (sqrt z)) (- (+ (sqrt y) (sqrt x)) (+ t_1 t_2))) t_8)
         (+
          (-
           (+ (+ (/ 1.0 (+ t_3 (sqrt t))) t_4) t_2)
           (+ (+ (sqrt y) (sqrt z)) (sqrt x)))
          1.0))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((1.0 + t));
	double t_4 = sqrt((z + 1.0));
	double t_5 = sqrt(z) - t_4;
	double t_6 = sqrt(y) - t_2;
	double t_7 = ((t_1 - sqrt(x)) - t_6) - t_5;
	double t_8 = sqrt(t) - t_3;
	double tmp;
	if (t_7 <= 0.0) {
		tmp = (((0.5 / sqrt(x)) - t_6) - t_5) - t_8;
	} else if (t_7 <= 1.0002) {
		tmp = ((((sqrt((1.0 / z)) + sqrt((1.0 / y))) * 0.5) + t_1) - sqrt(x)) - t_8;
	} else if (t_7 <= 2.002) {
		tmp = ((0.5 / sqrt(z)) - ((sqrt(y) + sqrt(x)) - (t_1 + t_2))) - t_8;
	} else {
		tmp = ((((1.0 / (t_3 + sqrt(t))) + t_4) + t_2) - ((sqrt(y) + sqrt(z)) + sqrt(x))) + 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: t_8
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt((y + 1.0d0))
    t_3 = sqrt((1.0d0 + t))
    t_4 = sqrt((z + 1.0d0))
    t_5 = sqrt(z) - t_4
    t_6 = sqrt(y) - t_2
    t_7 = ((t_1 - sqrt(x)) - t_6) - t_5
    t_8 = sqrt(t) - t_3
    if (t_7 <= 0.0d0) then
        tmp = (((0.5d0 / sqrt(x)) - t_6) - t_5) - t_8
    else if (t_7 <= 1.0002d0) then
        tmp = ((((sqrt((1.0d0 / z)) + sqrt((1.0d0 / y))) * 0.5d0) + t_1) - sqrt(x)) - t_8
    else if (t_7 <= 2.002d0) then
        tmp = ((0.5d0 / sqrt(z)) - ((sqrt(y) + sqrt(x)) - (t_1 + t_2))) - t_8
    else
        tmp = ((((1.0d0 / (t_3 + sqrt(t))) + t_4) + t_2) - ((sqrt(y) + sqrt(z)) + sqrt(x))) + 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double t_2 = Math.sqrt((y + 1.0));
	double t_3 = Math.sqrt((1.0 + t));
	double t_4 = Math.sqrt((z + 1.0));
	double t_5 = Math.sqrt(z) - t_4;
	double t_6 = Math.sqrt(y) - t_2;
	double t_7 = ((t_1 - Math.sqrt(x)) - t_6) - t_5;
	double t_8 = Math.sqrt(t) - t_3;
	double tmp;
	if (t_7 <= 0.0) {
		tmp = (((0.5 / Math.sqrt(x)) - t_6) - t_5) - t_8;
	} else if (t_7 <= 1.0002) {
		tmp = ((((Math.sqrt((1.0 / z)) + Math.sqrt((1.0 / y))) * 0.5) + t_1) - Math.sqrt(x)) - t_8;
	} else if (t_7 <= 2.002) {
		tmp = ((0.5 / Math.sqrt(z)) - ((Math.sqrt(y) + Math.sqrt(x)) - (t_1 + t_2))) - t_8;
	} else {
		tmp = ((((1.0 / (t_3 + Math.sqrt(t))) + t_4) + t_2) - ((Math.sqrt(y) + Math.sqrt(z)) + Math.sqrt(x))) + 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	t_2 = math.sqrt((y + 1.0))
	t_3 = math.sqrt((1.0 + t))
	t_4 = math.sqrt((z + 1.0))
	t_5 = math.sqrt(z) - t_4
	t_6 = math.sqrt(y) - t_2
	t_7 = ((t_1 - math.sqrt(x)) - t_6) - t_5
	t_8 = math.sqrt(t) - t_3
	tmp = 0
	if t_7 <= 0.0:
		tmp = (((0.5 / math.sqrt(x)) - t_6) - t_5) - t_8
	elif t_7 <= 1.0002:
		tmp = ((((math.sqrt((1.0 / z)) + math.sqrt((1.0 / y))) * 0.5) + t_1) - math.sqrt(x)) - t_8
	elif t_7 <= 2.002:
		tmp = ((0.5 / math.sqrt(z)) - ((math.sqrt(y) + math.sqrt(x)) - (t_1 + t_2))) - t_8
	else:
		tmp = ((((1.0 / (t_3 + math.sqrt(t))) + t_4) + t_2) - ((math.sqrt(y) + math.sqrt(z)) + math.sqrt(x))) + 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(1.0 + t))
	t_4 = sqrt(Float64(z + 1.0))
	t_5 = Float64(sqrt(z) - t_4)
	t_6 = Float64(sqrt(y) - t_2)
	t_7 = Float64(Float64(Float64(t_1 - sqrt(x)) - t_6) - t_5)
	t_8 = Float64(sqrt(t) - t_3)
	tmp = 0.0
	if (t_7 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(0.5 / sqrt(x)) - t_6) - t_5) - t_8);
	elseif (t_7 <= 1.0002)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / z)) + sqrt(Float64(1.0 / y))) * 0.5) + t_1) - sqrt(x)) - t_8);
	elseif (t_7 <= 2.002)
		tmp = Float64(Float64(Float64(0.5 / sqrt(z)) - Float64(Float64(sqrt(y) + sqrt(x)) - Float64(t_1 + t_2))) - t_8);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / Float64(t_3 + sqrt(t))) + t_4) + t_2) - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))) + 1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	t_2 = sqrt((y + 1.0));
	t_3 = sqrt((1.0 + t));
	t_4 = sqrt((z + 1.0));
	t_5 = sqrt(z) - t_4;
	t_6 = sqrt(y) - t_2;
	t_7 = ((t_1 - sqrt(x)) - t_6) - t_5;
	t_8 = sqrt(t) - t_3;
	tmp = 0.0;
	if (t_7 <= 0.0)
		tmp = (((0.5 / sqrt(x)) - t_6) - t_5) - t_8;
	elseif (t_7 <= 1.0002)
		tmp = ((((sqrt((1.0 / z)) + sqrt((1.0 / y))) * 0.5) + t_1) - sqrt(x)) - t_8;
	elseif (t_7 <= 2.002)
		tmp = ((0.5 / sqrt(z)) - ((sqrt(y) + sqrt(x)) - (t_1 + t_2))) - t_8;
	else
		tmp = ((((1.0 / (t_3 + sqrt(t))) + t_4) + t_2) - ((sqrt(y) + sqrt(z)) + sqrt(x))) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[z], $MachinePrecision] - t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[y], $MachinePrecision] - t$95$2), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision] - t$95$5), $MachinePrecision]}, Block[{t$95$8 = N[(N[Sqrt[t], $MachinePrecision] - t$95$3), $MachinePrecision]}, If[LessEqual[t$95$7, 0.0], N[(N[(N[(N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision] - t$95$5), $MachinePrecision] - t$95$8), $MachinePrecision], If[LessEqual[t$95$7, 1.0002], N[(N[(N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$8), $MachinePrecision], If[LessEqual[t$95$7, 2.002], N[(N[(N[(0.5 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$8), $MachinePrecision], N[(N[(N[(N[(N[(1.0 / N[(t$95$3 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0

   1. Initial program 38.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot \frac{1}{2} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 55.7

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{x}}} \cdot 0.5 + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 55.7%

      \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 55.7%

      \[\leadsto \left(\left(\color{blue}{\frac{0.5}{\sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 1.0002

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{\frac{1}{z}} \cdot \frac{1}{2}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 17.1

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 44.0%

      \[\leadsto \left(\left(\sqrt{1 + x} + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.0002 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2.0019999999999998

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{\frac{1}{z}} \cdot \frac{1}{2}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 42.1

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 41.5%

      \[\leadsto \left(\frac{0.5}{\sqrt{z}} + \color{blue}{\left(\left(\sqrt{y + 1} + \sqrt{x + 1}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2.0019999999999998 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 98.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)} \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\sqrt{t + 1}} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\sqrt{t + 1} \cdot \color{blue}{\sqrt{t + 1}} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{\sqrt{t}} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \sqrt{t} \cdot \color{blue}{\sqrt{t}}}{\sqrt{t + 1} + \sqrt{t}} \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right) - t}}{\sqrt{t + 1} + \sqrt{t}} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

      12. lower-+.f64 99.3

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

   4. Applied rewrites 99.3%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]

   7. Applied rewrites 95.9%

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{\sqrt{t} + \sqrt{1 + t}}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right) \leq 0:\\ \;\;\;\;\left(\left(\frac{0.5}{\sqrt{x}} - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right) \leq 1.0002:\\ \;\;\;\;\left(\left(\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right) \cdot 0.5 + \sqrt{x + 1}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right) \leq 2.002:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{z}} - \left(\left(\sqrt{y} + \sqrt{x}\right) - \left(\sqrt{x + 1} + \sqrt{y + 1}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{\sqrt{1 + t} + \sqrt{t}} + \sqrt{z + 1}\right) + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{x + 1}\\ t_4 := \sqrt{1 + t}\\ t_5 := \sqrt{t} - t\_4\\ t_6 := \sqrt{y} - t\_1\\ t_7 := \left(\left(\left(t\_3 - \sqrt{x}\right) - t\_6\right) - \left(\sqrt{z} - t\_2\right)\right) - t\_5\\ \mathbf{if}\;t\_7 \leq 1:\\ \;\;\;\;\left(t\_3 - \left(t\_6 + \sqrt{x}\right)\right) - t\_5\\ \mathbf{elif}\;t\_7 \leq 3:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{z} + t\_2} + t\_1\right) + t\_3\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right) - \left(\left(t\_1 + t\_2\right) + t\_4\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (sqrt (+ 1.0 t)))
        (t_5 (- (sqrt t) t_4))
        (t_6 (- (sqrt y) t_1))
        (t_7 (- (- (- (- t_3 (sqrt x)) t_6) (- (sqrt z) t_2)) t_5)))
   (if (<= t_7 1.0)
     (- (- t_3 (+ t_6 (sqrt x))) t_5)
     (if (<= t_7 3.0)
       (- (+ (+ (/ 1.0 (+ (sqrt z) t_2)) t_1) t_3) (+ (sqrt y) (sqrt x)))
       (-
        1.0
        (-
         (+ (+ (+ (sqrt y) (sqrt z)) (sqrt x)) (sqrt t))
         (+ (+ t_1 t_2) t_4)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((x + 1.0));
	double t_4 = sqrt((1.0 + t));
	double t_5 = sqrt(t) - t_4;
	double t_6 = sqrt(y) - t_1;
	double t_7 = (((t_3 - sqrt(x)) - t_6) - (sqrt(z) - t_2)) - t_5;
	double tmp;
	if (t_7 <= 1.0) {
		tmp = (t_3 - (t_6 + sqrt(x))) - t_5;
	} else if (t_7 <= 3.0) {
		tmp = (((1.0 / (sqrt(z) + t_2)) + t_1) + t_3) - (sqrt(y) + sqrt(x));
	} else {
		tmp = 1.0 - ((((sqrt(y) + sqrt(z)) + sqrt(x)) + sqrt(t)) - ((t_1 + t_2) + t_4));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((z + 1.0d0))
    t_3 = sqrt((x + 1.0d0))
    t_4 = sqrt((1.0d0 + t))
    t_5 = sqrt(t) - t_4
    t_6 = sqrt(y) - t_1
    t_7 = (((t_3 - sqrt(x)) - t_6) - (sqrt(z) - t_2)) - t_5
    if (t_7 <= 1.0d0) then
        tmp = (t_3 - (t_6 + sqrt(x))) - t_5
    else if (t_7 <= 3.0d0) then
        tmp = (((1.0d0 / (sqrt(z) + t_2)) + t_1) + t_3) - (sqrt(y) + sqrt(x))
    else
        tmp = 1.0d0 - ((((sqrt(y) + sqrt(z)) + sqrt(x)) + sqrt(t)) - ((t_1 + t_2) + t_4))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((z + 1.0));
	double t_3 = Math.sqrt((x + 1.0));
	double t_4 = Math.sqrt((1.0 + t));
	double t_5 = Math.sqrt(t) - t_4;
	double t_6 = Math.sqrt(y) - t_1;
	double t_7 = (((t_3 - Math.sqrt(x)) - t_6) - (Math.sqrt(z) - t_2)) - t_5;
	double tmp;
	if (t_7 <= 1.0) {
		tmp = (t_3 - (t_6 + Math.sqrt(x))) - t_5;
	} else if (t_7 <= 3.0) {
		tmp = (((1.0 / (Math.sqrt(z) + t_2)) + t_1) + t_3) - (Math.sqrt(y) + Math.sqrt(x));
	} else {
		tmp = 1.0 - ((((Math.sqrt(y) + Math.sqrt(z)) + Math.sqrt(x)) + Math.sqrt(t)) - ((t_1 + t_2) + t_4));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((z + 1.0))
	t_3 = math.sqrt((x + 1.0))
	t_4 = math.sqrt((1.0 + t))
	t_5 = math.sqrt(t) - t_4
	t_6 = math.sqrt(y) - t_1
	t_7 = (((t_3 - math.sqrt(x)) - t_6) - (math.sqrt(z) - t_2)) - t_5
	tmp = 0
	if t_7 <= 1.0:
		tmp = (t_3 - (t_6 + math.sqrt(x))) - t_5
	elif t_7 <= 3.0:
		tmp = (((1.0 / (math.sqrt(z) + t_2)) + t_1) + t_3) - (math.sqrt(y) + math.sqrt(x))
	else:
		tmp = 1.0 - ((((math.sqrt(y) + math.sqrt(z)) + math.sqrt(x)) + math.sqrt(t)) - ((t_1 + t_2) + t_4))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = sqrt(Float64(1.0 + t))
	t_5 = Float64(sqrt(t) - t_4)
	t_6 = Float64(sqrt(y) - t_1)
	t_7 = Float64(Float64(Float64(Float64(t_3 - sqrt(x)) - t_6) - Float64(sqrt(z) - t_2)) - t_5)
	tmp = 0.0
	if (t_7 <= 1.0)
		tmp = Float64(Float64(t_3 - Float64(t_6 + sqrt(x))) - t_5);
	elseif (t_7 <= 3.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(z) + t_2)) + t_1) + t_3) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x)) + sqrt(t)) - Float64(Float64(t_1 + t_2) + t_4)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((z + 1.0));
	t_3 = sqrt((x + 1.0));
	t_4 = sqrt((1.0 + t));
	t_5 = sqrt(t) - t_4;
	t_6 = sqrt(y) - t_1;
	t_7 = (((t_3 - sqrt(x)) - t_6) - (sqrt(z) - t_2)) - t_5;
	tmp = 0.0;
	if (t_7 <= 1.0)
		tmp = (t_3 - (t_6 + sqrt(x))) - t_5;
	elseif (t_7 <= 3.0)
		tmp = (((1.0 / (sqrt(z) + t_2)) + t_1) + t_3) - (sqrt(y) + sqrt(x));
	else
		tmp = 1.0 - ((((sqrt(y) + sqrt(z)) + sqrt(x)) + sqrt(t)) - ((t_1 + t_2) + t_4));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[t], $MachinePrecision] - t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[y], $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision]}, If[LessEqual[t$95$7, 1.0], N[(N[(t$95$3 - N[(t$95$6 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision], If[LessEqual[t$95$7, 3.0], N[(N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 + t$95$2), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 73.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sub-neg N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} + \left(\mathsf{neg}\left(\sqrt{y}\right)\right)\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{y + 1} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{y + 1}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\sqrt{y + 1} \cdot \sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-prod N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\sqrt{y + 1}} \cdot \sqrt{\sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\sqrt{\sqrt{y + 1}}}, \sqrt{\sqrt{y + 1}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \color{blue}{\sqrt{\sqrt{y + 1}}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{-1 \cdot \sqrt{y}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \sqrt{y} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(1\right), \sqrt{y}, \sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 29.2%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + x} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{y}\right)\right)}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sub-neg N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 43.3

          \[\leadsto \left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   1. Initial program 96.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(z + 1\right) - z, \mathsf{fma}\left(\sqrt{y + 1} - \sqrt{y}, \left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{1 + x} - \sqrt{x}\right), e^{\sinh^{-1} \left(-\sqrt{x}\right) \cdot 2}\right), \left(\sqrt{z} + \sqrt{z + 1}\right) \cdot \left({\left(e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}^{3} + {\left(e^{\sinh^{-1} \left(-\sqrt{x}\right)}\right)}^{3}\right)\right)}{\left(\sqrt{z} + \sqrt{z + 1}\right) \cdot \mathsf{fma}\left(\sqrt{y + 1} - \sqrt{y}, \left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{1 + x} - \sqrt{x}\right), e^{\sinh^{-1} \left(-\sqrt{x}\right) \cdot 2}\right)}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      2. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\color{blue}{\sqrt{1 + z}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{\color{blue}{1 + z}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{\color{blue}{z + 1}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{\color{blue}{z + 1}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \sqrt{\color{blue}{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \sqrt{\color{blue}{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \sqrt{\color{blue}{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      15. rem-square-sqrt N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      16. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      17. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      18. lower-+.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      20. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{\color{blue}{z + 1}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      21. lift-+.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{\color{blue}{z + 1}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   8. Applied rewrites 98.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   11. Applied rewrites 25.9%

       \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)} \]

   if 3 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 95.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 86.7%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 1:\\ \;\;\;\;\left(\sqrt{x + 1} - \left(\left(\sqrt{y} - \sqrt{y + 1}\right) + \sqrt{x}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 3:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right) - \left(\left(\sqrt{y + 1} + \sqrt{z + 1}\right) + \sqrt{1 + t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{x + 1}\\ t_4 := \sqrt{1 + t}\\ t_5 := \sqrt{t} - t\_4\\ t_6 := \sqrt{y} - t\_1\\ t_7 := \left(\left(\left(t\_3 - \sqrt{x}\right) - t\_6\right) - \left(\sqrt{z} - t\_2\right)\right) - t\_5\\ \mathbf{if}\;t\_7 \leq 1:\\ \;\;\;\;\left(t\_3 - \left(t\_6 + \sqrt{x}\right)\right) - t\_5\\ \mathbf{elif}\;t\_7 \leq 3:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{z} + t\_2} + t\_1\right) + t\_3\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, z, t\_4\right) + 3\right) - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (sqrt (+ 1.0 t)))
        (t_5 (- (sqrt t) t_4))
        (t_6 (- (sqrt y) t_1))
        (t_7 (- (- (- (- t_3 (sqrt x)) t_6) (- (sqrt z) t_2)) t_5)))
   (if (<= t_7 1.0)
     (- (- t_3 (+ t_6 (sqrt x))) t_5)
     (if (<= t_7 3.0)
       (- (+ (+ (/ 1.0 (+ (sqrt z) t_2)) t_1) t_3) (+ (sqrt y) (sqrt x)))
       (-
        (+ (fma 0.5 z t_4) 3.0)
        (+ (+ (+ (sqrt y) (sqrt z)) (sqrt x)) (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((x + 1.0));
	double t_4 = sqrt((1.0 + t));
	double t_5 = sqrt(t) - t_4;
	double t_6 = sqrt(y) - t_1;
	double t_7 = (((t_3 - sqrt(x)) - t_6) - (sqrt(z) - t_2)) - t_5;
	double tmp;
	if (t_7 <= 1.0) {
		tmp = (t_3 - (t_6 + sqrt(x))) - t_5;
	} else if (t_7 <= 3.0) {
		tmp = (((1.0 / (sqrt(z) + t_2)) + t_1) + t_3) - (sqrt(y) + sqrt(x));
	} else {
		tmp = (fma(0.5, z, t_4) + 3.0) - (((sqrt(y) + sqrt(z)) + sqrt(x)) + sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = sqrt(Float64(1.0 + t))
	t_5 = Float64(sqrt(t) - t_4)
	t_6 = Float64(sqrt(y) - t_1)
	t_7 = Float64(Float64(Float64(Float64(t_3 - sqrt(x)) - t_6) - Float64(sqrt(z) - t_2)) - t_5)
	tmp = 0.0
	if (t_7 <= 1.0)
		tmp = Float64(Float64(t_3 - Float64(t_6 + sqrt(x))) - t_5);
	elseif (t_7 <= 3.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(z) + t_2)) + t_1) + t_3) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(fma(0.5, z, t_4) + 3.0) - Float64(Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x)) + sqrt(t)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[t], $MachinePrecision] - t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[y], $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision]}, If[LessEqual[t$95$7, 1.0], N[(N[(t$95$3 - N[(t$95$6 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision], If[LessEqual[t$95$7, 3.0], N[(N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * z + t$95$4), $MachinePrecision] + 3.0), $MachinePrecision] - N[(N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 1

   1. Initial program 73.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sub-neg N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} + \left(\mathsf{neg}\left(\sqrt{y}\right)\right)\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{y + 1} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{y + 1}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\sqrt{y + 1} \cdot \sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-prod N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\sqrt{y + 1}} \cdot \sqrt{\sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\sqrt{\sqrt{y + 1}}}, \sqrt{\sqrt{y + 1}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \color{blue}{\sqrt{\sqrt{y + 1}}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{-1 \cdot \sqrt{y}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \sqrt{y} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(1\right), \sqrt{y}, \sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 29.2%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + x} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{y}\right)\right)}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sub-neg N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 43.3

          \[\leadsto \left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   1. Initial program 96.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   4. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(z + 1\right) - z, \mathsf{fma}\left(\sqrt{y + 1} - \sqrt{y}, \left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{1 + x} - \sqrt{x}\right), e^{\sinh^{-1} \left(-\sqrt{x}\right) \cdot 2}\right), \left(\sqrt{z} + \sqrt{z + 1}\right) \cdot \left({\left(e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}^{3} + {\left(e^{\sinh^{-1} \left(-\sqrt{x}\right)}\right)}^{3}\right)\right)}{\left(\sqrt{z} + \sqrt{z + 1}\right) \cdot \mathsf{fma}\left(\sqrt{y + 1} - \sqrt{y}, \left(\sqrt{y + 1} - \sqrt{y}\right) - \left(\sqrt{1 + x} - \sqrt{x}\right), e^{\sinh^{-1} \left(-\sqrt{x}\right) \cdot 2}\right)}} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\left(\sqrt{1 + z} - \sqrt{z}\right)} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      2. flip-- N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\frac{\sqrt{1 + z} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\color{blue}{\sqrt{1 + z}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{\color{blue}{1 + z}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{\color{blue}{z + 1}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{\color{blue}{z + 1}} \cdot \sqrt{1 + z} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \sqrt{\color{blue}{1 + z}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \sqrt{\color{blue}{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\sqrt{z + 1} \cdot \sqrt{\color{blue}{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      12. rem-square-sqrt N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      14. lift-sqrt.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      15. rem-square-sqrt N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      16. lower--.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{1 + z} + \sqrt{z}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      17. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      18. lower-+.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{1 + z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{\color{blue}{1 + z}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      20. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{\color{blue}{z + 1}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

      21. lift-+.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{\color{blue}{z + 1}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   8. Applied rewrites 98.6%

      \[\leadsto \frac{\mathsf{fma}\left(\left(t + 1\right) - t, \frac{1}{\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}, \left(\sqrt{t} + \sqrt{t + 1}\right) \cdot 1\right)}{\left(\sqrt{t} + \sqrt{t + 1}\right) \cdot \frac{1}{\color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}} + \left(e^{\sinh^{-1} \left(-\sqrt{x}\right)} + e^{\sinh^{-1} \left(-\sqrt{y}\right)}\right)}} \]

   9. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)} \]

   11. Applied rewrites 25.9%

       \[\leadsto \color{blue}{\left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)} \]

   if 3 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 95.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 86.7%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.3%

      \[\leadsto 2 + \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot z\right)\right) - \left(\sqrt{t} + \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 76.8%

       \[\leadsto \left(\mathsf{fma}\left(0.5, z, \sqrt{1 + t}\right) + 3\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \color{blue}{\sqrt{t}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 1:\\ \;\;\;\;\left(\sqrt{x + 1} - \left(\left(\sqrt{y} - \sqrt{y + 1}\right) + \sqrt{x}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 3:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, z, \sqrt{1 + t}\right) + 3\right) - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \sqrt{x + 1}\\ t_3 := \left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\\ t_4 := \sqrt{1 + t}\\ t_5 := \sqrt{t} - t\_4\\ t_6 := \sqrt{y} - \sqrt{y + 1}\\ t_7 := \left(\left(\left(t\_2 - \sqrt{x}\right) - t\_6\right) - \left(\sqrt{z} - t\_1\right)\right) - t\_5\\ \mathbf{if}\;t\_7 \leq 2:\\ \;\;\;\;\left(t\_2 - \left(t\_6 + \sqrt{x}\right)\right) - t\_5\\ \mathbf{elif}\;t\_7 \leq 3.002:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, 0.5, t\_1\right) - t\_3\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, z, t\_4\right) + 3\right) - \left(t\_3 + \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (+ (+ (sqrt y) (sqrt z)) (sqrt x)))
        (t_4 (sqrt (+ 1.0 t)))
        (t_5 (- (sqrt t) t_4))
        (t_6 (- (sqrt y) (sqrt (+ y 1.0))))
        (t_7 (- (- (- (- t_2 (sqrt x)) t_6) (- (sqrt z) t_1)) t_5)))
   (if (<= t_7 2.0)
     (- (- t_2 (+ t_6 (sqrt x))) t_5)
     (if (<= t_7 3.002)
       (+ (- (fma (sqrt (/ 1.0 t)) 0.5 t_1) t_3) 2.0)
       (- (+ (fma 0.5 z t_4) 3.0) (+ t_3 (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = sqrt((x + 1.0));
	double t_3 = (sqrt(y) + sqrt(z)) + sqrt(x);
	double t_4 = sqrt((1.0 + t));
	double t_5 = sqrt(t) - t_4;
	double t_6 = sqrt(y) - sqrt((y + 1.0));
	double t_7 = (((t_2 - sqrt(x)) - t_6) - (sqrt(z) - t_1)) - t_5;
	double tmp;
	if (t_7 <= 2.0) {
		tmp = (t_2 - (t_6 + sqrt(x))) - t_5;
	} else if (t_7 <= 3.002) {
		tmp = (fma(sqrt((1.0 / t)), 0.5, t_1) - t_3) + 2.0;
	} else {
		tmp = (fma(0.5, z, t_4) + 3.0) - (t_3 + sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))
	t_4 = sqrt(Float64(1.0 + t))
	t_5 = Float64(sqrt(t) - t_4)
	t_6 = Float64(sqrt(y) - sqrt(Float64(y + 1.0)))
	t_7 = Float64(Float64(Float64(Float64(t_2 - sqrt(x)) - t_6) - Float64(sqrt(z) - t_1)) - t_5)
	tmp = 0.0
	if (t_7 <= 2.0)
		tmp = Float64(Float64(t_2 - Float64(t_6 + sqrt(x))) - t_5);
	elseif (t_7 <= 3.002)
		tmp = Float64(Float64(fma(sqrt(Float64(1.0 / t)), 0.5, t_1) - t_3) + 2.0);
	else
		tmp = Float64(Float64(fma(0.5, z, t_4) + 3.0) - Float64(t_3 + sqrt(t)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[t], $MachinePrecision] - t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision]}, If[LessEqual[t$95$7, 2.0], N[(N[(t$95$2 - N[(t$95$6 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision], If[LessEqual[t$95$7, 3.002], N[(N[(N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] - t$95$3), $MachinePrecision] + 2.0), $MachinePrecision], N[(N[(N[(0.5 * z + t$95$4), $MachinePrecision] + 3.0), $MachinePrecision] - N[(t$95$3 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 87.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sub-neg N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} + \left(\mathsf{neg}\left(\sqrt{y}\right)\right)\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{y + 1} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{y + 1}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\sqrt{y + 1} \cdot \sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-prod N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\sqrt{y + 1}} \cdot \sqrt{\sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\sqrt{\sqrt{y + 1}}}, \sqrt{\sqrt{y + 1}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \color{blue}{\sqrt{\sqrt{y + 1}}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{-1 \cdot \sqrt{y}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \sqrt{y} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(1\right), \sqrt{y}, \sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 42.8%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + x} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right)\right) - \sqrt{x}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\color{blue}{\sqrt{1 + x}} + \left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\sqrt{\color{blue}{1 + x}} + \left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \color{blue}{\left(\left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right) - \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} + \color{blue}{\left(\mathsf{neg}\left(\sqrt{y}\right)\right)}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. sub-neg N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower--.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\color{blue}{\left(\sqrt{1 + y} - \sqrt{y}\right)} - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{1 + x} + \left(\left(\color{blue}{\sqrt{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\left(\sqrt{\color{blue}{1 + y}} - \sqrt{y}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 43.3

          \[\leadsto \left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.0019999999999998

   1. Initial program 95.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 18.4%

      \[\leadsto 2 + \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto 2 + \left(\left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 24.1%

       \[\leadsto 2 + \left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, 0.5, \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right)\right) \]

   if 3.0019999999999998 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 99.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 92.7%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.3%

      \[\leadsto 2 + \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot z\right)\right) - \left(\sqrt{t} + \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 86.9%

       \[\leadsto \left(\mathsf{fma}\left(0.5, z, \sqrt{1 + t}\right) + 3\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \color{blue}{\sqrt{t}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 2:\\ \;\;\;\;\left(\sqrt{x + 1} - \left(\left(\sqrt{y} - \sqrt{y + 1}\right) + \sqrt{x}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 3.002:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, 0.5, \sqrt{z + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, z, \sqrt{1 + t}\right) + 3\right) - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z + 1}\\ t_2 := \left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\\ t_3 := \sqrt{1 + t}\\ t_4 := \sqrt{t} - t\_3\\ t_5 := \sqrt{y} - \sqrt{y + 1}\\ t_6 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - t\_5\right) - \left(\sqrt{z} - t\_1\right)\right) - t\_4\\ \mathbf{if}\;t\_6 \leq 2:\\ \;\;\;\;\left(1 - \left(t\_5 + \sqrt{x}\right)\right) - t\_4\\ \mathbf{elif}\;t\_6 \leq 3.002:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, 0.5, t\_1\right) - t\_2\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, z, t\_3\right) + 3\right) - \left(t\_2 + \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z 1.0)))
        (t_2 (+ (+ (sqrt y) (sqrt z)) (sqrt x)))
        (t_3 (sqrt (+ 1.0 t)))
        (t_4 (- (sqrt t) t_3))
        (t_5 (- (sqrt y) (sqrt (+ y 1.0))))
        (t_6
         (- (- (- (- (sqrt (+ x 1.0)) (sqrt x)) t_5) (- (sqrt z) t_1)) t_4)))
   (if (<= t_6 2.0)
     (- (- 1.0 (+ t_5 (sqrt x))) t_4)
     (if (<= t_6 3.002)
       (+ (- (fma (sqrt (/ 1.0 t)) 0.5 t_1) t_2) 2.0)
       (- (+ (fma 0.5 z t_3) 3.0) (+ t_2 (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + 1.0));
	double t_2 = (sqrt(y) + sqrt(z)) + sqrt(x);
	double t_3 = sqrt((1.0 + t));
	double t_4 = sqrt(t) - t_3;
	double t_5 = sqrt(y) - sqrt((y + 1.0));
	double t_6 = (((sqrt((x + 1.0)) - sqrt(x)) - t_5) - (sqrt(z) - t_1)) - t_4;
	double tmp;
	if (t_6 <= 2.0) {
		tmp = (1.0 - (t_5 + sqrt(x))) - t_4;
	} else if (t_6 <= 3.002) {
		tmp = (fma(sqrt((1.0 / t)), 0.5, t_1) - t_2) + 2.0;
	} else {
		tmp = (fma(0.5, z, t_3) + 3.0) - (t_2 + sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(z + 1.0))
	t_2 = Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))
	t_3 = sqrt(Float64(1.0 + t))
	t_4 = Float64(sqrt(t) - t_3)
	t_5 = Float64(sqrt(y) - sqrt(Float64(y + 1.0)))
	t_6 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) - t_5) - Float64(sqrt(z) - t_1)) - t_4)
	tmp = 0.0
	if (t_6 <= 2.0)
		tmp = Float64(Float64(1.0 - Float64(t_5 + sqrt(x))) - t_4);
	elseif (t_6 <= 3.002)
		tmp = Float64(Float64(fma(sqrt(Float64(1.0 / t)), 0.5, t_1) - t_2) + 2.0);
	else
		tmp = Float64(Float64(fma(0.5, z, t_3) + 3.0) - Float64(t_2 + sqrt(t)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[t], $MachinePrecision] - t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]}, If[LessEqual[t$95$6, 2.0], N[(N[(1.0 - N[(t$95$5 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision], If[LessEqual[t$95$6, 3.002], N[(N[(N[(N[Sqrt[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision] + 2.0), $MachinePrecision], N[(N[(N[(0.5 * z + t$95$3), $MachinePrecision] + 3.0), $MachinePrecision] - N[(t$95$2 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 87.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sub-neg N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} + \left(\mathsf{neg}\left(\sqrt{y}\right)\right)\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{y + 1} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{y + 1}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\sqrt{y + 1} \cdot \sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-prod N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\sqrt{y + 1}} \cdot \sqrt{\sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\sqrt{\sqrt{y + 1}}}, \sqrt{\sqrt{y + 1}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \color{blue}{\sqrt{\sqrt{y + 1}}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{-1 \cdot \sqrt{y}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \sqrt{y} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(1\right), \sqrt{y}, \sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 42.8%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + x} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)} - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)} - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} + 1\right)} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} + 1\right)} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} + 1\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} + 1\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \color{blue}{\left(-1 \cdot \sqrt{y} + \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \color{blue}{\mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \color{blue}{\sqrt{y}}, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \color{blue}{\sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{\color{blue}{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{z} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{z} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{z}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 6.7

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \left(\sqrt{z} + \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 6.7%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \left(\sqrt{z} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in z around inf

      \[\leadsto \left(\left(1 + \left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right)\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 30.6%

       \[\leadsto \left(1 + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3.0019999999999998

   1. Initial program 95.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 20.0%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 18.4%

      \[\leadsto 2 + \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto 2 + \left(\left(\sqrt{1 + z} + \frac{1}{2} \cdot \sqrt{\frac{1}{t}}\right) - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 24.1%

       \[\leadsto 2 + \left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, 0.5, \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right)\right) \]

   if 3.0019999999999998 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 99.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 92.7%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.3%

      \[\leadsto 2 + \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot z\right)\right) - \left(\sqrt{t} + \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 86.9%

       \[\leadsto \left(\mathsf{fma}\left(0.5, z, \sqrt{1 + t}\right) + 3\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \color{blue}{\sqrt{t}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 2:\\ \;\;\;\;\left(1 - \left(\left(\sqrt{y} - \sqrt{y + 1}\right) + \sqrt{x}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 3.002:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{t}}, 0.5, \sqrt{z + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + 2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, z, \sqrt{1 + t}\right) + 3\right) - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{z + 1}\\ t_3 := \left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\\ t_4 := \sqrt{1 + t}\\ t_5 := \sqrt{t} - t\_4\\ t_6 := \sqrt{y} - t\_1\\ t_7 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - t\_6\right) - \left(\sqrt{z} - t\_2\right)\right) - t\_5\\ \mathbf{if}\;t\_7 \leq 2:\\ \;\;\;\;\left(1 - \left(t\_6 + \sqrt{x}\right)\right) - t\_5\\ \mathbf{elif}\;t\_7 \leq 3:\\ \;\;\;\;\left(\left(t\_1 + 1\right) + t\_2\right) - t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, z, t\_4\right) + 3\right) - \left(t\_3 + \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (+ (+ (sqrt y) (sqrt z)) (sqrt x)))
        (t_4 (sqrt (+ 1.0 t)))
        (t_5 (- (sqrt t) t_4))
        (t_6 (- (sqrt y) t_1))
        (t_7
         (- (- (- (- (sqrt (+ x 1.0)) (sqrt x)) t_6) (- (sqrt z) t_2)) t_5)))
   (if (<= t_7 2.0)
     (- (- 1.0 (+ t_6 (sqrt x))) t_5)
     (if (<= t_7 3.0)
       (- (+ (+ t_1 1.0) t_2) t_3)
       (- (+ (fma 0.5 z t_4) 3.0) (+ t_3 (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((z + 1.0));
	double t_3 = (sqrt(y) + sqrt(z)) + sqrt(x);
	double t_4 = sqrt((1.0 + t));
	double t_5 = sqrt(t) - t_4;
	double t_6 = sqrt(y) - t_1;
	double t_7 = (((sqrt((x + 1.0)) - sqrt(x)) - t_6) - (sqrt(z) - t_2)) - t_5;
	double tmp;
	if (t_7 <= 2.0) {
		tmp = (1.0 - (t_6 + sqrt(x))) - t_5;
	} else if (t_7 <= 3.0) {
		tmp = ((t_1 + 1.0) + t_2) - t_3;
	} else {
		tmp = (fma(0.5, z, t_4) + 3.0) - (t_3 + sqrt(t));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))
	t_4 = sqrt(Float64(1.0 + t))
	t_5 = Float64(sqrt(t) - t_4)
	t_6 = Float64(sqrt(y) - t_1)
	t_7 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) - t_6) - Float64(sqrt(z) - t_2)) - t_5)
	tmp = 0.0
	if (t_7 <= 2.0)
		tmp = Float64(Float64(1.0 - Float64(t_6 + sqrt(x))) - t_5);
	elseif (t_7 <= 3.0)
		tmp = Float64(Float64(Float64(t_1 + 1.0) + t_2) - t_3);
	else
		tmp = Float64(Float64(fma(0.5, z, t_4) + 3.0) - Float64(t_3 + sqrt(t)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[t], $MachinePrecision] - t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[y], $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision]}, If[LessEqual[t$95$7, 2.0], N[(N[(1.0 - N[(t$95$6 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision], If[LessEqual[t$95$7, 3.0], N[(N[(N[(t$95$1 + 1.0), $MachinePrecision] + t$95$2), $MachinePrecision] - t$95$3), $MachinePrecision], N[(N[(N[(0.5 * z + t$95$4), $MachinePrecision] + 3.0), $MachinePrecision] - N[(t$95$3 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 87.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sub-neg N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} + \left(\mathsf{neg}\left(\sqrt{y}\right)\right)\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{y + 1} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{y + 1}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\sqrt{y + 1} \cdot \sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-prod N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\sqrt{y + 1}} \cdot \sqrt{\sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\sqrt{\sqrt{y + 1}}}, \sqrt{\sqrt{y + 1}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \color{blue}{\sqrt{\sqrt{y + 1}}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{-1 \cdot \sqrt{y}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \sqrt{y} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(1\right), \sqrt{y}, \sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 42.8%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + x} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)} - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)} - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} + 1\right)} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} + 1\right)} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} + 1\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} + 1\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \color{blue}{\left(-1 \cdot \sqrt{y} + \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \color{blue}{\mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \color{blue}{\sqrt{y}}, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \color{blue}{\sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{\color{blue}{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{z} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{z} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{z}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 6.7

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \left(\sqrt{z} + \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 6.7%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \left(\sqrt{z} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in z around inf

      \[\leadsto \left(\left(1 + \left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right)\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 30.6%

       \[\leadsto \left(1 + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 26.2

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 26.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 24.5%

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

   if 3 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 95.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 86.7%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.3%

      \[\leadsto 2 + \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \left(3 + \left(\sqrt{1 + t} + \frac{1}{2} \cdot z\right)\right) - \left(\sqrt{t} + \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 76.8%

       \[\leadsto \left(\mathsf{fma}\left(0.5, z, \sqrt{1 + t}\right) + 3\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \color{blue}{\sqrt{t}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 2:\\ \;\;\;\;\left(1 - \left(\left(\sqrt{y} - \sqrt{y + 1}\right) + \sqrt{x}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 3:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + 1\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.5, z, \sqrt{1 + t}\right) + 3\right) - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{z + 1}\\ t_3 := \left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\\ t_4 := \sqrt{1 + t}\\ t_5 := \sqrt{t} - t\_4\\ t_6 := \sqrt{y} - t\_1\\ t_7 := \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - t\_6\right) - \left(\sqrt{z} - t\_2\right)\right) - t\_5\\ \mathbf{if}\;t\_7 \leq 2:\\ \;\;\;\;\left(1 - \left(t\_6 + \sqrt{x}\right)\right) - t\_5\\ \mathbf{elif}\;t\_7 \leq 3:\\ \;\;\;\;\left(\left(t\_1 + 1\right) + t\_2\right) - t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(3 + t\_4\right) - \left(t\_3 + \sqrt{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (+ (+ (sqrt y) (sqrt z)) (sqrt x)))
        (t_4 (sqrt (+ 1.0 t)))
        (t_5 (- (sqrt t) t_4))
        (t_6 (- (sqrt y) t_1))
        (t_7
         (- (- (- (- (sqrt (+ x 1.0)) (sqrt x)) t_6) (- (sqrt z) t_2)) t_5)))
   (if (<= t_7 2.0)
     (- (- 1.0 (+ t_6 (sqrt x))) t_5)
     (if (<= t_7 3.0)
       (- (+ (+ t_1 1.0) t_2) t_3)
       (- (+ 3.0 t_4) (+ t_3 (sqrt t)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((z + 1.0));
	double t_3 = (sqrt(y) + sqrt(z)) + sqrt(x);
	double t_4 = sqrt((1.0 + t));
	double t_5 = sqrt(t) - t_4;
	double t_6 = sqrt(y) - t_1;
	double t_7 = (((sqrt((x + 1.0)) - sqrt(x)) - t_6) - (sqrt(z) - t_2)) - t_5;
	double tmp;
	if (t_7 <= 2.0) {
		tmp = (1.0 - (t_6 + sqrt(x))) - t_5;
	} else if (t_7 <= 3.0) {
		tmp = ((t_1 + 1.0) + t_2) - t_3;
	} else {
		tmp = (3.0 + t_4) - (t_3 + sqrt(t));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((z + 1.0d0))
    t_3 = (sqrt(y) + sqrt(z)) + sqrt(x)
    t_4 = sqrt((1.0d0 + t))
    t_5 = sqrt(t) - t_4
    t_6 = sqrt(y) - t_1
    t_7 = (((sqrt((x + 1.0d0)) - sqrt(x)) - t_6) - (sqrt(z) - t_2)) - t_5
    if (t_7 <= 2.0d0) then
        tmp = (1.0d0 - (t_6 + sqrt(x))) - t_5
    else if (t_7 <= 3.0d0) then
        tmp = ((t_1 + 1.0d0) + t_2) - t_3
    else
        tmp = (3.0d0 + t_4) - (t_3 + sqrt(t))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((z + 1.0));
	double t_3 = (Math.sqrt(y) + Math.sqrt(z)) + Math.sqrt(x);
	double t_4 = Math.sqrt((1.0 + t));
	double t_5 = Math.sqrt(t) - t_4;
	double t_6 = Math.sqrt(y) - t_1;
	double t_7 = (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) - t_6) - (Math.sqrt(z) - t_2)) - t_5;
	double tmp;
	if (t_7 <= 2.0) {
		tmp = (1.0 - (t_6 + Math.sqrt(x))) - t_5;
	} else if (t_7 <= 3.0) {
		tmp = ((t_1 + 1.0) + t_2) - t_3;
	} else {
		tmp = (3.0 + t_4) - (t_3 + Math.sqrt(t));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((z + 1.0))
	t_3 = (math.sqrt(y) + math.sqrt(z)) + math.sqrt(x)
	t_4 = math.sqrt((1.0 + t))
	t_5 = math.sqrt(t) - t_4
	t_6 = math.sqrt(y) - t_1
	t_7 = (((math.sqrt((x + 1.0)) - math.sqrt(x)) - t_6) - (math.sqrt(z) - t_2)) - t_5
	tmp = 0
	if t_7 <= 2.0:
		tmp = (1.0 - (t_6 + math.sqrt(x))) - t_5
	elif t_7 <= 3.0:
		tmp = ((t_1 + 1.0) + t_2) - t_3
	else:
		tmp = (3.0 + t_4) - (t_3 + math.sqrt(t))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))
	t_4 = sqrt(Float64(1.0 + t))
	t_5 = Float64(sqrt(t) - t_4)
	t_6 = Float64(sqrt(y) - t_1)
	t_7 = Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) - t_6) - Float64(sqrt(z) - t_2)) - t_5)
	tmp = 0.0
	if (t_7 <= 2.0)
		tmp = Float64(Float64(1.0 - Float64(t_6 + sqrt(x))) - t_5);
	elseif (t_7 <= 3.0)
		tmp = Float64(Float64(Float64(t_1 + 1.0) + t_2) - t_3);
	else
		tmp = Float64(Float64(3.0 + t_4) - Float64(t_3 + sqrt(t)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((z + 1.0));
	t_3 = (sqrt(y) + sqrt(z)) + sqrt(x);
	t_4 = sqrt((1.0 + t));
	t_5 = sqrt(t) - t_4;
	t_6 = sqrt(y) - t_1;
	t_7 = (((sqrt((x + 1.0)) - sqrt(x)) - t_6) - (sqrt(z) - t_2)) - t_5;
	tmp = 0.0;
	if (t_7 <= 2.0)
		tmp = (1.0 - (t_6 + sqrt(x))) - t_5;
	elseif (t_7 <= 3.0)
		tmp = ((t_1 + 1.0) + t_2) - t_3;
	else
		tmp = (3.0 + t_4) - (t_3 + sqrt(t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[t], $MachinePrecision] - t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[y], $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision]}, If[LessEqual[t$95$7, 2.0], N[(N[(1.0 - N[(t$95$6 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision], If[LessEqual[t$95$7, 3.0], N[(N[(N[(t$95$1 + 1.0), $MachinePrecision] + t$95$2), $MachinePrecision] - t$95$3), $MachinePrecision], N[(N[(3.0 + t$95$4), $MachinePrecision] - N[(t$95$3 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 2

   1. Initial program 87.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. sub-neg N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} + \left(\mathsf{neg}\left(\sqrt{y}\right)\right)\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. associate-+l+ N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{y + 1} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{y + 1}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\sqrt{y + 1} \cdot \sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. sqrt-prod N/A

          \[\leadsto \left(\left(\color{blue}{\sqrt{\sqrt{y + 1}} \cdot \sqrt{\sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\sqrt{\sqrt{y + 1}}}, \sqrt{\sqrt{y + 1}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \color{blue}{\sqrt{\sqrt{y + 1}}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{-1 \cdot \sqrt{y}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \sqrt{y} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(1\right), \sqrt{y}, \sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 42.8%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + x} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate-+r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)} - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)} - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} + 1\right)} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} + 1\right)} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} + 1\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} + 1\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \color{blue}{\left(-1 \cdot \sqrt{y} + \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \color{blue}{\mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \color{blue}{\sqrt{y}}, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \color{blue}{\sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{\color{blue}{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{z} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{z} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{z}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 6.7

          \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \left(\sqrt{z} + \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 6.7%

      \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \left(\sqrt{z} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in z around inf

      \[\leadsto \left(\left(1 + \left(\sqrt{1 + y} + -1 \cdot \sqrt{y}\right)\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 30.6%

       \[\leadsto \left(1 + \color{blue}{\left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 2 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t))) < 3

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 26.2

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 26.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 24.5%

      \[\leadsto \left(\left(1 + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

   if 3 < (+.f64 (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) (-.f64 (sqrt.f64 (+.f64 t #s(literal 1 binary64))) (sqrt.f64 t)))

   1. Initial program 95.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 86.7%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.3%

      \[\leadsto 2 + \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto \left(3 + \sqrt{1 + t}\right) - \left(\sqrt{t} + \color{blue}{\left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 74.4%

       \[\leadsto \left(3 + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \color{blue}{\sqrt{t}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 2:\\ \;\;\;\;\left(1 - \left(\left(\sqrt{y} - \sqrt{y + 1}\right) + \sqrt{x}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{elif}\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \leq 3:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + 1\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{z} - t\_2\\ t_4 := \sqrt{1 + t}\\ \mathbf{if}\;t\_1 - t\_3 \leq 0:\\ \;\;\;\;\left(\left(\sqrt{\frac{1}{y}} \cdot 0.5 + 0.5 \cdot \sqrt{\frac{1}{x}}\right) - t\_3\right) - \left(\sqrt{t} - t\_4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + t\right) - t}{t\_4 + \sqrt{t}} - \left(\frac{z - \left(z + 1\right)}{\sqrt{z} + t\_2} - t\_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt y) (sqrt (+ y 1.0)))))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (- (sqrt z) t_2))
        (t_4 (sqrt (+ 1.0 t))))
   (if (<= (- t_1 t_3) 0.0)
     (-
      (- (+ (* (sqrt (/ 1.0 y)) 0.5) (* 0.5 (sqrt (/ 1.0 x)))) t_3)
      (- (sqrt t) t_4))
     (-
      (/ (- (+ 1.0 t) t) (+ t_4 (sqrt t)))
      (- (/ (- z (+ z 1.0)) (+ (sqrt z) t_2)) t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (sqrt((x + 1.0)) - sqrt(x)) - (sqrt(y) - sqrt((y + 1.0)));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt(z) - t_2;
	double t_4 = sqrt((1.0 + t));
	double tmp;
	if ((t_1 - t_3) <= 0.0) {
		tmp = (((sqrt((1.0 / y)) * 0.5) + (0.5 * sqrt((1.0 / x)))) - t_3) - (sqrt(t) - t_4);
	} else {
		tmp = (((1.0 + t) - t) / (t_4 + sqrt(t))) - (((z - (z + 1.0)) / (sqrt(z) + t_2)) - t_1);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (sqrt((x + 1.0d0)) - sqrt(x)) - (sqrt(y) - sqrt((y + 1.0d0)))
    t_2 = sqrt((z + 1.0d0))
    t_3 = sqrt(z) - t_2
    t_4 = sqrt((1.0d0 + t))
    if ((t_1 - t_3) <= 0.0d0) then
        tmp = (((sqrt((1.0d0 / y)) * 0.5d0) + (0.5d0 * sqrt((1.0d0 / x)))) - t_3) - (sqrt(t) - t_4)
    else
        tmp = (((1.0d0 + t) - t) / (t_4 + sqrt(t))) - (((z - (z + 1.0d0)) / (sqrt(z) + t_2)) - t_1)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = (Math.sqrt((x + 1.0)) - Math.sqrt(x)) - (Math.sqrt(y) - Math.sqrt((y + 1.0)));
	double t_2 = Math.sqrt((z + 1.0));
	double t_3 = Math.sqrt(z) - t_2;
	double t_4 = Math.sqrt((1.0 + t));
	double tmp;
	if ((t_1 - t_3) <= 0.0) {
		tmp = (((Math.sqrt((1.0 / y)) * 0.5) + (0.5 * Math.sqrt((1.0 / x)))) - t_3) - (Math.sqrt(t) - t_4);
	} else {
		tmp = (((1.0 + t) - t) / (t_4 + Math.sqrt(t))) - (((z - (z + 1.0)) / (Math.sqrt(z) + t_2)) - t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (math.sqrt((x + 1.0)) - math.sqrt(x)) - (math.sqrt(y) - math.sqrt((y + 1.0)))
	t_2 = math.sqrt((z + 1.0))
	t_3 = math.sqrt(z) - t_2
	t_4 = math.sqrt((1.0 + t))
	tmp = 0
	if (t_1 - t_3) <= 0.0:
		tmp = (((math.sqrt((1.0 / y)) * 0.5) + (0.5 * math.sqrt((1.0 / x)))) - t_3) - (math.sqrt(t) - t_4)
	else:
		tmp = (((1.0 + t) - t) / (t_4 + math.sqrt(t))) - (((z - (z + 1.0)) / (math.sqrt(z) + t_2)) - t_1)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) - Float64(sqrt(y) - sqrt(Float64(y + 1.0))))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(sqrt(z) - t_2)
	t_4 = sqrt(Float64(1.0 + t))
	tmp = 0.0
	if (Float64(t_1 - t_3) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / y)) * 0.5) + Float64(0.5 * sqrt(Float64(1.0 / x)))) - t_3) - Float64(sqrt(t) - t_4));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + t) - t) / Float64(t_4 + sqrt(t))) - Float64(Float64(Float64(z - Float64(z + 1.0)) / Float64(sqrt(z) + t_2)) - t_1));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = (sqrt((x + 1.0)) - sqrt(x)) - (sqrt(y) - sqrt((y + 1.0)));
	t_2 = sqrt((z + 1.0));
	t_3 = sqrt(z) - t_2;
	t_4 = sqrt((1.0 + t));
	tmp = 0.0;
	if ((t_1 - t_3) <= 0.0)
		tmp = (((sqrt((1.0 / y)) * 0.5) + (0.5 * sqrt((1.0 / x)))) - t_3) - (sqrt(t) - t_4);
	else
		tmp = (((1.0 + t) - t) / (t_4 + sqrt(t))) - (((z - (z + 1.0)) / (sqrt(z) + t_2)) - t_1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[z], $MachinePrecision] - t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 - t$95$3), $MachinePrecision], 0.0], N[(N[(N[(N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 + t), $MachinePrecision] - t), $MachinePrecision] / N[(t$95$4 + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(z - N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 0.0

   1. Initial program 38.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot \frac{1}{2} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 55.7

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{x}}} \cdot 0.5 + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 55.7%

      \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{2} + \color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{2} + \color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{2} + \color{blue}{\sqrt{\frac{1}{y}}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 63.5

         \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot 0.5 + \sqrt{\color{blue}{\frac{1}{y}}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 63.5%

      \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot 0.5 + \color{blue}{\sqrt{\frac{1}{y}} \cdot 0.5}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 96.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)} \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\sqrt{t + 1}} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\sqrt{t + 1} \cdot \color{blue}{\sqrt{t + 1}} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{\sqrt{t}} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \sqrt{t} \cdot \color{blue}{\sqrt{t}}}{\sqrt{t + 1} + \sqrt{t}} \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\color{blue}{\left(t + 1\right) - t}}{\sqrt{t + 1} + \sqrt{t}} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

      12. lower-+.f64 96.9

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

   4. Applied rewrites 96.9%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      5. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      8. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      9. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      11. lift--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z} + \sqrt{z + 1}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      12. lower-/.f64 97.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      13. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - z}{\sqrt{z} + \sqrt{z + 1}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      14. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z} + \sqrt{z + 1}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      15. lift-+.f64 97.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(1 + z\right)} - z}{\sqrt{z} + \sqrt{z + 1}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      16. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      17. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      18. lower-+.f64 97.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      19. lift-+.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{z + 1}} + \sqrt{z}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      20. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

      21. lift-+.f64 97.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(1 + z\right) - z}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]

   6. Applied rewrites 97.5%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(1 + z\right) - z}{\sqrt{1 + z} + \sqrt{z}}}\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right) \leq 0:\\ \;\;\;\;\left(\left(\sqrt{\frac{1}{y}} \cdot 0.5 + 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + t\right) - t}{\sqrt{1 + t} + \sqrt{t}} - \left(\frac{z - \left(z + 1\right)}{\sqrt{z} + \sqrt{z + 1}} - \left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t} - \sqrt{1 + t}\\ t_2 := \sqrt{\frac{1}{y}}\\ t_3 := \sqrt{x + 1}\\ t_4 := \sqrt{y + 1}\\ t_5 := \left(t\_3 - \sqrt{x}\right) - \left(\sqrt{y} - t\_4\right)\\ t_6 := \sqrt{z + 1}\\ \mathbf{if}\;t\_5 \leq 0:\\ \;\;\;\;\left(\left(t\_2 \cdot 0.5 + 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \left(\sqrt{z} - t\_6\right)\right) - t\_1\\ \mathbf{elif}\;t\_5 \leq 1.0002:\\ \;\;\;\;\left(\left(\left(\sqrt{\frac{1}{z}} + t\_2\right) \cdot 0.5 + t\_3\right) - \sqrt{x}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(\left(\sqrt{y} + \sqrt{x}\right) - \left(\frac{1}{\sqrt{z} + t\_6} + t\_4\right)\right)\right) - t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt t) (sqrt (+ 1.0 t))))
        (t_2 (sqrt (/ 1.0 y)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (sqrt (+ y 1.0)))
        (t_5 (- (- t_3 (sqrt x)) (- (sqrt y) t_4)))
        (t_6 (sqrt (+ z 1.0))))
   (if (<= t_5 0.0)
     (- (- (+ (* t_2 0.5) (* 0.5 (sqrt (/ 1.0 x)))) (- (sqrt z) t_6)) t_1)
     (if (<= t_5 1.0002)
       (- (- (+ (* (+ (sqrt (/ 1.0 z)) t_2) 0.5) t_3) (sqrt x)) t_1)
       (-
        (- 1.0 (- (+ (sqrt y) (sqrt x)) (+ (/ 1.0 (+ (sqrt z) t_6)) t_4)))
        t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(t) - sqrt((1.0 + t));
	double t_2 = sqrt((1.0 / y));
	double t_3 = sqrt((x + 1.0));
	double t_4 = sqrt((y + 1.0));
	double t_5 = (t_3 - sqrt(x)) - (sqrt(y) - t_4);
	double t_6 = sqrt((z + 1.0));
	double tmp;
	if (t_5 <= 0.0) {
		tmp = (((t_2 * 0.5) + (0.5 * sqrt((1.0 / x)))) - (sqrt(z) - t_6)) - t_1;
	} else if (t_5 <= 1.0002) {
		tmp = ((((sqrt((1.0 / z)) + t_2) * 0.5) + t_3) - sqrt(x)) - t_1;
	} else {
		tmp = (1.0 - ((sqrt(y) + sqrt(x)) - ((1.0 / (sqrt(z) + t_6)) + t_4))) - t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = sqrt(t) - sqrt((1.0d0 + t))
    t_2 = sqrt((1.0d0 / y))
    t_3 = sqrt((x + 1.0d0))
    t_4 = sqrt((y + 1.0d0))
    t_5 = (t_3 - sqrt(x)) - (sqrt(y) - t_4)
    t_6 = sqrt((z + 1.0d0))
    if (t_5 <= 0.0d0) then
        tmp = (((t_2 * 0.5d0) + (0.5d0 * sqrt((1.0d0 / x)))) - (sqrt(z) - t_6)) - t_1
    else if (t_5 <= 1.0002d0) then
        tmp = ((((sqrt((1.0d0 / z)) + t_2) * 0.5d0) + t_3) - sqrt(x)) - t_1
    else
        tmp = (1.0d0 - ((sqrt(y) + sqrt(x)) - ((1.0d0 / (sqrt(z) + t_6)) + t_4))) - t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(t) - Math.sqrt((1.0 + t));
	double t_2 = Math.sqrt((1.0 / y));
	double t_3 = Math.sqrt((x + 1.0));
	double t_4 = Math.sqrt((y + 1.0));
	double t_5 = (t_3 - Math.sqrt(x)) - (Math.sqrt(y) - t_4);
	double t_6 = Math.sqrt((z + 1.0));
	double tmp;
	if (t_5 <= 0.0) {
		tmp = (((t_2 * 0.5) + (0.5 * Math.sqrt((1.0 / x)))) - (Math.sqrt(z) - t_6)) - t_1;
	} else if (t_5 <= 1.0002) {
		tmp = ((((Math.sqrt((1.0 / z)) + t_2) * 0.5) + t_3) - Math.sqrt(x)) - t_1;
	} else {
		tmp = (1.0 - ((Math.sqrt(y) + Math.sqrt(x)) - ((1.0 / (Math.sqrt(z) + t_6)) + t_4))) - t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(t) - math.sqrt((1.0 + t))
	t_2 = math.sqrt((1.0 / y))
	t_3 = math.sqrt((x + 1.0))
	t_4 = math.sqrt((y + 1.0))
	t_5 = (t_3 - math.sqrt(x)) - (math.sqrt(y) - t_4)
	t_6 = math.sqrt((z + 1.0))
	tmp = 0
	if t_5 <= 0.0:
		tmp = (((t_2 * 0.5) + (0.5 * math.sqrt((1.0 / x)))) - (math.sqrt(z) - t_6)) - t_1
	elif t_5 <= 1.0002:
		tmp = ((((math.sqrt((1.0 / z)) + t_2) * 0.5) + t_3) - math.sqrt(x)) - t_1
	else:
		tmp = (1.0 - ((math.sqrt(y) + math.sqrt(x)) - ((1.0 / (math.sqrt(z) + t_6)) + t_4))) - t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(t) - sqrt(Float64(1.0 + t)))
	t_2 = sqrt(Float64(1.0 / y))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = sqrt(Float64(y + 1.0))
	t_5 = Float64(Float64(t_3 - sqrt(x)) - Float64(sqrt(y) - t_4))
	t_6 = sqrt(Float64(z + 1.0))
	tmp = 0.0
	if (t_5 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(t_2 * 0.5) + Float64(0.5 * sqrt(Float64(1.0 / x)))) - Float64(sqrt(z) - t_6)) - t_1);
	elseif (t_5 <= 1.0002)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / z)) + t_2) * 0.5) + t_3) - sqrt(x)) - t_1);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(sqrt(y) + sqrt(x)) - Float64(Float64(1.0 / Float64(sqrt(z) + t_6)) + t_4))) - t_1);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(t) - sqrt((1.0 + t));
	t_2 = sqrt((1.0 / y));
	t_3 = sqrt((x + 1.0));
	t_4 = sqrt((y + 1.0));
	t_5 = (t_3 - sqrt(x)) - (sqrt(y) - t_4);
	t_6 = sqrt((z + 1.0));
	tmp = 0.0;
	if (t_5 <= 0.0)
		tmp = (((t_2 * 0.5) + (0.5 * sqrt((1.0 / x)))) - (sqrt(z) - t_6)) - t_1;
	elseif (t_5 <= 1.0002)
		tmp = ((((sqrt((1.0 / z)) + t_2) * 0.5) + t_3) - sqrt(x)) - t_1;
	else
		tmp = (1.0 - ((sqrt(y) + sqrt(x)) - ((1.0 / (sqrt(z) + t_6)) + t_4))) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, 0.0], N[(N[(N[(N[(t$95$2 * 0.5), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$6), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t$95$5, 1.0002], N[(N[(N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision] * 0.5), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(1.0 - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$6), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 0.0

   1. Initial program 64.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot \frac{1}{2} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 74.3

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{x}}} \cdot 0.5 + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 74.3%

      \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{2} + \color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{2} + \color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{2} + \color{blue}{\sqrt{\frac{1}{y}}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 79.3

         \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot 0.5 + \sqrt{\color{blue}{\frac{1}{y}}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 79.3%

      \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot 0.5 + \color{blue}{\sqrt{\frac{1}{y}} \cdot 0.5}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 1.0002

   1. Initial program 95.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{\frac{1}{z}} \cdot \frac{1}{2}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 13.4

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 13.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \left(\left(\sqrt{1 + x} + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.0002 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)))

   1. Initial program 97.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 98.3

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 98.3%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(1 + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(1 + \left(\color{blue}{\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\color{blue}{\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(1 + \left(\left(\color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(1 + \left(\left(\frac{1}{\color{blue}{\sqrt{1 + z} + \sqrt{z}}} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\left(\frac{1}{\color{blue}{\sqrt{1 + z} + \sqrt{z}}} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\left(\frac{1}{\color{blue}{\sqrt{1 + z}} + \sqrt{z}} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lower-sqrt.f64 95.7

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{1 + y}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right) \leq 0:\\ \;\;\;\;\left(\left(\sqrt{\frac{1}{y}} \cdot 0.5 + 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{elif}\;\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right) \leq 1.0002:\\ \;\;\;\;\left(\left(\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right) \cdot 0.5 + \sqrt{x + 1}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(\left(\sqrt{y} + \sqrt{x}\right) - \left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \sqrt{y + 1}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{z + 1}\\ t_4 := \sqrt{y} - t\_1\\ t_5 := \left(t\_2 - \sqrt{x}\right) - t\_4\\ t_6 := \sqrt{t} - \sqrt{1 + t}\\ \mathbf{if}\;t\_5 \leq 0:\\ \;\;\;\;\left(\left(\frac{0.5}{\sqrt{x}} - t\_4\right) - \left(\sqrt{z} - t\_3\right)\right) - t\_6\\ \mathbf{elif}\;t\_5 \leq 1.0002:\\ \;\;\;\;\left(\left(\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right) \cdot 0.5 + t\_2\right) - \sqrt{x}\right) - t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(\left(\sqrt{y} + \sqrt{x}\right) - \left(\frac{1}{\sqrt{z} + t\_3} + t\_1\right)\right)\right) - t\_6\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (+ z 1.0)))
        (t_4 (- (sqrt y) t_1))
        (t_5 (- (- t_2 (sqrt x)) t_4))
        (t_6 (- (sqrt t) (sqrt (+ 1.0 t)))))
   (if (<= t_5 0.0)
     (- (- (- (/ 0.5 (sqrt x)) t_4) (- (sqrt z) t_3)) t_6)
     (if (<= t_5 1.0002)
       (-
        (- (+ (* (+ (sqrt (/ 1.0 z)) (sqrt (/ 1.0 y))) 0.5) t_2) (sqrt x))
        t_6)
       (-
        (- 1.0 (- (+ (sqrt y) (sqrt x)) (+ (/ 1.0 (+ (sqrt z) t_3)) t_1)))
        t_6)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((x + 1.0));
	double t_3 = sqrt((z + 1.0));
	double t_4 = sqrt(y) - t_1;
	double t_5 = (t_2 - sqrt(x)) - t_4;
	double t_6 = sqrt(t) - sqrt((1.0 + t));
	double tmp;
	if (t_5 <= 0.0) {
		tmp = (((0.5 / sqrt(x)) - t_4) - (sqrt(z) - t_3)) - t_6;
	} else if (t_5 <= 1.0002) {
		tmp = ((((sqrt((1.0 / z)) + sqrt((1.0 / y))) * 0.5) + t_2) - sqrt(x)) - t_6;
	} else {
		tmp = (1.0 - ((sqrt(y) + sqrt(x)) - ((1.0 / (sqrt(z) + t_3)) + t_1))) - t_6;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((x + 1.0d0))
    t_3 = sqrt((z + 1.0d0))
    t_4 = sqrt(y) - t_1
    t_5 = (t_2 - sqrt(x)) - t_4
    t_6 = sqrt(t) - sqrt((1.0d0 + t))
    if (t_5 <= 0.0d0) then
        tmp = (((0.5d0 / sqrt(x)) - t_4) - (sqrt(z) - t_3)) - t_6
    else if (t_5 <= 1.0002d0) then
        tmp = ((((sqrt((1.0d0 / z)) + sqrt((1.0d0 / y))) * 0.5d0) + t_2) - sqrt(x)) - t_6
    else
        tmp = (1.0d0 - ((sqrt(y) + sqrt(x)) - ((1.0d0 / (sqrt(z) + t_3)) + t_1))) - t_6
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((x + 1.0));
	double t_3 = Math.sqrt((z + 1.0));
	double t_4 = Math.sqrt(y) - t_1;
	double t_5 = (t_2 - Math.sqrt(x)) - t_4;
	double t_6 = Math.sqrt(t) - Math.sqrt((1.0 + t));
	double tmp;
	if (t_5 <= 0.0) {
		tmp = (((0.5 / Math.sqrt(x)) - t_4) - (Math.sqrt(z) - t_3)) - t_6;
	} else if (t_5 <= 1.0002) {
		tmp = ((((Math.sqrt((1.0 / z)) + Math.sqrt((1.0 / y))) * 0.5) + t_2) - Math.sqrt(x)) - t_6;
	} else {
		tmp = (1.0 - ((Math.sqrt(y) + Math.sqrt(x)) - ((1.0 / (Math.sqrt(z) + t_3)) + t_1))) - t_6;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((x + 1.0))
	t_3 = math.sqrt((z + 1.0))
	t_4 = math.sqrt(y) - t_1
	t_5 = (t_2 - math.sqrt(x)) - t_4
	t_6 = math.sqrt(t) - math.sqrt((1.0 + t))
	tmp = 0
	if t_5 <= 0.0:
		tmp = (((0.5 / math.sqrt(x)) - t_4) - (math.sqrt(z) - t_3)) - t_6
	elif t_5 <= 1.0002:
		tmp = ((((math.sqrt((1.0 / z)) + math.sqrt((1.0 / y))) * 0.5) + t_2) - math.sqrt(x)) - t_6
	else:
		tmp = (1.0 - ((math.sqrt(y) + math.sqrt(x)) - ((1.0 / (math.sqrt(z) + t_3)) + t_1))) - t_6
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(x + 1.0))
	t_3 = sqrt(Float64(z + 1.0))
	t_4 = Float64(sqrt(y) - t_1)
	t_5 = Float64(Float64(t_2 - sqrt(x)) - t_4)
	t_6 = Float64(sqrt(t) - sqrt(Float64(1.0 + t)))
	tmp = 0.0
	if (t_5 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(0.5 / sqrt(x)) - t_4) - Float64(sqrt(z) - t_3)) - t_6);
	elseif (t_5 <= 1.0002)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / z)) + sqrt(Float64(1.0 / y))) * 0.5) + t_2) - sqrt(x)) - t_6);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(sqrt(y) + sqrt(x)) - Float64(Float64(1.0 / Float64(sqrt(z) + t_3)) + t_1))) - t_6);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((x + 1.0));
	t_3 = sqrt((z + 1.0));
	t_4 = sqrt(y) - t_1;
	t_5 = (t_2 - sqrt(x)) - t_4;
	t_6 = sqrt(t) - sqrt((1.0 + t));
	tmp = 0.0;
	if (t_5 <= 0.0)
		tmp = (((0.5 / sqrt(x)) - t_4) - (sqrt(z) - t_3)) - t_6;
	elseif (t_5 <= 1.0002)
		tmp = ((((sqrt((1.0 / z)) + sqrt((1.0 / y))) * 0.5) + t_2) - sqrt(x)) - t_6;
	else
		tmp = (1.0 - ((sqrt(y) + sqrt(x)) - ((1.0 / (sqrt(z) + t_3)) + t_1))) - t_6;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[y], $MachinePrecision] - t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.0], N[(N[(N[(N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision], If[LessEqual[t$95$5, 1.0002], N[(N[(N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision], N[(N[(1.0 - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 0.0

   1. Initial program 64.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot \frac{1}{2} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 74.3

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{x}}} \cdot 0.5 + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 74.3%

      \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 74.3%

      \[\leadsto \left(\left(\color{blue}{\frac{0.5}{\sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 1.0002

   1. Initial program 95.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{\frac{1}{z}} \cdot \frac{1}{2}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 13.4

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 13.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \left(\left(\sqrt{1 + x} + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.0002 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)))

   1. Initial program 97.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 98.3

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 98.3%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{\left(1 + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{\sqrt{z} + \sqrt{1 + z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(1 + \left(\color{blue}{\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\color{blue}{\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \sqrt{1 + y}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(1 + \left(\left(\color{blue}{\frac{1}{\sqrt{z} + \sqrt{1 + z}}} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(1 + \left(\left(\frac{1}{\color{blue}{\sqrt{1 + z} + \sqrt{z}}} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(1 + \left(\left(\frac{1}{\color{blue}{\sqrt{1 + z} + \sqrt{z}}} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(1 + \left(\left(\frac{1}{\color{blue}{\sqrt{1 + z}} + \sqrt{z}} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{\color{blue}{1 + z}} + \sqrt{z}} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \color{blue}{\sqrt{z}}} + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \color{blue}{\sqrt{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{\color{blue}{1 + y}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{1 + y}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lower-sqrt.f64 95.7

          \[\leadsto \left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{1 + y}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\left(1 + \left(\left(\frac{1}{\sqrt{1 + z} + \sqrt{z}} + \sqrt{1 + y}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right) \leq 0:\\ \;\;\;\;\left(\left(\frac{0.5}{\sqrt{x}} - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{elif}\;\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right) \leq 1.0002:\\ \;\;\;\;\left(\left(\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right) \cdot 0.5 + \sqrt{x + 1}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(\left(\sqrt{y} + \sqrt{x}\right) - \left(\frac{1}{\sqrt{z} + \sqrt{z + 1}} + \sqrt{y + 1}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{z} - \sqrt{z + 1}\\ t_3 := \sqrt{y} - \sqrt{y + 1}\\ t_4 := \left(t\_1 - \sqrt{x}\right) - t\_3\\ t_5 := \sqrt{t} - \sqrt{1 + t}\\ \mathbf{if}\;t\_4 \leq 0:\\ \;\;\;\;\left(\left(\frac{0.5}{\sqrt{x}} - t\_3\right) - t\_2\right) - t\_5\\ \mathbf{elif}\;t\_4 \leq 1.0002:\\ \;\;\;\;\left(\left(\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right) \cdot 0.5 + t\_1\right) - \sqrt{x}\right) - t\_5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) - t\_3\right) - t\_2\right) - t\_5\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0)))
        (t_2 (- (sqrt z) (sqrt (+ z 1.0))))
        (t_3 (- (sqrt y) (sqrt (+ y 1.0))))
        (t_4 (- (- t_1 (sqrt x)) t_3))
        (t_5 (- (sqrt t) (sqrt (+ 1.0 t)))))
   (if (<= t_4 0.0)
     (- (- (- (/ 0.5 (sqrt x)) t_3) t_2) t_5)
     (if (<= t_4 1.0002)
       (-
        (- (+ (* (+ (sqrt (/ 1.0 z)) (sqrt (/ 1.0 y))) 0.5) t_1) (sqrt x))
        t_5)
       (- (- (- (- 1.0 (sqrt x)) t_3) t_2) t_5)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt(z) - sqrt((z + 1.0));
	double t_3 = sqrt(y) - sqrt((y + 1.0));
	double t_4 = (t_1 - sqrt(x)) - t_3;
	double t_5 = sqrt(t) - sqrt((1.0 + t));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = (((0.5 / sqrt(x)) - t_3) - t_2) - t_5;
	} else if (t_4 <= 1.0002) {
		tmp = ((((sqrt((1.0 / z)) + sqrt((1.0 / y))) * 0.5) + t_1) - sqrt(x)) - t_5;
	} else {
		tmp = (((1.0 - sqrt(x)) - t_3) - t_2) - t_5;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt(z) - sqrt((z + 1.0d0))
    t_3 = sqrt(y) - sqrt((y + 1.0d0))
    t_4 = (t_1 - sqrt(x)) - t_3
    t_5 = sqrt(t) - sqrt((1.0d0 + t))
    if (t_4 <= 0.0d0) then
        tmp = (((0.5d0 / sqrt(x)) - t_3) - t_2) - t_5
    else if (t_4 <= 1.0002d0) then
        tmp = ((((sqrt((1.0d0 / z)) + sqrt((1.0d0 / y))) * 0.5d0) + t_1) - sqrt(x)) - t_5
    else
        tmp = (((1.0d0 - sqrt(x)) - t_3) - t_2) - t_5
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double t_2 = Math.sqrt(z) - Math.sqrt((z + 1.0));
	double t_3 = Math.sqrt(y) - Math.sqrt((y + 1.0));
	double t_4 = (t_1 - Math.sqrt(x)) - t_3;
	double t_5 = Math.sqrt(t) - Math.sqrt((1.0 + t));
	double tmp;
	if (t_4 <= 0.0) {
		tmp = (((0.5 / Math.sqrt(x)) - t_3) - t_2) - t_5;
	} else if (t_4 <= 1.0002) {
		tmp = ((((Math.sqrt((1.0 / z)) + Math.sqrt((1.0 / y))) * 0.5) + t_1) - Math.sqrt(x)) - t_5;
	} else {
		tmp = (((1.0 - Math.sqrt(x)) - t_3) - t_2) - t_5;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	t_2 = math.sqrt(z) - math.sqrt((z + 1.0))
	t_3 = math.sqrt(y) - math.sqrt((y + 1.0))
	t_4 = (t_1 - math.sqrt(x)) - t_3
	t_5 = math.sqrt(t) - math.sqrt((1.0 + t))
	tmp = 0
	if t_4 <= 0.0:
		tmp = (((0.5 / math.sqrt(x)) - t_3) - t_2) - t_5
	elif t_4 <= 1.0002:
		tmp = ((((math.sqrt((1.0 / z)) + math.sqrt((1.0 / y))) * 0.5) + t_1) - math.sqrt(x)) - t_5
	else:
		tmp = (((1.0 - math.sqrt(x)) - t_3) - t_2) - t_5
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = Float64(sqrt(z) - sqrt(Float64(z + 1.0)))
	t_3 = Float64(sqrt(y) - sqrt(Float64(y + 1.0)))
	t_4 = Float64(Float64(t_1 - sqrt(x)) - t_3)
	t_5 = Float64(sqrt(t) - sqrt(Float64(1.0 + t)))
	tmp = 0.0
	if (t_4 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(0.5 / sqrt(x)) - t_3) - t_2) - t_5);
	elseif (t_4 <= 1.0002)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / z)) + sqrt(Float64(1.0 / y))) * 0.5) + t_1) - sqrt(x)) - t_5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) - t_3) - t_2) - t_5);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	t_2 = sqrt(z) - sqrt((z + 1.0));
	t_3 = sqrt(y) - sqrt((y + 1.0));
	t_4 = (t_1 - sqrt(x)) - t_3;
	t_5 = sqrt(t) - sqrt((1.0 + t));
	tmp = 0.0;
	if (t_4 <= 0.0)
		tmp = (((0.5 / sqrt(x)) - t_3) - t_2) - t_5;
	elseif (t_4 <= 1.0002)
		tmp = ((((sqrt((1.0 / z)) + sqrt((1.0 / y))) * 0.5) + t_1) - sqrt(x)) - t_5;
	else
		tmp = (((1.0 - sqrt(x)) - t_3) - t_2) - t_5;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 0.0], N[(N[(N[(N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision] - t$95$2), $MachinePrecision] - t$95$5), $MachinePrecision], If[LessEqual[t$95$4, 1.0002], N[(N[(N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision] - t$95$2), $MachinePrecision] - t$95$5), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 0.0

   1. Initial program 64.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot \frac{1}{2} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 74.3

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{x}}} \cdot 0.5 + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 74.3%

      \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 74.3%

      \[\leadsto \left(\left(\color{blue}{\frac{0.5}{\sqrt{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 1.0002

   1. Initial program 95.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{\frac{1}{z}} \cdot \frac{1}{2}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 13.4

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 13.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 29.4%

      \[\leadsto \left(\left(\sqrt{1 + x} + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.0002 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)))

   1. Initial program 97.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 94.1

         \[\leadsto \left(\left(\left(1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 94.1%

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right) \leq 0:\\ \;\;\;\;\left(\left(\frac{0.5}{\sqrt{x}} - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{elif}\;\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right) \leq 1.0002:\\ \;\;\;\;\left(\left(\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right) \cdot 0.5 + \sqrt{x + 1}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{y} - \sqrt{y + 1}\\ t_3 := \sqrt{t} - \sqrt{1 + t}\\ \mathbf{if}\;\left(t\_1 - \sqrt{x}\right) - t\_2 \leq 1.0002:\\ \;\;\;\;\left(\left(\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right) \cdot 0.5 + t\_1\right) - \sqrt{x}\right) - t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) - t\_2\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0)))
        (t_2 (- (sqrt y) (sqrt (+ y 1.0))))
        (t_3 (- (sqrt t) (sqrt (+ 1.0 t)))))
   (if (<= (- (- t_1 (sqrt x)) t_2) 1.0002)
     (- (- (+ (* (+ (sqrt (/ 1.0 z)) (sqrt (/ 1.0 y))) 0.5) t_1) (sqrt x)) t_3)
     (- (- (- (- 1.0 (sqrt x)) t_2) (- (sqrt z) (sqrt (+ z 1.0)))) t_3))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0));
	double t_2 = sqrt(y) - sqrt((y + 1.0));
	double t_3 = sqrt(t) - sqrt((1.0 + t));
	double tmp;
	if (((t_1 - sqrt(x)) - t_2) <= 1.0002) {
		tmp = ((((sqrt((1.0 / z)) + sqrt((1.0 / y))) * 0.5) + t_1) - sqrt(x)) - t_3;
	} else {
		tmp = (((1.0 - sqrt(x)) - t_2) - (sqrt(z) - sqrt((z + 1.0)))) - t_3;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    t_2 = sqrt(y) - sqrt((y + 1.0d0))
    t_3 = sqrt(t) - sqrt((1.0d0 + t))
    if (((t_1 - sqrt(x)) - t_2) <= 1.0002d0) then
        tmp = ((((sqrt((1.0d0 / z)) + sqrt((1.0d0 / y))) * 0.5d0) + t_1) - sqrt(x)) - t_3
    else
        tmp = (((1.0d0 - sqrt(x)) - t_2) - (sqrt(z) - sqrt((z + 1.0d0)))) - t_3
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double t_2 = Math.sqrt(y) - Math.sqrt((y + 1.0));
	double t_3 = Math.sqrt(t) - Math.sqrt((1.0 + t));
	double tmp;
	if (((t_1 - Math.sqrt(x)) - t_2) <= 1.0002) {
		tmp = ((((Math.sqrt((1.0 / z)) + Math.sqrt((1.0 / y))) * 0.5) + t_1) - Math.sqrt(x)) - t_3;
	} else {
		tmp = (((1.0 - Math.sqrt(x)) - t_2) - (Math.sqrt(z) - Math.sqrt((z + 1.0)))) - t_3;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	t_2 = math.sqrt(y) - math.sqrt((y + 1.0))
	t_3 = math.sqrt(t) - math.sqrt((1.0 + t))
	tmp = 0
	if ((t_1 - math.sqrt(x)) - t_2) <= 1.0002:
		tmp = ((((math.sqrt((1.0 / z)) + math.sqrt((1.0 / y))) * 0.5) + t_1) - math.sqrt(x)) - t_3
	else:
		tmp = (((1.0 - math.sqrt(x)) - t_2) - (math.sqrt(z) - math.sqrt((z + 1.0)))) - t_3
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	t_2 = Float64(sqrt(y) - sqrt(Float64(y + 1.0)))
	t_3 = Float64(sqrt(t) - sqrt(Float64(1.0 + t)))
	tmp = 0.0
	if (Float64(Float64(t_1 - sqrt(x)) - t_2) <= 1.0002)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / z)) + sqrt(Float64(1.0 / y))) * 0.5) + t_1) - sqrt(x)) - t_3);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) - t_2) - Float64(sqrt(z) - sqrt(Float64(z + 1.0)))) - t_3);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	t_2 = sqrt(y) - sqrt((y + 1.0));
	t_3 = sqrt(t) - sqrt((1.0 + t));
	tmp = 0.0;
	if (((t_1 - sqrt(x)) - t_2) <= 1.0002)
		tmp = ((((sqrt((1.0 / z)) + sqrt((1.0 / y))) * 0.5) + t_1) - sqrt(x)) - t_3;
	else
		tmp = (((1.0 - sqrt(x)) - t_2) - (sqrt(z) - sqrt((z + 1.0)))) - t_3;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], 1.0002], N[(N[(N[(N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) < 1.0002

   1. Initial program 87.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{\frac{1}{z}} \cdot \frac{1}{2}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 17.0

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 17.0%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\left(\sqrt{1 + x} + \left(\frac{1}{2} \cdot \sqrt{\frac{1}{y}} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 28.8%

      \[\leadsto \left(\left(\sqrt{1 + x} + 0.5 \cdot \left(\sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.0002 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)))

   1. Initial program 97.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-sqrt.f64 94.1

         \[\leadsto \left(\left(\left(1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 94.1%

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right) \leq 1.0002:\\ \;\;\;\;\left(\left(\left(\sqrt{\frac{1}{z}} + \sqrt{\frac{1}{y}}\right) \cdot 0.5 + \sqrt{x + 1}\right) - \sqrt{x}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 97.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{t} - \sqrt{1 + t}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\left(\sqrt{\frac{1}{y}} \cdot 0.5 + 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \left(\sqrt{z} - t\_2\right)\right) - t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_1 - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \frac{z - \left(z + 1\right)}{\sqrt{z} + t\_2}\right) - t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ x 1.0)) (sqrt x)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (- (sqrt t) (sqrt (+ 1.0 t)))))
   (if (<= t_1 0.0)
     (-
      (-
       (+ (* (sqrt (/ 1.0 y)) 0.5) (* 0.5 (sqrt (/ 1.0 x))))
       (- (sqrt z) t_2))
      t_3)
     (-
      (-
       (- t_1 (- (sqrt y) (sqrt (+ y 1.0))))
       (/ (- z (+ z 1.0)) (+ (sqrt z) t_2)))
      t_3))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0)) - sqrt(x);
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt(t) - sqrt((1.0 + t));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (((sqrt((1.0 / y)) * 0.5) + (0.5 * sqrt((1.0 / x)))) - (sqrt(z) - t_2)) - t_3;
	} else {
		tmp = ((t_1 - (sqrt(y) - sqrt((y + 1.0)))) - ((z - (z + 1.0)) / (sqrt(z) + t_2))) - t_3;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0)) - sqrt(x)
    t_2 = sqrt((z + 1.0d0))
    t_3 = sqrt(t) - sqrt((1.0d0 + t))
    if (t_1 <= 0.0d0) then
        tmp = (((sqrt((1.0d0 / y)) * 0.5d0) + (0.5d0 * sqrt((1.0d0 / x)))) - (sqrt(z) - t_2)) - t_3
    else
        tmp = ((t_1 - (sqrt(y) - sqrt((y + 1.0d0)))) - ((z - (z + 1.0d0)) / (sqrt(z) + t_2))) - t_3
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double t_2 = Math.sqrt((z + 1.0));
	double t_3 = Math.sqrt(t) - Math.sqrt((1.0 + t));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (((Math.sqrt((1.0 / y)) * 0.5) + (0.5 * Math.sqrt((1.0 / x)))) - (Math.sqrt(z) - t_2)) - t_3;
	} else {
		tmp = ((t_1 - (Math.sqrt(y) - Math.sqrt((y + 1.0)))) - ((z - (z + 1.0)) / (Math.sqrt(z) + t_2))) - t_3;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0)) - math.sqrt(x)
	t_2 = math.sqrt((z + 1.0))
	t_3 = math.sqrt(t) - math.sqrt((1.0 + t))
	tmp = 0
	if t_1 <= 0.0:
		tmp = (((math.sqrt((1.0 / y)) * 0.5) + (0.5 * math.sqrt((1.0 / x)))) - (math.sqrt(z) - t_2)) - t_3
	else:
		tmp = ((t_1 - (math.sqrt(y) - math.sqrt((y + 1.0)))) - ((z - (z + 1.0)) / (math.sqrt(z) + t_2))) - t_3
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = Float64(sqrt(t) - sqrt(Float64(1.0 + t)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / y)) * 0.5) + Float64(0.5 * sqrt(Float64(1.0 / x)))) - Float64(sqrt(z) - t_2)) - t_3);
	else
		tmp = Float64(Float64(Float64(t_1 - Float64(sqrt(y) - sqrt(Float64(y + 1.0)))) - Float64(Float64(z - Float64(z + 1.0)) / Float64(sqrt(z) + t_2))) - t_3);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0)) - sqrt(x);
	t_2 = sqrt((z + 1.0));
	t_3 = sqrt(t) - sqrt((1.0 + t));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = (((sqrt((1.0 / y)) * 0.5) + (0.5 * sqrt((1.0 / x)))) - (sqrt(z) - t_2)) - t_3;
	else
		tmp = ((t_1 - (sqrt(y) - sqrt((y + 1.0)))) - ((z - (z + 1.0)) / (sqrt(z) + t_2))) - t_3;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision], N[(N[(N[(t$95$1 - N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(z - N[(z + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0

   1. Initial program 82.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \left(\left(\color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}}} \cdot \frac{1}{2} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 87.2

         \[\leadsto \left(\left(\sqrt{\color{blue}{\frac{1}{x}}} \cdot 0.5 + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 87.2%

      \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in y around inf

      \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{2} + \color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{2} + \color{blue}{\sqrt{\frac{1}{y}} \cdot \frac{1}{2}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot \frac{1}{2} + \color{blue}{\sqrt{\frac{1}{y}}} \cdot \frac{1}{2}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-/.f64 40.2

         \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot 0.5 + \sqrt{\color{blue}{\frac{1}{y}}} \cdot 0.5\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Applied rewrites 40.2%

      \[\leadsto \left(\left(\sqrt{\frac{1}{x}} \cdot 0.5 + \color{blue}{\sqrt{\frac{1}{y}} \cdot 0.5}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 96.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 97.5

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 97.5%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z} + \sqrt{z + 1}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0:\\ \;\;\;\;\left(\left(\sqrt{\frac{1}{y}} \cdot 0.5 + 0.5 \cdot \sqrt{\frac{1}{x}}\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \frac{z - \left(z + 1\right)}{\sqrt{z} + \sqrt{z + 1}}\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 68.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{x + 1}\\ \mathbf{if}\;\left(\left(t\_3 - \sqrt{x}\right) - \left(\sqrt{y} - t\_1\right)\right) - \left(\sqrt{z} - t\_2\right) \leq 2:\\ \;\;\;\;\left(t\_3 + t\_1\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_2 - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + 2\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ y 1.0))) (t_2 (sqrt (+ z 1.0))) (t_3 (sqrt (+ x 1.0))))
   (if (<= (- (- (- t_3 (sqrt x)) (- (sqrt y) t_1)) (- (sqrt z) t_2)) 2.0)
     (- (+ t_3 t_1) (+ (sqrt y) (sqrt x)))
     (+ (- (- t_2 (sqrt x)) (+ (sqrt y) (sqrt z))) 2.0))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((y + 1.0));
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((x + 1.0));
	double tmp;
	if ((((t_3 - sqrt(x)) - (sqrt(y) - t_1)) - (sqrt(z) - t_2)) <= 2.0) {
		tmp = (t_3 + t_1) - (sqrt(y) + sqrt(x));
	} else {
		tmp = ((t_2 - sqrt(x)) - (sqrt(y) + sqrt(z))) + 2.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt((y + 1.0d0))
    t_2 = sqrt((z + 1.0d0))
    t_3 = sqrt((x + 1.0d0))
    if ((((t_3 - sqrt(x)) - (sqrt(y) - t_1)) - (sqrt(z) - t_2)) <= 2.0d0) then
        tmp = (t_3 + t_1) - (sqrt(y) + sqrt(x))
    else
        tmp = ((t_2 - sqrt(x)) - (sqrt(y) + sqrt(z))) + 2.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((y + 1.0));
	double t_2 = Math.sqrt((z + 1.0));
	double t_3 = Math.sqrt((x + 1.0));
	double tmp;
	if ((((t_3 - Math.sqrt(x)) - (Math.sqrt(y) - t_1)) - (Math.sqrt(z) - t_2)) <= 2.0) {
		tmp = (t_3 + t_1) - (Math.sqrt(y) + Math.sqrt(x));
	} else {
		tmp = ((t_2 - Math.sqrt(x)) - (Math.sqrt(y) + Math.sqrt(z))) + 2.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((y + 1.0))
	t_2 = math.sqrt((z + 1.0))
	t_3 = math.sqrt((x + 1.0))
	tmp = 0
	if (((t_3 - math.sqrt(x)) - (math.sqrt(y) - t_1)) - (math.sqrt(z) - t_2)) <= 2.0:
		tmp = (t_3 + t_1) - (math.sqrt(y) + math.sqrt(x))
	else:
		tmp = ((t_2 - math.sqrt(x)) - (math.sqrt(y) + math.sqrt(z))) + 2.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(y + 1.0))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 - sqrt(x)) - Float64(sqrt(y) - t_1)) - Float64(sqrt(z) - t_2)) <= 2.0)
		tmp = Float64(Float64(t_3 + t_1) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(t_2 - sqrt(x)) - Float64(sqrt(y) + sqrt(z))) + 2.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((y + 1.0));
	t_2 = sqrt((z + 1.0));
	t_3 = sqrt((x + 1.0));
	tmp = 0.0;
	if ((((t_3 - sqrt(x)) - (sqrt(y) - t_1)) - (sqrt(z) - t_2)) <= 2.0)
		tmp = (t_3 + t_1) - (sqrt(y) + sqrt(x));
	else
		tmp = ((t_2 - sqrt(x)) - (sqrt(y) + sqrt(z))) + 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(t$95$3 + t$95$1), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))) < 2

   1. Initial program 89.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 5.3

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 5.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 14.1%

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

   if 2 < (+.f64 (+.f64 (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))) (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)))

   1. Initial program 95.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 59.6%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.7%

      \[\leadsto 2 + \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 45.0%

       \[\leadsto 2 + \left(\left(\sqrt{1 + z} - \sqrt{x}\right) - \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{x + 1} - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right) \leq 2:\\ \;\;\;\;\left(\sqrt{x + 1} + \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{z + 1} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + 2\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 91.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y} + \sqrt{x}\\ t_2 := \sqrt{z + 1}\\ t_3 := \sqrt{y + 1}\\ t_4 := \sqrt{t} - \sqrt{1 + t}\\ \mathbf{if}\;t\_2 - \sqrt{z} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\left(1 - \left(t\_1 - \left(\sqrt{\frac{1}{z}} \cdot 0.5 + t\_3\right)\right)\right) - t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t\_3 + 1\right) - t\_1\right) - \left(\sqrt{z} - t\_2\right)\right) - t\_4\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt y) (sqrt x)))
        (t_2 (sqrt (+ z 1.0)))
        (t_3 (sqrt (+ y 1.0)))
        (t_4 (- (sqrt t) (sqrt (+ 1.0 t)))))
   (if (<= (- t_2 (sqrt z)) 5e-6)
     (- (- 1.0 (- t_1 (+ (* (sqrt (/ 1.0 z)) 0.5) t_3))) t_4)
     (- (- (- (+ t_3 1.0) t_1) (- (sqrt z) t_2)) t_4))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(y) + sqrt(x);
	double t_2 = sqrt((z + 1.0));
	double t_3 = sqrt((y + 1.0));
	double t_4 = sqrt(t) - sqrt((1.0 + t));
	double tmp;
	if ((t_2 - sqrt(z)) <= 5e-6) {
		tmp = (1.0 - (t_1 - ((sqrt((1.0 / z)) * 0.5) + t_3))) - t_4;
	} else {
		tmp = (((t_3 + 1.0) - t_1) - (sqrt(z) - t_2)) - t_4;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sqrt(y) + sqrt(x)
    t_2 = sqrt((z + 1.0d0))
    t_3 = sqrt((y + 1.0d0))
    t_4 = sqrt(t) - sqrt((1.0d0 + t))
    if ((t_2 - sqrt(z)) <= 5d-6) then
        tmp = (1.0d0 - (t_1 - ((sqrt((1.0d0 / z)) * 0.5d0) + t_3))) - t_4
    else
        tmp = (((t_3 + 1.0d0) - t_1) - (sqrt(z) - t_2)) - t_4
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(y) + Math.sqrt(x);
	double t_2 = Math.sqrt((z + 1.0));
	double t_3 = Math.sqrt((y + 1.0));
	double t_4 = Math.sqrt(t) - Math.sqrt((1.0 + t));
	double tmp;
	if ((t_2 - Math.sqrt(z)) <= 5e-6) {
		tmp = (1.0 - (t_1 - ((Math.sqrt((1.0 / z)) * 0.5) + t_3))) - t_4;
	} else {
		tmp = (((t_3 + 1.0) - t_1) - (Math.sqrt(z) - t_2)) - t_4;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(y) + math.sqrt(x)
	t_2 = math.sqrt((z + 1.0))
	t_3 = math.sqrt((y + 1.0))
	t_4 = math.sqrt(t) - math.sqrt((1.0 + t))
	tmp = 0
	if (t_2 - math.sqrt(z)) <= 5e-6:
		tmp = (1.0 - (t_1 - ((math.sqrt((1.0 / z)) * 0.5) + t_3))) - t_4
	else:
		tmp = (((t_3 + 1.0) - t_1) - (math.sqrt(z) - t_2)) - t_4
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(y) + sqrt(x))
	t_2 = sqrt(Float64(z + 1.0))
	t_3 = sqrt(Float64(y + 1.0))
	t_4 = Float64(sqrt(t) - sqrt(Float64(1.0 + t)))
	tmp = 0.0
	if (Float64(t_2 - sqrt(z)) <= 5e-6)
		tmp = Float64(Float64(1.0 - Float64(t_1 - Float64(Float64(sqrt(Float64(1.0 / z)) * 0.5) + t_3))) - t_4);
	else
		tmp = Float64(Float64(Float64(Float64(t_3 + 1.0) - t_1) - Float64(sqrt(z) - t_2)) - t_4);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(y) + sqrt(x);
	t_2 = sqrt((z + 1.0));
	t_3 = sqrt((y + 1.0));
	t_4 = sqrt(t) - sqrt((1.0 + t));
	tmp = 0.0;
	if ((t_2 - sqrt(z)) <= 5e-6)
		tmp = (1.0 - (t_1 - ((sqrt((1.0 / z)) * 0.5) + t_3))) - t_4;
	else
		tmp = (((t_3 + 1.0) - t_1) - (sqrt(z) - t_2)) - t_4;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[(1.0 - N[(t$95$1 - N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision], N[(N[(N[(N[(t$95$3 + 1.0), $MachinePrecision] - t$95$1), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 5.00000000000000041e-6

   1. Initial program 84.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{2} \cdot \sqrt{\frac{1}{z}} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{\frac{1}{z}} \cdot \frac{1}{2}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{z}}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + x}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, \frac{1}{2}, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 48.5

          \[\leadsto \left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 48.5%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\left(1 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 54.8%

      \[\leadsto \left(1 + \color{blue}{\left(\left(\sqrt{1 + y} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 5.00000000000000041e-6 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 96.9%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} + 1\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} + 1\right)} - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} + 1\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} + 1\right) - \left(\sqrt{x} + \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) - \left(\color{blue}{\sqrt{y}} + \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 40.9

         \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) - \left(\sqrt{y} + \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 40.9%

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{1 + y} + 1\right) - \left(\sqrt{y} + \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{z + 1} - \sqrt{z} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\left(1 - \left(\left(\sqrt{y} + \sqrt{x}\right) - \left(\sqrt{\frac{1}{z}} \cdot 0.5 + \sqrt{y + 1}\right)\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt{y + 1} + 1\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 90.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\left(\left(1 - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (-
  (-
   (- (- 1.0 (sqrt x)) (- (sqrt y) (sqrt (+ y 1.0))))
   (- (sqrt z) (sqrt (+ z 1.0))))
  (- (sqrt t) (sqrt (+ 1.0 t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (((1.0 - sqrt(x)) - (sqrt(y) - sqrt((y + 1.0)))) - (sqrt(z) - sqrt((z + 1.0)))) - (sqrt(t) - sqrt((1.0 + t)));
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 - sqrt(x)) - (sqrt(y) - sqrt((y + 1.0d0)))) - (sqrt(z) - sqrt((z + 1.0d0)))) - (sqrt(t) - sqrt((1.0d0 + t)))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return (((1.0 - Math.sqrt(x)) - (Math.sqrt(y) - Math.sqrt((y + 1.0)))) - (Math.sqrt(z) - Math.sqrt((z + 1.0)))) - (Math.sqrt(t) - Math.sqrt((1.0 + t)));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (((1.0 - math.sqrt(x)) - (math.sqrt(y) - math.sqrt((y + 1.0)))) - (math.sqrt(z) - math.sqrt((z + 1.0)))) - (math.sqrt(t) - math.sqrt((1.0 + t)))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 - sqrt(x)) - Float64(sqrt(y) - sqrt(Float64(y + 1.0)))) - Float64(sqrt(z) - sqrt(Float64(z + 1.0)))) - Float64(sqrt(t) - sqrt(Float64(1.0 + t))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (((1.0 - sqrt(x)) - (sqrt(y) - sqrt((y + 1.0)))) - (sqrt(z) - sqrt((z + 1.0)))) - (sqrt(t) - sqrt((1.0 + t)));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] - N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[t], $MachinePrecision] - N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. lower-sqrt.f64 53.1

      \[\leadsto \left(\left(\left(1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Applied rewrites 53.1%

   \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

6. Final simplification 53.1%

   \[\leadsto \left(\left(\left(1 - \sqrt{x}\right) - \left(\sqrt{y} - \sqrt{y + 1}\right)\right) - \left(\sqrt{z} - \sqrt{z + 1}\right)\right) - \left(\sqrt{t} - \sqrt{1 + t}\right) \]
7. Add Preprocessing

Alternative 22: 78.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y} + \sqrt{z}\\ \mathbf{if}\;y \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;\left(\left(\sqrt{z + 1} - \sqrt{x}\right) - t\_1\right) + 2\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+20}:\\ \;\;\;\;\left(\sqrt{x + 1} + \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t} - \left(\left(t\_1 + \sqrt{x}\right) + \sqrt{t}\right)\right) + 1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (sqrt y) (sqrt z))))
   (if (<= y 2.9e-24)
     (+ (- (- (sqrt (+ z 1.0)) (sqrt x)) t_1) 2.0)
     (if (<= y 5.2e+20)
       (- (+ (sqrt (+ x 1.0)) (sqrt (+ y 1.0))) (+ (sqrt y) (sqrt x)))
       (+ (- (sqrt t) (+ (+ t_1 (sqrt x)) (sqrt t))) 1.0)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(y) + sqrt(z);
	double tmp;
	if (y <= 2.9e-24) {
		tmp = ((sqrt((z + 1.0)) - sqrt(x)) - t_1) + 2.0;
	} else if (y <= 5.2e+20) {
		tmp = (sqrt((x + 1.0)) + sqrt((y + 1.0))) - (sqrt(y) + sqrt(x));
	} else {
		tmp = (sqrt(t) - ((t_1 + sqrt(x)) + sqrt(t))) + 1.0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(y) + sqrt(z)
    if (y <= 2.9d-24) then
        tmp = ((sqrt((z + 1.0d0)) - sqrt(x)) - t_1) + 2.0d0
    else if (y <= 5.2d+20) then
        tmp = (sqrt((x + 1.0d0)) + sqrt((y + 1.0d0))) - (sqrt(y) + sqrt(x))
    else
        tmp = (sqrt(t) - ((t_1 + sqrt(x)) + sqrt(t))) + 1.0d0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(y) + Math.sqrt(z);
	double tmp;
	if (y <= 2.9e-24) {
		tmp = ((Math.sqrt((z + 1.0)) - Math.sqrt(x)) - t_1) + 2.0;
	} else if (y <= 5.2e+20) {
		tmp = (Math.sqrt((x + 1.0)) + Math.sqrt((y + 1.0))) - (Math.sqrt(y) + Math.sqrt(x));
	} else {
		tmp = (Math.sqrt(t) - ((t_1 + Math.sqrt(x)) + Math.sqrt(t))) + 1.0;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(y) + math.sqrt(z)
	tmp = 0
	if y <= 2.9e-24:
		tmp = ((math.sqrt((z + 1.0)) - math.sqrt(x)) - t_1) + 2.0
	elif y <= 5.2e+20:
		tmp = (math.sqrt((x + 1.0)) + math.sqrt((y + 1.0))) - (math.sqrt(y) + math.sqrt(x))
	else:
		tmp = (math.sqrt(t) - ((t_1 + math.sqrt(x)) + math.sqrt(t))) + 1.0
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(y) + sqrt(z))
	tmp = 0.0
	if (y <= 2.9e-24)
		tmp = Float64(Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(x)) - t_1) + 2.0);
	elseif (y <= 5.2e+20)
		tmp = Float64(Float64(sqrt(Float64(x + 1.0)) + sqrt(Float64(y + 1.0))) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(sqrt(t) - Float64(Float64(t_1 + sqrt(x)) + sqrt(t))) + 1.0);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(y) + sqrt(z);
	tmp = 0.0;
	if (y <= 2.9e-24)
		tmp = ((sqrt((z + 1.0)) - sqrt(x)) - t_1) + 2.0;
	elseif (y <= 5.2e+20)
		tmp = (sqrt((x + 1.0)) + sqrt((y + 1.0))) - (sqrt(y) + sqrt(x));
	else
		tmp = (sqrt(t) - ((t_1 + sqrt(x)) + sqrt(t))) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.9e-24], N[(N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 5.2e+20], N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[t], $MachinePrecision] - N[(N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.8999999999999999e-24

   1. Initial program 96.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 35.1%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.2%

      \[\leadsto 2 + \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 33.2%

       \[\leadsto 2 + \left(\left(\sqrt{1 + z} - \sqrt{x}\right) - \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) \]

   if 2.8999999999999999e-24 < y < 5.2e20

   1. Initial program 92.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)} \]

      2. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + x} + \sqrt{1 + y}\right)} + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{1 + x}} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + z}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + z}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

      16. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]

      17. lower-sqrt.f64 20.3

          \[\leadsto \left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \color{blue}{\sqrt{x}}\right) \]

   5. Applied rewrites 20.3%

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + x} + \sqrt{1 + y}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 19.5%

      \[\leadsto \left(\sqrt{1 + x} + \sqrt{1 + y}\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]

   if 5.2e20 < y

   1. Initial program 84.1%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      3. lower--.f64 N/A

         \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      7. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

      13. lower-+.f64 N/A

          \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. Applied rewrites 31.9%

      \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto 1 + \left(\sqrt{t} - \left(\color{blue}{\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} + \sqrt{t}\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 7.5%

      \[\leadsto 1 + \left(\sqrt{t} - \left(\color{blue}{\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} + \sqrt{t}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 21.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;\left(\left(\sqrt{z + 1} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+20}:\\ \;\;\;\;\left(\sqrt{x + 1} + \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{t} - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 63.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\left(\sqrt{z + 1} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + 2 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (+ (- (- (sqrt (+ z 1.0)) (sqrt x)) (+ (sqrt y) (sqrt z))) 2.0))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return ((sqrt((z + 1.0)) - sqrt(x)) - (sqrt(y) + sqrt(z))) + 2.0;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((sqrt((z + 1.0d0)) - sqrt(x)) - (sqrt(y) + sqrt(z))) + 2.0d0
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return ((Math.sqrt((z + 1.0)) - Math.sqrt(x)) - (Math.sqrt(y) + Math.sqrt(z))) + 2.0;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return ((math.sqrt((z + 1.0)) - math.sqrt(x)) - (math.sqrt(y) + math.sqrt(z))) + 2.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(Float64(sqrt(Float64(z + 1.0)) - sqrt(x)) - Float64(sqrt(y) + sqrt(z))) + 2.0)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = ((sqrt((z + 1.0)) - sqrt(x)) - (sqrt(y) + sqrt(z))) + 2.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]

Derivation

1. Initial program 90.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   3. lower--.f64 N/A

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   7. lower-+.f64 N/A

      \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   9. lower-+.f64 N/A

      \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   10. lower-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   13. lower-+.f64 N/A

       \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

5. Applied rewrites 33.0%

   \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \left(2 + \left(\sqrt{1 + t} + \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 28.2%

   \[\leadsto 2 + \color{blue}{\left(\left(\sqrt{1 + t} + \sqrt{1 + z}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

9. Taylor expanded in t around inf

   \[\leadsto 2 + \left(\sqrt{1 + z} - \left(\sqrt{x} + \color{blue}{\left(\sqrt{y} + \sqrt{z}\right)}\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 20.4%

    \[\leadsto 2 + \left(\left(\sqrt{1 + z} - \sqrt{x}\right) - \left(\sqrt{z} + \color{blue}{\sqrt{y}}\right)\right) \]

12. Final simplification 20.4%

    \[\leadsto \left(\left(\sqrt{z + 1} - \sqrt{x}\right) - \left(\sqrt{y} + \sqrt{z}\right)\right) + 2 \]
13. Add Preprocessing

Alternative 24: 3.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \left(\sqrt{\frac{1}{y}} \cdot 0\right) \cdot y \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (* (* (sqrt (/ 1.0 y)) 0.0) y))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return (sqrt((1.0 / y)) * 0.0) * y;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (sqrt((1.0d0 / y)) * 0.0d0) * y
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return (Math.sqrt((1.0 / y)) * 0.0) * y;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return (math.sqrt((1.0 / y)) * 0.0) * y

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(Float64(sqrt(Float64(1.0 / y)) * 0.0) * y)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (sqrt((1.0 / y)) * 0.0) * y;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] * 0.0), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 90.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. lift--.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. sub-neg N/A

      \[\leadsto \left(\left(\color{blue}{\left(\sqrt{y + 1} + \left(\mathsf{neg}\left(\sqrt{y}\right)\right)\right)} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. associate-+l+ N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{y + 1} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\sqrt{y + 1}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. rem-square-sqrt N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{\sqrt{y + 1} \cdot \sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. sqrt-prod N/A

       \[\leadsto \left(\left(\color{blue}{\sqrt{\sqrt{y + 1}} \cdot \sqrt{\sqrt{y + 1}}} + \left(\left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\sqrt{\sqrt{y + 1}}}, \sqrt{\sqrt{y + 1}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   13. lower-sqrt.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \color{blue}{\sqrt{\sqrt{y + 1}}}, \left(\mathsf{neg}\left(\sqrt{y}\right)\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   14. neg-mul-1 N/A

       \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{-1 \cdot \sqrt{y}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   15. metadata-eval N/A

       \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \sqrt{y} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(1\right), \sqrt{y}, \sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

4. Applied rewrites 56.9%

   \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{y + 1}}, \sqrt{\sqrt{y + 1}}, \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + x} - \sqrt{x}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
6. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   2. associate-+r+ N/A

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)} - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\left(1 + \sqrt{1 + y}\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right)} - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} + 1\right)} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\left(\sqrt{1 + y} + 1\right)} + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\left(\color{blue}{\sqrt{1 + y}} + 1\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{\color{blue}{1 + y}} + 1\right) + \left(\sqrt{1 + z} + -1 \cdot \sqrt{y}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. +-commutative N/A

      \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \color{blue}{\left(-1 \cdot \sqrt{y} + \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \color{blue}{\mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)}\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \color{blue}{\sqrt{y}}, \sqrt{1 + z}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \color{blue}{\sqrt{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   12. lower-+.f64 N/A

       \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{\color{blue}{1 + z}}\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   13. +-commutative N/A

       \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{z} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \color{blue}{\left(\sqrt{z} + \sqrt{x}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   15. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \left(\color{blue}{\sqrt{z}} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   16. lower-sqrt.f64 22.8

       \[\leadsto \left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \left(\sqrt{z} + \color{blue}{\sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

7. Applied rewrites 22.8%

   \[\leadsto \color{blue}{\left(\left(\left(\sqrt{1 + y} + 1\right) + \mathsf{fma}\left(-1, \sqrt{y}, \sqrt{1 + z}\right)\right) - \left(\sqrt{z} + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

8. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\sqrt{\frac{1}{y}} + -1 \cdot \sqrt{\frac{1}{y}}\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{y}} + -1 \cdot \sqrt{\frac{1}{y}}\right) \cdot y} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{y}} + -1 \cdot \sqrt{\frac{1}{y}}\right) \cdot y} \]

   3. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(\left(-1 + 1\right) \cdot \sqrt{\frac{1}{y}}\right)} \cdot y \]

   4. metadata-eval N/A

      \[\leadsto \left(\color{blue}{0} \cdot \sqrt{\frac{1}{y}}\right) \cdot y \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(0 \cdot \sqrt{\frac{1}{y}}\right)} \cdot y \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(0 \cdot \color{blue}{\sqrt{\frac{1}{y}}}\right) \cdot y \]

   7. lower-/.f64 3.1

      \[\leadsto \left(0 \cdot \sqrt{\color{blue}{\frac{1}{y}}}\right) \cdot y \]

10. Applied rewrites 3.1%

    \[\leadsto \color{blue}{\left(0 \cdot \sqrt{\frac{1}{y}}\right) \cdot y} \]

11. Final simplification 3.1%

    \[\leadsto \left(\sqrt{\frac{1}{y}} \cdot 0\right) \cdot y \]
12. Add Preprocessing

Alternative 25: 1.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ -\sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return -sqrt(x);
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -sqrt(x)
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return -Math.sqrt(x);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return -math.sqrt(x)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(-sqrt(x))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = -sqrt(x);
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := (-N[Sqrt[x], $MachinePrecision])

Derivation

1. Initial program 90.7%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + \left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{1 + \left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   3. lower--.f64 N/A

      \[\leadsto 1 + \color{blue}{\left(\left(\sqrt{1 + t} + \left(\sqrt{1 + y} + \sqrt{1 + z}\right)\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto 1 + \left(\color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + t}\right)} - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   7. lower-+.f64 N/A

      \[\leadsto 1 + \left(\left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto 1 + \left(\left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   9. lower-+.f64 N/A

      \[\leadsto 1 + \left(\left(\left(\sqrt{\color{blue}{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   10. lower-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{\color{blue}{1 + y}}\right) + \sqrt{1 + t}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \color{blue}{\sqrt{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

   13. lower-+.f64 N/A

       \[\leadsto 1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{\color{blue}{1 + t}}\right) - \left(\sqrt{t} + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\right) \]

5. Applied rewrites 33.0%

   \[\leadsto \color{blue}{1 + \left(\left(\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \sqrt{1 + t}\right) - \left(\left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right) + \sqrt{t}\right)\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto -1 \cdot \color{blue}{\sqrt{x}} \]
7. Step-by-step derivation

8. Applied rewrites 1.6%

   \[\leadsto -\sqrt{x} \]
9. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))
    (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z))))
  (- (sqrt (+ t 1.0)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (1.0d0 / (sqrt((y + 1.0d0)) + sqrt(y)))) + (1.0d0 / (sqrt((z + 1.0d0)) + sqrt(z)))) + (sqrt((t + 1.0d0)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (1.0 / (Math.sqrt((y + 1.0)) + Math.sqrt(y)))) + (1.0 / (Math.sqrt((z + 1.0)) + Math.sqrt(z)))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (1.0 / (math.sqrt((y + 1.0)) + math.sqrt(y)))) + (1.0 / (math.sqrt((z + 1.0)) + math.sqrt(z)))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(1.0 / Float64(sqrt(Float64(y + 1.0)) + sqrt(y)))) + Float64(1.0 / Float64(sqrt(Float64(z + 1.0)) + sqrt(z)))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (1.0 / (sqrt((y + 1.0)) + sqrt(y)))) + (1.0 / (sqrt((z + 1.0)) + sqrt(z)))) + (sqrt((t + 1.0)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024298 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ (/ 1 (+ (sqrt (+ x 1)) (sqrt x))) (/ 1 (+ (sqrt (+ y 1)) (sqrt y)))) (/ 1 (+ (sqrt (+ z 1)) (sqrt z)))) (- (sqrt (+ t 1)) (sqrt t))))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))