Main:z from

Percentage Accurate: 91.9% → 99.3%

Time: 24.8s

Alternatives: 23

Speedup: 1.1×

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: 91.9% 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, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := t\_1 + \sqrt{x}\\ t_3 := \frac{1}{t\_2}\\ t_4 := \sqrt{1 + z} - \sqrt{z}\\ t_5 := \sqrt{y + 1}\\ t_6 := \sqrt{y} + t\_5\\ t_7 := \sqrt{t + 1}\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\left(t\_7 - \sqrt{t}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \mathsf{fma}\left(\frac{\sqrt{x}}{t\_6}, t\_3, \mathsf{fma}\left(\frac{\sqrt{y}}{t\_2}, \frac{1}{t\_6}, \left(t\_1 + t\_5\right) \cdot \frac{t\_3}{t\_6}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t + 1\right) - t}{\sqrt{t} + t\_7} + \left(\left(\left(t\_5 - \sqrt{y}\right) + \left(t\_1 - \sqrt{x}\right)\right) + t\_4\right)\\ \end{array} \end{array} \]

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 (+ x 1.0)))
        (t_2 (+ t_1 (sqrt x)))
        (t_3 (/ 1.0 t_2))
        (t_4 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_5 (sqrt (+ y 1.0)))
        (t_6 (+ (sqrt y) t_5))
        (t_7 (sqrt (+ t 1.0))))
   (if (<= t_4 5e-6)
     (+
      (- t_7 (sqrt t))
      (fma
       (sqrt (/ 1.0 z))
       0.5
       (fma
        (/ (sqrt x) t_6)
        t_3
        (fma (/ (sqrt y) t_2) (/ 1.0 t_6) (* (+ t_1 t_5) (/ t_3 t_6))))))
     (+
      (/ (- (+ t 1.0) t) (+ (sqrt t) t_7))
      (+ (+ (- t_5 (sqrt y)) (- t_1 (sqrt x))) t_4)))))

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 = t_1 + sqrt(x);
	double t_3 = 1.0 / t_2;
	double t_4 = sqrt((1.0 + z)) - sqrt(z);
	double t_5 = sqrt((y + 1.0));
	double t_6 = sqrt(y) + t_5;
	double t_7 = sqrt((t + 1.0));
	double tmp;
	if (t_4 <= 5e-6) {
		tmp = (t_7 - sqrt(t)) + fma(sqrt((1.0 / z)), 0.5, fma((sqrt(x) / t_6), t_3, fma((sqrt(y) / t_2), (1.0 / t_6), ((t_1 + t_5) * (t_3 / t_6)))));
	} else {
		tmp = (((t + 1.0) - t) / (sqrt(t) + t_7)) + (((t_5 - sqrt(y)) + (t_1 - sqrt(x))) + t_4);
	}
	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(t_1 + sqrt(x))
	t_3 = Float64(1.0 / t_2)
	t_4 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_5 = sqrt(Float64(y + 1.0))
	t_6 = Float64(sqrt(y) + t_5)
	t_7 = sqrt(Float64(t + 1.0))
	tmp = 0.0
	if (t_4 <= 5e-6)
		tmp = Float64(Float64(t_7 - sqrt(t)) + fma(sqrt(Float64(1.0 / z)), 0.5, fma(Float64(sqrt(x) / t_6), t_3, fma(Float64(sqrt(y) / t_2), Float64(1.0 / t_6), Float64(Float64(t_1 + t_5) * Float64(t_3 / t_6))))));
	else
		tmp = Float64(Float64(Float64(Float64(t + 1.0) - t) / Float64(sqrt(t) + t_7)) + Float64(Float64(Float64(t_5 - sqrt(y)) + Float64(t_1 - sqrt(x))) + t_4));
	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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[y], $MachinePrecision] + t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 5e-6], N[(N[(t$95$7 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(N[(N[Sqrt[x], $MachinePrecision] / t$95$6), $MachinePrecision] * t$95$3 + N[(N[(N[Sqrt[y], $MachinePrecision] / t$95$2), $MachinePrecision] * N[(1.0 / t$95$6), $MachinePrecision] + N[(N[(t$95$1 + t$95$5), $MachinePrecision] * N[(t$95$3 / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t + 1.0), $MachinePrecision] - t), $MachinePrecision] / N[(N[Sqrt[t], $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$5 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $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 84.3%

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

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 68.1

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

   7. Applied rewrites 68.1%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 94.2%

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

   if 5.00000000000000041e-6 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   1. Initial program 97.4%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. 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 97.5

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

   4. Applied rewrites 97.5%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\left(\sqrt{t + 1} - \sqrt{t}\right) + \mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \mathsf{fma}\left(\frac{\sqrt{x}}{\sqrt{y} + \sqrt{y + 1}}, \frac{1}{\sqrt{x + 1} + \sqrt{x}}, \mathsf{fma}\left(\frac{\sqrt{y}}{\sqrt{x + 1} + \sqrt{x}}, \frac{1}{\sqrt{y} + \sqrt{y + 1}}, \left(\sqrt{x + 1} + \sqrt{y + 1}\right) \cdot \frac{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}{\sqrt{y} + \sqrt{y + 1}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}} + \left(\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := t\_1 - \sqrt{z}\\ t_3 := \sqrt{y + 1}\\ t_4 := \sqrt{x + 1}\\ t_5 := \left(\left(t\_3 - \sqrt{y}\right) + \left(t\_4 - \sqrt{x}\right)\right) + t\_2\\ t_6 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;t\_5 \leq 0.0001:\\ \;\;\;\;\left(\sqrt{\frac{1}{x}} \cdot 0.5 + t\_2\right) + t\_6\\ \mathbf{elif}\;t\_5 \leq 1.001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, t\_4\right) - \sqrt{x}\\ \mathbf{elif}\;t\_5 \leq 2.00005:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{z}} + \left(t\_4 + t\_3\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, t\_1\right) + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + t\_6\\ \end{array} \end{array} \]

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 (+ 1.0 z)))
        (t_2 (- t_1 (sqrt z)))
        (t_3 (sqrt (+ y 1.0)))
        (t_4 (sqrt (+ x 1.0)))
        (t_5 (+ (+ (- t_3 (sqrt y)) (- t_4 (sqrt x))) t_2))
        (t_6 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= t_5 0.0001)
     (+ (+ (* (sqrt (/ 1.0 x)) 0.5) t_2) t_6)
     (if (<= t_5 1.001)
       (- (fma 0.5 (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z))) t_4) (sqrt x))
       (if (<= t_5 2.00005)
         (- (+ (/ 0.5 (sqrt z)) (+ t_4 t_3)) (+ (sqrt y) (sqrt x)))
         (+
          (- (+ (fma 0.5 x t_1) 2.0) (+ (+ (sqrt y) (sqrt z)) (sqrt x)))
          t_6))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = t_1 - sqrt(z);
	double t_3 = sqrt((y + 1.0));
	double t_4 = sqrt((x + 1.0));
	double t_5 = ((t_3 - sqrt(y)) + (t_4 - sqrt(x))) + t_2;
	double t_6 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if (t_5 <= 0.0001) {
		tmp = ((sqrt((1.0 / x)) * 0.5) + t_2) + t_6;
	} else if (t_5 <= 1.001) {
		tmp = fma(0.5, (sqrt((1.0 / y)) + sqrt((1.0 / z))), t_4) - sqrt(x);
	} else if (t_5 <= 2.00005) {
		tmp = ((0.5 / sqrt(z)) + (t_4 + t_3)) - (sqrt(y) + sqrt(x));
	} else {
		tmp = ((fma(0.5, x, t_1) + 2.0) - ((sqrt(y) + sqrt(z)) + sqrt(x))) + t_6;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = Float64(t_1 - sqrt(z))
	t_3 = sqrt(Float64(y + 1.0))
	t_4 = sqrt(Float64(x + 1.0))
	t_5 = Float64(Float64(Float64(t_3 - sqrt(y)) + Float64(t_4 - sqrt(x))) + t_2)
	t_6 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (t_5 <= 0.0001)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / x)) * 0.5) + t_2) + t_6);
	elseif (t_5 <= 1.001)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z))), t_4) - sqrt(x));
	elseif (t_5 <= 2.00005)
		tmp = Float64(Float64(Float64(0.5 / sqrt(z)) + Float64(t_4 + t_3)) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(fma(0.5, x, t_1) + 2.0) - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))) + t_6);
	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[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.0001], N[(N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$6), $MachinePrecision], If[LessEqual[t$95$5, 1.001], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.00005], N[(N[(N[(0.5 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 + t$95$3), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * x + t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$6), $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))) < 1.00000000000000005e-4

   1. Initial program 53.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 57.0%

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

   if 1.00000000000000005e-4 < (+.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))) < 1.0009999999999999

   1. Initial program 96.2%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 3.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 22.3%

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

   if 1.0009999999999999 < (+.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))) < 2.0000499999999999

   1. Initial program 97.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 22.7%

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

   10. Applied rewrites 22.7%

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

   if 2.0000499999999999 < (+.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 98.9%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.7%

      \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + 2\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 0.0001:\\ \;\;\;\;\left(\sqrt{\frac{1}{x}} \cdot 0.5 + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 1.001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{x + 1}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2.00005:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{z}} + \left(\sqrt{x + 1} + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{1 + z}\right) + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{x + 1}\\ t_3 := \sqrt{1 + z}\\ t_4 := t\_3 - \sqrt{z}\\ t_5 := \left(\left(t\_1 - \sqrt{y}\right) + \left(t\_2 - \sqrt{x}\right)\right) + t\_4\\ t_6 := t\_2 + t\_1\\ \mathbf{if}\;t\_5 \leq 0.0001:\\ \;\;\;\;\left(\sqrt{\frac{1}{x}} \cdot 0.5 + t\_4\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;t\_5 \leq 1.001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, t\_2\right) - \sqrt{x}\\ \mathbf{elif}\;t\_5 \leq 2.00005:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{z}} + t\_6\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_6 - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + t\_3\\ \end{array} \end{array} \]

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 (+ y 1.0)))
        (t_2 (sqrt (+ x 1.0)))
        (t_3 (sqrt (+ 1.0 z)))
        (t_4 (- t_3 (sqrt z)))
        (t_5 (+ (+ (- t_1 (sqrt y)) (- t_2 (sqrt x))) t_4))
        (t_6 (+ t_2 t_1)))
   (if (<= t_5 0.0001)
     (+ (+ (* (sqrt (/ 1.0 x)) 0.5) t_4) (- (sqrt (+ t 1.0)) (sqrt t)))
     (if (<= t_5 1.001)
       (- (fma 0.5 (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z))) t_2) (sqrt x))
       (if (<= t_5 2.00005)
         (- (+ (/ 0.5 (sqrt z)) t_6) (+ (sqrt y) (sqrt x)))
         (+ (- t_6 (+ (+ (sqrt y) (sqrt z)) (sqrt x))) t_3))))))

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((1.0 + z));
	double t_4 = t_3 - sqrt(z);
	double t_5 = ((t_1 - sqrt(y)) + (t_2 - sqrt(x))) + t_4;
	double t_6 = t_2 + t_1;
	double tmp;
	if (t_5 <= 0.0001) {
		tmp = ((sqrt((1.0 / x)) * 0.5) + t_4) + (sqrt((t + 1.0)) - sqrt(t));
	} else if (t_5 <= 1.001) {
		tmp = fma(0.5, (sqrt((1.0 / y)) + sqrt((1.0 / z))), t_2) - sqrt(x);
	} else if (t_5 <= 2.00005) {
		tmp = ((0.5 / sqrt(z)) + t_6) - (sqrt(y) + sqrt(x));
	} else {
		tmp = (t_6 - ((sqrt(y) + sqrt(z)) + sqrt(x))) + t_3;
	}
	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(1.0 + z))
	t_4 = Float64(t_3 - sqrt(z))
	t_5 = Float64(Float64(Float64(t_1 - sqrt(y)) + Float64(t_2 - sqrt(x))) + t_4)
	t_6 = Float64(t_2 + t_1)
	tmp = 0.0
	if (t_5 <= 0.0001)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / x)) * 0.5) + t_4) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)));
	elseif (t_5 <= 1.001)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z))), t_2) - sqrt(x));
	elseif (t_5 <= 2.00005)
		tmp = Float64(Float64(Float64(0.5 / sqrt(z)) + t_6) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(t_6 - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))) + t_3);
	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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$5, 0.0001], N[(N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] + t$95$4), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 1.001], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.00005], N[(N[(N[(0.5 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$6 - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $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))) < 1.00000000000000005e-4

   1. Initial program 53.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 57.0%

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

   if 1.00000000000000005e-4 < (+.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))) < 1.0009999999999999

   1. Initial program 96.2%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 3.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 22.3%

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

   if 1.0009999999999999 < (+.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))) < 2.0000499999999999

   1. Initial program 97.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 22.7%

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

   10. Applied rewrites 22.7%

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

   if 2.0000499999999999 < (+.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 98.9%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 58.5%

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

   7. Applied rewrites 58.6%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 0.0001:\\ \;\;\;\;\left(\sqrt{\frac{1}{x}} \cdot 0.5 + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 1.001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{x + 1}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2.00005:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{z}} + \left(\sqrt{x + 1} + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x + 1} + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + \sqrt{1 + z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{y + 1}\\ t_4 := \sqrt{x + 1}\\ t_5 := \left(\left(t\_3 - \sqrt{y}\right) + \left(t\_4 - \sqrt{x}\right)\right) + \left(t\_2 - \sqrt{z}\right)\\ \mathbf{if}\;t\_5 \leq 10^{-5}:\\ \;\;\;\;\left(\frac{1}{t\_4 + \sqrt{x}} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + t\_1\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{y} + t\_3} + t\_4\right) - \sqrt{x}\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, t\_2\right) + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + t\_1\\ \end{array} \end{array} \]

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 1.0)) (sqrt t)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (sqrt (+ y 1.0)))
        (t_4 (sqrt (+ x 1.0)))
        (t_5 (+ (+ (- t_3 (sqrt y)) (- t_4 (sqrt x))) (- t_2 (sqrt z)))))
   (if (<= t_5 1e-5)
     (+ (+ (/ 1.0 (+ t_4 (sqrt x))) (* 0.5 (sqrt (/ 1.0 z)))) t_1)
     (if (<= t_5 2.0)
       (+ (- (+ (/ 1.0 (+ (sqrt y) t_3)) t_4) (sqrt x)) t_1)
       (+
        (- (+ (fma 0.5 x t_2) 2.0) (+ (+ (sqrt y) (sqrt z)) (sqrt x)))
        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 + 1.0)) - sqrt(t);
	double t_2 = sqrt((1.0 + z));
	double t_3 = sqrt((y + 1.0));
	double t_4 = sqrt((x + 1.0));
	double t_5 = ((t_3 - sqrt(y)) + (t_4 - sqrt(x))) + (t_2 - sqrt(z));
	double tmp;
	if (t_5 <= 1e-5) {
		tmp = ((1.0 / (t_4 + sqrt(x))) + (0.5 * sqrt((1.0 / z)))) + t_1;
	} else if (t_5 <= 2.0) {
		tmp = (((1.0 / (sqrt(y) + t_3)) + t_4) - sqrt(x)) + t_1;
	} else {
		tmp = ((fma(0.5, x, t_2) + 2.0) - ((sqrt(y) + sqrt(z)) + sqrt(x))) + t_1;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_2 = sqrt(Float64(1.0 + z))
	t_3 = sqrt(Float64(y + 1.0))
	t_4 = sqrt(Float64(x + 1.0))
	t_5 = Float64(Float64(Float64(t_3 - sqrt(y)) + Float64(t_4 - sqrt(x))) + Float64(t_2 - sqrt(z)))
	tmp = 0.0
	if (t_5 <= 1e-5)
		tmp = Float64(Float64(Float64(1.0 / Float64(t_4 + sqrt(x))) + Float64(0.5 * sqrt(Float64(1.0 / z)))) + t_1);
	elseif (t_5 <= 2.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_3)) + t_4) - sqrt(x)) + t_1);
	else
		tmp = Float64(Float64(Float64(fma(0.5, x, t_2) + 2.0) - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))) + t_1);
	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[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 1e-5], N[(N[(N[(1.0 / N[(t$95$4 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(N[(0.5 * x + t$95$2), $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 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))) < 1.00000000000000008e-5

   1. Initial program 51.9%

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

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 53.4%

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

   5. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 60.2

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

   7. Applied rewrites 60.2%

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

   8. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 59.1

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

   10. Applied rewrites 59.1%

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

   11. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 73.3

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

   13. Applied rewrites 73.3%

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

   if 1.00000000000000008e-5 < (+.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 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

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 40.8

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

   7. Applied rewrites 40.8%

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

   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 97.8%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 90.3%

      \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + 2\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 10^{-5}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \sqrt{x + 1}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{1 + z}\right) + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := t\_1 - \sqrt{z}\\ t_3 := \sqrt{y + 1}\\ t_4 := \sqrt{x + 1}\\ t_5 := \left(\left(t\_3 - \sqrt{y}\right) + \left(t\_4 - \sqrt{x}\right)\right) + t\_2\\ t_6 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;t\_5 \leq 0.0001:\\ \;\;\;\;\left(\sqrt{\frac{1}{x}} \cdot 0.5 + t\_2\right) + t\_6\\ \mathbf{elif}\;t\_5 \leq 2:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{y} + t\_3} + t\_4\right) - \sqrt{x}\right) + t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, t\_1\right) + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + t\_6\\ \end{array} \end{array} \]

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 (+ 1.0 z)))
        (t_2 (- t_1 (sqrt z)))
        (t_3 (sqrt (+ y 1.0)))
        (t_4 (sqrt (+ x 1.0)))
        (t_5 (+ (+ (- t_3 (sqrt y)) (- t_4 (sqrt x))) t_2))
        (t_6 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= t_5 0.0001)
     (+ (+ (* (sqrt (/ 1.0 x)) 0.5) t_2) t_6)
     (if (<= t_5 2.0)
       (+ (- (+ (/ 1.0 (+ (sqrt y) t_3)) t_4) (sqrt x)) t_6)
       (+
        (- (+ (fma 0.5 x t_1) 2.0) (+ (+ (sqrt y) (sqrt z)) (sqrt x)))
        t_6)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = t_1 - sqrt(z);
	double t_3 = sqrt((y + 1.0));
	double t_4 = sqrt((x + 1.0));
	double t_5 = ((t_3 - sqrt(y)) + (t_4 - sqrt(x))) + t_2;
	double t_6 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if (t_5 <= 0.0001) {
		tmp = ((sqrt((1.0 / x)) * 0.5) + t_2) + t_6;
	} else if (t_5 <= 2.0) {
		tmp = (((1.0 / (sqrt(y) + t_3)) + t_4) - sqrt(x)) + t_6;
	} else {
		tmp = ((fma(0.5, x, t_1) + 2.0) - ((sqrt(y) + sqrt(z)) + sqrt(x))) + t_6;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = Float64(t_1 - sqrt(z))
	t_3 = sqrt(Float64(y + 1.0))
	t_4 = sqrt(Float64(x + 1.0))
	t_5 = Float64(Float64(Float64(t_3 - sqrt(y)) + Float64(t_4 - sqrt(x))) + t_2)
	t_6 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (t_5 <= 0.0001)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / x)) * 0.5) + t_2) + t_6);
	elseif (t_5 <= 2.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_3)) + t_4) - sqrt(x)) + t_6);
	else
		tmp = Float64(Float64(Float64(fma(0.5, x, t_1) + 2.0) - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))) + t_6);
	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[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.0001], N[(N[(N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$6), $MachinePrecision], If[LessEqual[t$95$5, 2.0], N[(N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision], N[(N[(N[(N[(0.5 * x + t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 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))) < 1.00000000000000005e-4

   1. Initial program 53.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 57.0%

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

   if 1.00000000000000005e-4 < (+.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 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-sqrt.f64 41.0

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

   7. Applied rewrites 41.0%

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

   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 97.8%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 90.3%

      \[\leadsto \left(\left(\mathsf{fma}\left(0.5, x, \sqrt{z + 1}\right) + 2\right) - \left(\color{blue}{\left(\sqrt{z} + \sqrt{y}\right)} + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 0.0001:\\ \;\;\;\;\left(\sqrt{\frac{1}{x}} \cdot 0.5 + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{elif}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2:\\ \;\;\;\;\left(\left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \sqrt{x + 1}\right) - \sqrt{x}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.5, x, \sqrt{1 + z}\right) + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{x + 1}\\ t_4 := \left(\left(t\_2 - \sqrt{y}\right) + \left(t\_3 - \sqrt{x}\right)\right) + \left(t\_1 - \sqrt{z}\right)\\ t_5 := t\_3 + t\_2\\ \mathbf{if}\;t\_4 \leq 1.001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, t\_3\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2.00005:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{z}} + t\_5\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_5 - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + t\_1\\ \end{array} \end{array} \]

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 (+ 1.0 z)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (+ (+ (- t_2 (sqrt y)) (- t_3 (sqrt x))) (- t_1 (sqrt z))))
        (t_5 (+ t_3 t_2)))
   (if (<= t_4 1.001)
     (- (fma 0.5 (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z))) t_3) (sqrt x))
     (if (<= t_4 2.00005)
       (- (+ (/ 0.5 (sqrt z)) t_5) (+ (sqrt y) (sqrt x)))
       (+ (- t_5 (+ (+ (sqrt y) (sqrt z)) (sqrt x))) t_1)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((x + 1.0));
	double t_4 = ((t_2 - sqrt(y)) + (t_3 - sqrt(x))) + (t_1 - sqrt(z));
	double t_5 = t_3 + t_2;
	double tmp;
	if (t_4 <= 1.001) {
		tmp = fma(0.5, (sqrt((1.0 / y)) + sqrt((1.0 / z))), t_3) - sqrt(x);
	} else if (t_4 <= 2.00005) {
		tmp = ((0.5 / sqrt(z)) + t_5) - (sqrt(y) + sqrt(x));
	} else {
		tmp = (t_5 - ((sqrt(y) + sqrt(z)) + sqrt(x))) + t_1;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(Float64(Float64(t_2 - sqrt(y)) + Float64(t_3 - sqrt(x))) + Float64(t_1 - sqrt(z)))
	t_5 = Float64(t_3 + t_2)
	tmp = 0.0
	if (t_4 <= 1.001)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z))), t_3) - sqrt(x));
	elseif (t_4 <= 2.00005)
		tmp = Float64(Float64(Float64(0.5 / sqrt(z)) + t_5) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(t_5 - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))) + t_1);
	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[(1.0 + z), $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[(N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 + t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, 1.001], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.00005], N[(N[(N[(0.5 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$5 - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 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))) < 1.0009999999999999

   1. Initial program 83.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 16.6%

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

   if 1.0009999999999999 < (+.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))) < 2.0000499999999999

   1. Initial program 97.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 22.7%

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

   10. Applied rewrites 22.7%

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

   if 2.0000499999999999 < (+.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 98.9%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 58.5%

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

   7. Applied rewrites 58.6%

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

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 1.001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{x + 1}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2.00005:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{z}} + \left(\sqrt{x + 1} + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{x + 1} + \sqrt{y + 1}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + \sqrt{1 + z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{x + 1}\\ t_4 := \left(\left(t\_2 - \sqrt{y}\right) + \left(t\_3 - \sqrt{x}\right)\right) + \left(t\_1 - \sqrt{z}\right)\\ \mathbf{if}\;t\_4 \leq 1.001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, t\_3\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2.00005:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{z}} + \left(t\_3 + t\_2\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_2 + t\_1\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

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 (+ 1.0 z)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (+ (+ (- t_2 (sqrt y)) (- t_3 (sqrt x))) (- t_1 (sqrt z)))))
   (if (<= t_4 1.001)
     (- (fma 0.5 (+ (sqrt (/ 1.0 y)) (sqrt (/ 1.0 z))) t_3) (sqrt x))
     (if (<= t_4 2.00005)
       (- (+ (/ 0.5 (sqrt z)) (+ t_3 t_2)) (+ (sqrt y) (sqrt x)))
       (+ (- (+ t_2 t_1) (+ (+ (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((1.0 + z));
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((x + 1.0));
	double t_4 = ((t_2 - sqrt(y)) + (t_3 - sqrt(x))) + (t_1 - sqrt(z));
	double tmp;
	if (t_4 <= 1.001) {
		tmp = fma(0.5, (sqrt((1.0 / y)) + sqrt((1.0 / z))), t_3) - sqrt(x);
	} else if (t_4 <= 2.00005) {
		tmp = ((0.5 / sqrt(z)) + (t_3 + t_2)) - (sqrt(y) + sqrt(x));
	} else {
		tmp = ((t_2 + t_1) - ((sqrt(y) + sqrt(z)) + 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(1.0 + z))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(Float64(Float64(t_2 - sqrt(y)) + Float64(t_3 - sqrt(x))) + Float64(t_1 - sqrt(z)))
	tmp = 0.0
	if (t_4 <= 1.001)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / y)) + sqrt(Float64(1.0 / z))), t_3) - sqrt(x));
	elseif (t_4 <= 2.00005)
		tmp = Float64(Float64(Float64(0.5 / sqrt(z)) + Float64(t_3 + t_2)) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(t_2 + t_1) - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))) + 1.0);
	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[(1.0 + z), $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[(N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.001], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.00005], N[(N[(N[(0.5 / N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + t$95$2), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 + t$95$1), $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 3 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))) < 1.0009999999999999

   1. Initial program 83.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 16.6%

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

   if 1.0009999999999999 < (+.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))) < 2.0000499999999999

   1. Initial program 97.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 22.7%

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

   10. Applied rewrites 22.7%

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

   if 2.0000499999999999 < (+.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 98.9%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 54.6%

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

4. Final simplification 25.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 1.001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{x + 1}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2.00005:\\ \;\;\;\;\left(\frac{0.5}{\sqrt{z}} + \left(\sqrt{x + 1} + \sqrt{y + 1}\right)\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + \sqrt{1 + z}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{x + 1}\\ t_4 := \left(\left(t\_2 - \sqrt{y}\right) + \left(t\_3 - \sqrt{x}\right)\right) + \left(t\_1 - \sqrt{z}\right)\\ t_5 := \sqrt{\frac{1}{z}}\\ \mathbf{if}\;t\_4 \leq 1.001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + t\_5, t\_3\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2.00005:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_5, 0.5, t\_2\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_2 + t\_1\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

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 (+ 1.0 z)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (+ (+ (- t_2 (sqrt y)) (- t_3 (sqrt x))) (- t_1 (sqrt z))))
        (t_5 (sqrt (/ 1.0 z))))
   (if (<= t_4 1.001)
     (- (fma 0.5 (+ (sqrt (/ 1.0 y)) t_5) t_3) (sqrt x))
     (if (<= t_4 2.00005)
       (+ (- (fma t_5 0.5 t_2) (+ (sqrt y) (sqrt x))) 1.0)
       (+ (- (+ t_2 t_1) (+ (+ (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((1.0 + z));
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((x + 1.0));
	double t_4 = ((t_2 - sqrt(y)) + (t_3 - sqrt(x))) + (t_1 - sqrt(z));
	double t_5 = sqrt((1.0 / z));
	double tmp;
	if (t_4 <= 1.001) {
		tmp = fma(0.5, (sqrt((1.0 / y)) + t_5), t_3) - sqrt(x);
	} else if (t_4 <= 2.00005) {
		tmp = (fma(t_5, 0.5, t_2) - (sqrt(y) + sqrt(x))) + 1.0;
	} else {
		tmp = ((t_2 + t_1) - ((sqrt(y) + sqrt(z)) + 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(1.0 + z))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(Float64(Float64(t_2 - sqrt(y)) + Float64(t_3 - sqrt(x))) + Float64(t_1 - sqrt(z)))
	t_5 = sqrt(Float64(1.0 / z))
	tmp = 0.0
	if (t_4 <= 1.001)
		tmp = Float64(fma(0.5, Float64(sqrt(Float64(1.0 / y)) + t_5), t_3) - sqrt(x));
	elseif (t_4 <= 2.00005)
		tmp = Float64(Float64(fma(t_5, 0.5, t_2) - Float64(sqrt(y) + sqrt(x))) + 1.0);
	else
		tmp = Float64(Float64(Float64(t_2 + t_1) - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))) + 1.0);
	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[(1.0 + z), $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[(N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 1.001], N[(N[(0.5 * N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision] + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.00005], N[(N[(N[(t$95$5 * 0.5 + t$95$2), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(t$95$2 + t$95$1), $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 3 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))) < 1.0009999999999999

   1. Initial program 83.3%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 3.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 16.6%

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

   if 1.0009999999999999 < (+.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))) < 2.0000499999999999

   1. Initial program 97.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 4.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 22.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 25.0%

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

   if 2.0000499999999999 < (+.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 98.9%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 54.6%

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

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 1.001:\\ \;\;\;\;\mathsf{fma}\left(0.5, \sqrt{\frac{1}{y}} + \sqrt{\frac{1}{z}}, \sqrt{x + 1}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2.00005:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + \sqrt{1 + z}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{\frac{1}{z}}\\ t_4 := \sqrt{x + 1}\\ t_5 := \left(\left(t\_2 - \sqrt{y}\right) + \left(t\_4 - \sqrt{x}\right)\right) + \left(t\_1 - \sqrt{z}\right)\\ \mathbf{if}\;t\_5 \leq 0.9999999999982103:\\ \;\;\;\;\mathsf{fma}\left(t\_3, 0.5, t\_4\right) - \sqrt{x}\\ \mathbf{elif}\;t\_5 \leq 2.00005:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_3, 0.5, t\_2\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_2 + t\_1\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \end{array} \]

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 (+ 1.0 z)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (/ 1.0 z)))
        (t_4 (sqrt (+ x 1.0)))
        (t_5 (+ (+ (- t_2 (sqrt y)) (- t_4 (sqrt x))) (- t_1 (sqrt z)))))
   (if (<= t_5 0.9999999999982103)
     (- (fma t_3 0.5 t_4) (sqrt x))
     (if (<= t_5 2.00005)
       (+ (- (fma t_3 0.5 t_2) (+ (sqrt y) (sqrt x))) 1.0)
       (+ (- (+ t_2 t_1) (+ (+ (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((1.0 + z));
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((1.0 / z));
	double t_4 = sqrt((x + 1.0));
	double t_5 = ((t_2 - sqrt(y)) + (t_4 - sqrt(x))) + (t_1 - sqrt(z));
	double tmp;
	if (t_5 <= 0.9999999999982103) {
		tmp = fma(t_3, 0.5, t_4) - sqrt(x);
	} else if (t_5 <= 2.00005) {
		tmp = (fma(t_3, 0.5, t_2) - (sqrt(y) + sqrt(x))) + 1.0;
	} else {
		tmp = ((t_2 + t_1) - ((sqrt(y) + sqrt(z)) + 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(1.0 + z))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(1.0 / z))
	t_4 = sqrt(Float64(x + 1.0))
	t_5 = Float64(Float64(Float64(t_2 - sqrt(y)) + Float64(t_4 - sqrt(x))) + Float64(t_1 - sqrt(z)))
	tmp = 0.0
	if (t_5 <= 0.9999999999982103)
		tmp = Float64(fma(t_3, 0.5, t_4) - sqrt(x));
	elseif (t_5 <= 2.00005)
		tmp = Float64(Float64(fma(t_3, 0.5, t_2) - Float64(sqrt(y) + sqrt(x))) + 1.0);
	else
		tmp = Float64(Float64(Float64(t_2 + t_1) - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))) + 1.0);
	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[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.9999999999982103], N[(N[(t$95$3 * 0.5 + t$95$4), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.00005], N[(N[(N[(t$95$3 * 0.5 + t$95$2), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(t$95$2 + t$95$1), $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 3 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))) < 0.99999999999821032

   1. Initial program 59.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 3.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 7.9%

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

   if 0.99999999999821032 < (+.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))) < 2.0000499999999999

   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 14.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 24.0%

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

   if 2.0000499999999999 < (+.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 98.9%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 2.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 54.6%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 0.9999999999982103:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{x + 1}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2.00005:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{y + 1} + \sqrt{1 + z}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{\frac{1}{z}}\\ t_4 := \sqrt{x + 1}\\ t_5 := \left(\left(t\_2 - \sqrt{y}\right) + \left(t\_4 - \sqrt{x}\right)\right) + \left(t\_1 - \sqrt{z}\right)\\ \mathbf{if}\;t\_5 \leq 0.9999999999982103:\\ \;\;\;\;\mathsf{fma}\left(t\_3, 0.5, t\_4\right) - \sqrt{x}\\ \mathbf{elif}\;t\_5 \leq 2.00005:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_3, 0.5, t\_2\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, y, 0.5\right), y, t\_1\right) + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\\ \end{array} \end{array} \]

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 (+ 1.0 z)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (/ 1.0 z)))
        (t_4 (sqrt (+ x 1.0)))
        (t_5 (+ (+ (- t_2 (sqrt y)) (- t_4 (sqrt x))) (- t_1 (sqrt z)))))
   (if (<= t_5 0.9999999999982103)
     (- (fma t_3 0.5 t_4) (sqrt x))
     (if (<= t_5 2.00005)
       (+ (- (fma t_3 0.5 t_2) (+ (sqrt y) (sqrt x))) 1.0)
       (-
        (+ (fma (fma -0.125 y 0.5) y t_1) 2.0)
        (+ (+ (sqrt y) (sqrt z)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((1.0 / z));
	double t_4 = sqrt((x + 1.0));
	double t_5 = ((t_2 - sqrt(y)) + (t_4 - sqrt(x))) + (t_1 - sqrt(z));
	double tmp;
	if (t_5 <= 0.9999999999982103) {
		tmp = fma(t_3, 0.5, t_4) - sqrt(x);
	} else if (t_5 <= 2.00005) {
		tmp = (fma(t_3, 0.5, t_2) - (sqrt(y) + sqrt(x))) + 1.0;
	} else {
		tmp = (fma(fma(-0.125, y, 0.5), y, t_1) + 2.0) - ((sqrt(y) + sqrt(z)) + sqrt(x));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(1.0 / z))
	t_4 = sqrt(Float64(x + 1.0))
	t_5 = Float64(Float64(Float64(t_2 - sqrt(y)) + Float64(t_4 - sqrt(x))) + Float64(t_1 - sqrt(z)))
	tmp = 0.0
	if (t_5 <= 0.9999999999982103)
		tmp = Float64(fma(t_3, 0.5, t_4) - sqrt(x));
	elseif (t_5 <= 2.00005)
		tmp = Float64(Float64(fma(t_3, 0.5, t_2) - Float64(sqrt(y) + sqrt(x))) + 1.0);
	else
		tmp = Float64(Float64(fma(fma(-0.125, y, 0.5), y, t_1) + 2.0) - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x)));
	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[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, 0.9999999999982103], N[(N[(t$95$3 * 0.5 + t$95$4), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2.00005], N[(N[(N[(t$95$3 * 0.5 + t$95$2), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(-0.125 * y + 0.5), $MachinePrecision] * y + t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 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))) < 0.99999999999821032

   1. Initial program 59.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 3.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 7.9%

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

   if 0.99999999999821032 < (+.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))) < 2.0000499999999999

   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 14.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 24.0%

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

   if 2.0000499999999999 < (+.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 98.9%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.6%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + z} + y \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot y\right)\right)\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 53.3%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, y, 0.5\right), y, \sqrt{z + 1}\right) + 2\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 0.9999999999982103:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{x + 1}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2.00005:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\right) + 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, y, 0.5\right), y, \sqrt{1 + z}\right) + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{y + 1}\\ t_3 := \sqrt{x + 1}\\ t_4 := \left(\left(t\_2 - \sqrt{y}\right) + \left(t\_3 - \sqrt{x}\right)\right) + \left(t\_1 - \sqrt{z}\right)\\ \mathbf{if}\;t\_4 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, t\_3\right) - \sqrt{x}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\left(t\_3 + t\_2\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, y, 0.5\right), y, t\_1\right) + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\\ \end{array} \end{array} \]

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 (+ 1.0 z)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (+ (+ (- t_2 (sqrt y)) (- t_3 (sqrt x))) (- t_1 (sqrt z)))))
   (if (<= t_4 1.0)
     (- (fma (sqrt (/ 1.0 z)) 0.5 t_3) (sqrt x))
     (if (<= t_4 2.0)
       (- (+ t_3 t_2) (+ (sqrt y) (sqrt x)))
       (-
        (+ (fma (fma -0.125 y 0.5) y t_1) 2.0)
        (+ (+ (sqrt y) (sqrt z)) (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z));
	double t_2 = sqrt((y + 1.0));
	double t_3 = sqrt((x + 1.0));
	double t_4 = ((t_2 - sqrt(y)) + (t_3 - sqrt(x))) + (t_1 - sqrt(z));
	double tmp;
	if (t_4 <= 1.0) {
		tmp = fma(sqrt((1.0 / z)), 0.5, t_3) - sqrt(x);
	} else if (t_4 <= 2.0) {
		tmp = (t_3 + t_2) - (sqrt(y) + sqrt(x));
	} else {
		tmp = (fma(fma(-0.125, y, 0.5), y, t_1) + 2.0) - ((sqrt(y) + sqrt(z)) + sqrt(x));
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(Float64(Float64(t_2 - sqrt(y)) + Float64(t_3 - sqrt(x))) + Float64(t_1 - sqrt(z)))
	tmp = 0.0
	if (t_4 <= 1.0)
		tmp = Float64(fma(sqrt(Float64(1.0 / z)), 0.5, t_3) - sqrt(x));
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(t_3 + t_2) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(fma(fma(-0.125, y, 0.5), y, t_1) + 2.0) - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x)));
	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[(1.0 + z), $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[(N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0], N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(t$95$3 + t$95$2), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.125 * y + 0.5), $MachinePrecision] * y + t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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))) < 1

   1. Initial program 83.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 3.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 15.7%

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

   if 1 < (+.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 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 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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 4.4%

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

   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 21.4%

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

   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 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.0%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(2 + \left(\sqrt{1 + z} + y \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot y\right)\right)\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 51.4%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, y, 0.5\right), y, \sqrt{z + 1}\right) + 2\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{x + 1}\right) - \sqrt{x}\\ \mathbf{elif}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 2:\\ \;\;\;\;\left(\sqrt{x + 1} + \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, y, 0.5\right), y, \sqrt{1 + z}\right) + 2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.0% accurate, 0.5× speedup

Math

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

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 (+ 1.0 z)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (+ (+ (- t_2 (sqrt y)) (- t_3 (sqrt x))) (- t_1 (sqrt z)))))
   (if (<= t_4 1.0)
     (- (fma (sqrt (/ 1.0 z)) 0.5 t_3) (sqrt x))
     (if (<= t_4 2.0)
       (- (+ t_3 t_2) (+ (sqrt y) (sqrt x)))
       (- (+ (fma 0.5 y t_1) 2.0) (+ (+ (sqrt y) (sqrt z)) (sqrt x)))))))

C

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

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(Float64(Float64(t_2 - sqrt(y)) + Float64(t_3 - sqrt(x))) + Float64(t_1 - sqrt(z)))
	tmp = 0.0
	if (t_4 <= 1.0)
		tmp = Float64(fma(sqrt(Float64(1.0 / z)), 0.5, t_3) - sqrt(x));
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(t_3 + t_2) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(fma(0.5, y, t_1) + 2.0) - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x)));
	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[(1.0 + z), $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[(N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0], N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(t$95$3 + t$95$2), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * y + t$95$1), $MachinePrecision] + 2.0), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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))) < 1

   1. Initial program 83.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 3.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 15.7%

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

   if 1 < (+.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 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 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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 4.4%

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

   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 21.4%

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

   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 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 51.4%

       \[\leadsto \left(\mathsf{fma}\left(0.5, y, \sqrt{z + 1}\right) + 2\right) - \left(\left(\sqrt{z} + \color{blue}{\sqrt{y}}\right) + \sqrt{x}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.0%

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

Alternative 13: 86.0% accurate, 0.5× speedup

Math

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

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 (+ 1.0 z)))
        (t_2 (sqrt (+ y 1.0)))
        (t_3 (sqrt (+ x 1.0)))
        (t_4 (+ (+ (- t_2 (sqrt y)) (- t_3 (sqrt x))) (- t_1 (sqrt z)))))
   (if (<= t_4 1.0)
     (- (fma (sqrt (/ 1.0 z)) 0.5 t_3) (sqrt x))
     (if (<= t_4 2.0)
       (- (+ t_3 t_2) (+ (sqrt y) (sqrt x)))
       (- (+ 2.0 t_1) (+ (+ (sqrt y) (sqrt z)) (sqrt x)))))))

C

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

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(1.0 + z))
	t_2 = sqrt(Float64(y + 1.0))
	t_3 = sqrt(Float64(x + 1.0))
	t_4 = Float64(Float64(Float64(t_2 - sqrt(y)) + Float64(t_3 - sqrt(x))) + Float64(t_1 - sqrt(z)))
	tmp = 0.0
	if (t_4 <= 1.0)
		tmp = Float64(fma(sqrt(Float64(1.0 / z)), 0.5, t_3) - sqrt(x));
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(t_3 + t_2) - Float64(sqrt(y) + sqrt(x)));
	else
		tmp = Float64(Float64(2.0 + t_1) - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x)));
	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[(1.0 + z), $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[(N[(N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 1.0], N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$3), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(t$95$3 + t$95$2), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + t$95$1), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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))) < 1

   1. Initial program 83.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 3.4%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 15.7%

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

   if 1 < (+.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 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 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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 4.4%

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

   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 21.4%

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

   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 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 50.9%

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

4. Final simplification 23.9%

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

Alternative 14: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{y + 1}\\ t_4 := \sqrt{x + 1}\\ \mathbf{if}\;\left(\left(t\_3 - \sqrt{y}\right) + \left(t\_4 - \sqrt{x}\right)\right) + t\_1 \leq 0.9998:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{z}} + \mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{t\_4 + \sqrt{x}}\right)\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{\sqrt{y} + t\_3} + \mathsf{fma}\left(0.5, x, 1\right)\right) - \sqrt{x}\right) + t\_1\right) + t\_2\\ \end{array} \end{array} \]

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 (+ 1.0 z)) (sqrt z)))
        (t_2 (- (sqrt (+ t 1.0)) (sqrt t)))
        (t_3 (sqrt (+ y 1.0)))
        (t_4 (sqrt (+ x 1.0))))
   (if (<= (+ (+ (- t_3 (sqrt y)) (- t_4 (sqrt x))) t_1) 0.9998)
     (+
      (+
       (* 0.5 (sqrt (/ 1.0 z)))
       (fma (sqrt (/ 1.0 y)) 0.5 (/ 1.0 (+ t_4 (sqrt x)))))
      t_2)
     (+
      (+ (- (+ (/ 1.0 (+ (sqrt y) t_3)) (fma 0.5 x 1.0)) (sqrt x)) t_1)
      t_2))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((1.0 + z)) - sqrt(z);
	double t_2 = sqrt((t + 1.0)) - sqrt(t);
	double t_3 = sqrt((y + 1.0));
	double t_4 = sqrt((x + 1.0));
	double tmp;
	if ((((t_3 - sqrt(y)) + (t_4 - sqrt(x))) + t_1) <= 0.9998) {
		tmp = ((0.5 * sqrt((1.0 / z))) + fma(sqrt((1.0 / y)), 0.5, (1.0 / (t_4 + sqrt(x))))) + t_2;
	} else {
		tmp = ((((1.0 / (sqrt(y) + t_3)) + fma(0.5, x, 1.0)) - sqrt(x)) + t_1) + t_2;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_2 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_3 = sqrt(Float64(y + 1.0))
	t_4 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(Float64(t_3 - sqrt(y)) + Float64(t_4 - sqrt(x))) + t_1) <= 0.9998)
		tmp = Float64(Float64(Float64(0.5 * sqrt(Float64(1.0 / z))) + fma(sqrt(Float64(1.0 / y)), 0.5, Float64(1.0 / Float64(t_4 + sqrt(x))))) + t_2);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_3)) + fma(0.5, x, 1.0)) - sqrt(x)) + t_1) + t_2);
	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[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$3 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], 0.9998], N[(N[(N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(1.0 / N[(t$95$4 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.5 * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $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))) < 0.99980000000000002

   1. Initial program 57.2%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   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. flip-- N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. frac-add N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 59.4%

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

   5. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 61.1

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

   7. Applied rewrites 61.1%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-sqrt.f64 70.0

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

   10. Applied rewrites 70.0%

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

   11. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 78.6

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

   13. Applied rewrites 78.6%

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

   if 0.99980000000000002 < (+.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 97.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \frac{1}{\color{blue}{\sqrt{y} + \sqrt{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \frac{1}{\color{blue}{\sqrt{y}} + \sqrt{1 + y}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \frac{1}{\sqrt{y} + \color{blue}{\sqrt{1 + y}}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \frac{1}{\sqrt{y} + \sqrt{\color{blue}{y + 1}}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \frac{1}{\sqrt{y} + \sqrt{\color{blue}{y + 1}}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-sqrt.f64 60.9

          \[\leadsto \left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 60.9%

      \[\leadsto \left(\color{blue}{\left(\left(\mathsf{fma}\left(0.5, x, 1\right) + \frac{1}{\sqrt{y} + \sqrt{y + 1}}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right) \leq 0.9998:\\ \;\;\;\;\left(0.5 \cdot \sqrt{\frac{1}{z}} + \mathsf{fma}\left(\sqrt{\frac{1}{y}}, 0.5, \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \mathsf{fma}\left(0.5, x, 1\right)\right) - \sqrt{x}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{t + 1} - \sqrt{t}\\ t_3 := \sqrt{x + 1}\\ \mathbf{if}\;\left(t\_1 - \sqrt{y}\right) + \left(t\_3 - \sqrt{x}\right) \leq 1.5:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + t\_1} + \frac{1}{t\_3 + \sqrt{x}}\right) + t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) + t\_1\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + t\_2\\ \end{array} \end{array} \]

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 (+ t 1.0)) (sqrt t)))
        (t_3 (sqrt (+ x 1.0))))
   (if (<= (+ (- t_1 (sqrt y)) (- t_3 (sqrt x))) 1.5)
     (+ (+ (/ 1.0 (+ (sqrt y) t_1)) (/ 1.0 (+ t_3 (sqrt x)))) t_2)
     (+
      (+
       (- (- (+ (fma 0.5 x 1.0) t_1) (sqrt y)) (sqrt x))
       (- (sqrt (+ 1.0 z)) (sqrt z)))
      t_2))))

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((t + 1.0)) - sqrt(t);
	double t_3 = sqrt((x + 1.0));
	double tmp;
	if (((t_1 - sqrt(y)) + (t_3 - sqrt(x))) <= 1.5) {
		tmp = ((1.0 / (sqrt(y) + t_1)) + (1.0 / (t_3 + sqrt(x)))) + t_2;
	} else {
		tmp = ((((fma(0.5, x, 1.0) + t_1) - sqrt(y)) - sqrt(x)) + (sqrt((1.0 + z)) - sqrt(z))) + t_2;
	}
	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 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	t_3 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(t_1 - sqrt(y)) + Float64(t_3 - sqrt(x))) <= 1.5)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + Float64(1.0 / Float64(t_3 + sqrt(x)))) + t_2);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(0.5, x, 1.0) + t_1) - sqrt(y)) - sqrt(x)) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z))) + t_2);
	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[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(t$95$3 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.5 * x + 1.0), $MachinePrecision] + t$95$1), $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $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))) < 1.5

   1. Initial program 89.0%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 89.7

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 89.7%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{\color{blue}{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\frac{\sqrt{\color{blue}{1 + x}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{\color{blue}{1 + x}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{1 + x}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \sqrt{\color{blue}{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \color{blue}{\sqrt{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. rem-square-sqrt N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. +-commutative N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied rewrites 91.9%

      \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + y}} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \color{blue}{\sqrt{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 48.4

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 1.5 < (+.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 97.7%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \left(\sqrt{x} + \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(1 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--r+ N/A

         \[\leadsto \left(\color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \sqrt{y}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\left(1 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \sqrt{y}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \frac{1}{2} \cdot x\right)\right) - \sqrt{y}\right)} - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\left(\left(1 + \color{blue}{\left(\frac{1}{2} \cdot x + \sqrt{1 + y}\right)}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. associate-+r+ N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) + \sqrt{1 + y}\right)} - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) + \sqrt{1 + y}\right)} - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\left(\left(\color{blue}{\left(\frac{1}{2} \cdot x + 1\right)} + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2}, x, 1\right)} + \sqrt{1 + y}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \color{blue}{\sqrt{1 + y}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \sqrt{\color{blue}{y + 1}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \sqrt{\color{blue}{y + 1}}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \sqrt{y + 1}\right) - \color{blue}{\sqrt{y}}\right) - \sqrt{x}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. lower-sqrt.f64 96.1

          \[\leadsto \left(\left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) + \sqrt{y + 1}\right) - \sqrt{y}\right) - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 96.1%

      \[\leadsto \left(\color{blue}{\left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) + \sqrt{y + 1}\right) - \sqrt{y}\right) - \sqrt{x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right) \leq 1.5:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) + \sqrt{y + 1}\right) - \sqrt{y}\right) - \sqrt{x}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;t\_2 - \sqrt{z} \leq 0:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + t\_1} + \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(1 + z\right) - z}{\sqrt{z} + t\_2} + \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right)\right) + t\_3\\ \end{array} \end{array} \]

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 (+ y 1.0)))
        (t_2 (sqrt (+ 1.0 z)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= (- t_2 (sqrt z)) 0.0)
     (+ (+ (/ 1.0 (+ (sqrt y) t_1)) (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))) t_3)
     (+
      (+
       (/ (- (+ 1.0 z) z) (+ (sqrt z) t_2))
       (+ (fma (fma -0.125 x 0.5) x (- 1.0 (sqrt x))) (- t_1 (sqrt y))))
      t_3))))

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((1.0 + z));
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if ((t_2 - sqrt(z)) <= 0.0) {
		tmp = ((1.0 / (sqrt(y) + t_1)) + (1.0 / (sqrt((x + 1.0)) + sqrt(x)))) + t_3;
	} else {
		tmp = ((((1.0 + z) - z) / (sqrt(z) + t_2)) + (fma(fma(-0.125, x, 0.5), x, (1.0 - sqrt(x))) + (t_1 - sqrt(y)))) + t_3;
	}
	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(1.0 + z))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (Float64(t_2 - sqrt(z)) <= 0.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x)))) + t_3);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 + z) - z) / Float64(sqrt(z) + t_2)) + Float64(fma(fma(-0.125, x, 0.5), x, Float64(1.0 - sqrt(x))) + Float64(t_1 - sqrt(y)))) + t_3);
	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[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(N[(N[(N[(1.0 + z), $MachinePrecision] - z), $MachinePrecision] / N[(N[Sqrt[z], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.125 * x + 0.5), $MachinePrecision] * x + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.0

   1. Initial program 84.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 85.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 85.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{\color{blue}{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\frac{\sqrt{\color{blue}{1 + x}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{\color{blue}{1 + x}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{1 + x}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \sqrt{\color{blue}{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \color{blue}{\sqrt{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. rem-square-sqrt N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. +-commutative N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied rewrites 87.6%

      \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + y}} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \color{blue}{\sqrt{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 89.1

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   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

      \[\leadsto \left(\left(\color{blue}{\left(\left(1 + x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)\right) - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + 1\right)} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. associate--l+ N/A

         \[\leadsto \left(\left(\color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + \left(1 - \sqrt{x}\right)\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) \cdot x} + \left(1 - \sqrt{x}\right)\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{-1}{8} \cdot x, x, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot x + \frac{1}{2}}, x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right)}, x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right), x, \color{blue}{1 - \sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-sqrt.f64 54.0

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 54.0%

      \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right), x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\left(\sqrt{z + 1} - \sqrt{z}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right), x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right), x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\sqrt{z + 1} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right), x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\sqrt{z + 1}} \cdot \sqrt{z + 1} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right), x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\sqrt{z + 1} \cdot \color{blue}{\sqrt{z + 1}} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right), x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right)} - \sqrt{z} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right), x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{\sqrt{z}} \cdot \sqrt{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right), x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \sqrt{z} \cdot \color{blue}{\sqrt{z}}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right), x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - \color{blue}{z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{8}, x, \frac{1}{2}\right), x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\color{blue}{\left(z + 1\right) - z}}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. lower-+.f64 54.5

          \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \frac{\left(z + 1\right) - z}{\color{blue}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 54.5%

      \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \color{blue}{\frac{\left(z + 1\right) - z}{\sqrt{z + 1} + \sqrt{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 0:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(1 + z\right) - z}{\sqrt{z} + \sqrt{1 + z}} + \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{t + 1}\\ \mathbf{if}\;t\_2 - \sqrt{z} \leq 0:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + t\_1} + \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) + \left(t\_3 - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) + t\_1\right) + t\_2\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + t\_3}\\ \end{array} \end{array} \]

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 (+ y 1.0))) (t_2 (sqrt (+ 1.0 z))) (t_3 (sqrt (+ t 1.0))))
   (if (<= (- t_2 (sqrt z)) 0.0)
     (+
      (+ (/ 1.0 (+ (sqrt y) t_1)) (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))))
      (- t_3 (sqrt t)))
     (+
      (- (+ (+ (fma 0.5 x 1.0) t_1) t_2) (+ (+ (sqrt y) (sqrt z)) (sqrt x)))
      (/ (- (+ t 1.0) t) (+ (sqrt t) t_3))))))

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((1.0 + z));
	double t_3 = sqrt((t + 1.0));
	double tmp;
	if ((t_2 - sqrt(z)) <= 0.0) {
		tmp = ((1.0 / (sqrt(y) + t_1)) + (1.0 / (sqrt((x + 1.0)) + sqrt(x)))) + (t_3 - sqrt(t));
	} else {
		tmp = (((fma(0.5, x, 1.0) + t_1) + t_2) - ((sqrt(y) + sqrt(z)) + sqrt(x))) + (((t + 1.0) - t) / (sqrt(t) + t_3));
	}
	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(1.0 + z))
	t_3 = sqrt(Float64(t + 1.0))
	tmp = 0.0
	if (Float64(t_2 - sqrt(z)) <= 0.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x)))) + Float64(t_3 - sqrt(t)));
	else
		tmp = Float64(Float64(Float64(Float64(fma(0.5, x, 1.0) + t_1) + t_2) - Float64(Float64(sqrt(y) + sqrt(z)) + sqrt(x))) + Float64(Float64(Float64(t + 1.0) - t) / Float64(sqrt(t) + t_3)));
	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[(1.0 + z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[z], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * x + 1.0), $MachinePrecision] + t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t + 1.0), $MachinePrecision] - t), $MachinePrecision] / N[(N[Sqrt[t], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.0

   1. Initial program 84.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 85.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 85.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{\color{blue}{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\frac{\sqrt{\color{blue}{1 + x}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{\color{blue}{1 + x}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{1 + x}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \sqrt{\color{blue}{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \color{blue}{\sqrt{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. rem-square-sqrt N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. +-commutative N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied rewrites 87.6%

      \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + y}} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \color{blue}{\sqrt{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 89.1

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   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

      \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 + \left(\sqrt{1 + y} + \left(\sqrt{1 + z} + \frac{1}{2} \cdot x\right)\right)\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 33.6%

      \[\leadsto \color{blue}{\left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right)} \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + \color{blue}{\frac{\sqrt{t + 1} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + \frac{\color{blue}{\sqrt{t + 1}} \cdot \sqrt{t + 1} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + \frac{\sqrt{t + 1} \cdot \color{blue}{\sqrt{t + 1}} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + \frac{\color{blue}{\left(t + 1\right)} - \sqrt{t} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{\sqrt{t}} \cdot \sqrt{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + \frac{\left(t + 1\right) - \sqrt{t} \cdot \color{blue}{\sqrt{t}}}{\sqrt{t + 1} + \sqrt{t}} \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + \frac{\left(t + 1\right) - \color{blue}{t}}{\sqrt{t + 1} + \sqrt{t}} \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + \frac{\color{blue}{\left(t + 1\right) - t}}{\sqrt{t + 1} + \sqrt{t}} \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

      12. lower-+.f64 33.6

          \[\leadsto \left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + \frac{\left(t + 1\right) - t}{\color{blue}{\sqrt{t} + \sqrt{t + 1}}} \]

   7. Applied rewrites 33.6%

      \[\leadsto \left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{z + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)\right) + \color{blue}{\frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 0:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(0.5, x, 1\right) + \sqrt{y + 1}\right) + \sqrt{1 + z}\right) - \left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)\right) + \frac{\left(t + 1\right) - t}{\sqrt{t} + \sqrt{t + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{y + 1}\\ t_2 := \sqrt{1 + z} - \sqrt{z}\\ t_3 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + t\_1} + \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) + t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(t\_1 - \sqrt{y}\right)\right) + t\_2\right) + t\_3\\ \end{array} \end{array} \]

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 (+ y 1.0)))
        (t_2 (- (sqrt (+ 1.0 z)) (sqrt z)))
        (t_3 (- (sqrt (+ t 1.0)) (sqrt t))))
   (if (<= t_2 0.0)
     (+ (+ (/ 1.0 (+ (sqrt y) t_1)) (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))) t_3)
     (+ (+ (+ (- 1.0 (sqrt x)) (- t_1 (sqrt y))) 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((y + 1.0));
	double t_2 = sqrt((1.0 + z)) - sqrt(z);
	double t_3 = sqrt((t + 1.0)) - sqrt(t);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = ((1.0 / (sqrt(y) + t_1)) + (1.0 / (sqrt((x + 1.0)) + sqrt(x)))) + t_3;
	} else {
		tmp = (((1.0 - sqrt(x)) + (t_1 - sqrt(y))) + 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((y + 1.0d0))
    t_2 = sqrt((1.0d0 + z)) - sqrt(z)
    t_3 = sqrt((t + 1.0d0)) - sqrt(t)
    if (t_2 <= 0.0d0) then
        tmp = ((1.0d0 / (sqrt(y) + t_1)) + (1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x)))) + t_3
    else
        tmp = (((1.0d0 - sqrt(x)) + (t_1 - sqrt(y))) + 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((y + 1.0));
	double t_2 = Math.sqrt((1.0 + z)) - Math.sqrt(z);
	double t_3 = Math.sqrt((t + 1.0)) - Math.sqrt(t);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = ((1.0 / (Math.sqrt(y) + t_1)) + (1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x)))) + t_3;
	} else {
		tmp = (((1.0 - Math.sqrt(x)) + (t_1 - Math.sqrt(y))) + 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((y + 1.0))
	t_2 = math.sqrt((1.0 + z)) - math.sqrt(z)
	t_3 = math.sqrt((t + 1.0)) - math.sqrt(t)
	tmp = 0
	if t_2 <= 0.0:
		tmp = ((1.0 / (math.sqrt(y) + t_1)) + (1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x)))) + t_3
	else:
		tmp = (((1.0 - math.sqrt(x)) + (t_1 - math.sqrt(y))) + t_2) + t_3
	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 = Float64(sqrt(Float64(1.0 + z)) - sqrt(z))
	t_3 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(y) + t_1)) + Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x)))) + t_3);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(t_1 - sqrt(y))) + 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((y + 1.0));
	t_2 = sqrt((1.0 + z)) - sqrt(z);
	t_3 = sqrt((t + 1.0)) - sqrt(t);
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = ((1.0 / (sqrt(y) + t_1)) + (1.0 / (sqrt((x + 1.0)) + sqrt(x)))) + t_3;
	else
		tmp = (((1.0 - sqrt(x)) + (t_1 - sqrt(y))) + 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[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[(N[(1.0 / N[(N[Sqrt[y], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z)) < 0.0

   1. Initial program 84.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\left(\sqrt{y + 1} - \sqrt{y}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\sqrt{y + 1}} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\sqrt{y + 1} \cdot \color{blue}{\sqrt{y + 1}} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right)} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{\sqrt{y}} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \sqrt{y} \cdot \color{blue}{\sqrt{y}}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - \color{blue}{y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\color{blue}{\left(y + 1\right) - y}}{\sqrt{y + 1} + \sqrt{y}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 85.2

          \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \frac{\left(y + 1\right) - y}{\color{blue}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 85.2%

      \[\leadsto \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \color{blue}{\frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{\color{blue}{x + 1}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\frac{\sqrt{\color{blue}{1 + x}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{\color{blue}{1 + x}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\frac{\color{blue}{\sqrt{1 + x}} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \sqrt{\color{blue}{x + 1}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \sqrt{\color{blue}{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\sqrt{1 + x} \cdot \color{blue}{\sqrt{1 + x}} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\left(\frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - \color{blue}{\sqrt{x}} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - \sqrt{x} \cdot \color{blue}{\sqrt{x}}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. rem-square-sqrt N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{x + 1} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. +-commutative N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      17. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      18. +-commutative N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      19. lift-+.f64 N/A

          \[\leadsto \left(\left(\frac{\left(1 + x\right) - x}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   6. Applied rewrites 87.6%

      \[\leadsto \left(\left(\color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \frac{\left(y + 1\right) - y}{\sqrt{y} + \sqrt{y + 1}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{x} + \sqrt{1 + x}} + \frac{1}{\sqrt{y} + \sqrt{1 + y}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y} + \sqrt{1 + y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{y} + \sqrt{1 + y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + y} + \sqrt{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + y}} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{y + 1}} + \sqrt{y}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \color{blue}{\sqrt{y}}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      11. +-commutative N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      14. +-commutative N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      15. lower-+.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      16. lower-sqrt.f64 89.1

          \[\leadsto \left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   9. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{y + 1} + \sqrt{y}} + \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right)} + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 z #s(literal 1 binary64))) (sqrt.f64 z))

   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 52.2

         \[\leadsto \left(\left(\left(1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 52.2%

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + z} - \sqrt{z} \leq 0:\\ \;\;\;\;\left(\frac{1}{\sqrt{y} + \sqrt{y + 1}} + \frac{1}{\sqrt{x + 1} + \sqrt{x}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 96.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{t + 1} - \sqrt{t}\\ \mathbf{if}\;x \leq 4.1 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + t\_1\\ \end{array} \end{array} \]

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 1.0)) (sqrt t))))
   (if (<= x 4.1e-9)
     (+
      (+
       (+ (- 1.0 (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
       (- (sqrt (+ 1.0 z)) (sqrt z)))
      t_1)
     (+
      (+ (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))) (* 0.5 (sqrt (/ 1.0 z))))
      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 + 1.0)) - sqrt(t);
	double tmp;
	if (x <= 4.1e-9) {
		tmp = (((1.0 - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((1.0 + z)) - sqrt(z))) + t_1;
	} else {
		tmp = ((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (0.5 * sqrt((1.0 / z)))) + 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) :: tmp
    t_1 = sqrt((t + 1.0d0)) - sqrt(t)
    if (x <= 4.1d-9) then
        tmp = (((1.0d0 - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((1.0d0 + z)) - sqrt(z))) + t_1
    else
        tmp = ((1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))) + (0.5d0 * sqrt((1.0d0 / z)))) + 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 + 1.0)) - Math.sqrt(t);
	double tmp;
	if (x <= 4.1e-9) {
		tmp = (((1.0 - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((1.0 + z)) - Math.sqrt(z))) + t_1;
	} else {
		tmp = ((1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x))) + (0.5 * Math.sqrt((1.0 / z)))) + 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 + 1.0)) - math.sqrt(t)
	tmp = 0
	if x <= 4.1e-9:
		tmp = (((1.0 - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((1.0 + z)) - math.sqrt(z))) + t_1
	else:
		tmp = ((1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))) + (0.5 * math.sqrt((1.0 / z)))) + t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(Float64(t + 1.0)) - sqrt(t))
	tmp = 0.0
	if (x <= 4.1e-9)
		tmp = Float64(Float64(Float64(Float64(1.0 - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z))) + t_1);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x))) + Float64(0.5 * sqrt(Float64(1.0 / z)))) + 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 + 1.0)) - sqrt(t);
	tmp = 0.0;
	if (x <= 4.1e-9)
		tmp = (((1.0 - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((1.0 + z)) - sqrt(z))) + t_1;
	else
		tmp = ((1.0 / (sqrt((x + 1.0)) + sqrt(x))) + (0.5 * sqrt((1.0 / z)))) + 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[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4.1e-9], N[(N[(N[(N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.1000000000000003e-9

   1. Initial program 97.8%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 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 97.7

         \[\leadsto \left(\left(\left(1 - \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Applied rewrites 97.7%

      \[\leadsto \left(\left(\color{blue}{\left(1 - \sqrt{x}\right)} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   if 4.1000000000000003e-9 < x

   1. Initial program 85.5%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      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. flip-- N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}}} + \left(\sqrt{x + 1} - \sqrt{x}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. flip-- N/A

         \[\leadsto \left(\left(\frac{\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}}{\sqrt{y + 1} + \sqrt{y}} + \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. frac-add N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\left(\sqrt{y + 1} \cdot \sqrt{y + 1} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right) + \left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right)}{\left(\sqrt{y + 1} + \sqrt{y}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   4. Applied rewrites 86.7%

      \[\leadsto \left(\color{blue}{\frac{\mathsf{fma}\left(\left(y + 1\right) - y, \sqrt{x} + \sqrt{1 + x}, \left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\left(1 + x\right) - x\right)\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\frac{\color{blue}{1 + \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{1 + y}\right)\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right)} + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \color{blue}{\sqrt{x}}\right) + \left(\sqrt{y} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \color{blue}{\left(\sqrt{y} + \sqrt{1 + y}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\color{blue}{\sqrt{y}} + \sqrt{1 + y}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \color{blue}{\sqrt{1 + y}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      8. +-commutative N/A

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{y + 1}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      9. lower-+.f64 51.9

         \[\leadsto \left(\frac{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{\color{blue}{y + 1}}\right)}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   7. Applied rewrites 51.9%

      \[\leadsto \left(\frac{\color{blue}{\left(1 + \sqrt{x}\right) + \left(\sqrt{y} + \sqrt{y + 1}\right)}}{\left(\sqrt{y} + \sqrt{y + 1}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{1 + x}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      5. +-commutative N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{\sqrt{\color{blue}{x + 1}} + \sqrt{x}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

      7. lower-sqrt.f64 48.7

         \[\leadsto \left(\frac{1}{\sqrt{x + 1} + \color{blue}{\sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   10. Applied rewrites 48.7%

       \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}}} + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   11. Taylor expanded in z around inf

       \[\leadsto \left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{z}}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\sqrt{\frac{1}{z}} \cdot \frac{1}{2}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\sqrt{\frac{1}{z}} \cdot \frac{1}{2}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\sqrt{\frac{1}{z}}} \cdot \frac{1}{2}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

       4. lower-/.f64 27.0

          \[\leadsto \left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \sqrt{\color{blue}{\frac{1}{z}}} \cdot 0.5\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

   13. Applied rewrites 27.0%

       \[\leadsto \left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \color{blue}{\sqrt{\frac{1}{z}} \cdot 0.5}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(\left(1 - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + 0.5 \cdot \sqrt{\frac{1}{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 65.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ t_2 := \sqrt{y + 1}\\ \mathbf{if}\;t\_2 - \sqrt{y} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, t\_1\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + t\_2\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \end{array} \end{array} \]

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))) (t_2 (sqrt (+ y 1.0))))
   (if (<= (- t_2 (sqrt y)) 0.0)
     (- (fma (sqrt (/ 1.0 z)) 0.5 t_1) (sqrt x))
     (- (+ t_1 t_2) (+ (sqrt y) (sqrt x))))))

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 tmp;
	if ((t_2 - sqrt(y)) <= 0.0) {
		tmp = fma(sqrt((1.0 / z)), 0.5, t_1) - sqrt(x);
	} else {
		tmp = (t_1 + t_2) - (sqrt(y) + sqrt(x));
	}
	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))
	tmp = 0.0
	if (Float64(t_2 - sqrt(y)) <= 0.0)
		tmp = Float64(fma(sqrt(Float64(1.0 / z)), 0.5, t_1) - sqrt(x));
	else
		tmp = Float64(Float64(t_1 + t_2) - Float64(sqrt(y) + sqrt(x)));
	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[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + t$95$2), $MachinePrecision] - N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y)) < 0.0

   1. Initial program 85.6%

      \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 3.7%

      \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 16.6%

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{x + 1}\right) - \sqrt{x} \]

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 y #s(literal 1 binary64))) (sqrt.f64 y))

   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. 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. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      12. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      14. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   5. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

   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 22.4%

      \[\leadsto \left(\sqrt{y + 1} + \sqrt{x + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{y + 1} - \sqrt{y} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{x + 1}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \sqrt{y + 1}\right) - \left(\sqrt{y} + \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 32.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{x + 1}\right) - \sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (- (fma (sqrt (/ 1.0 z)) 0.5 (sqrt (+ x 1.0))) (sqrt x)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return fma(sqrt((1.0 / z)), 0.5, sqrt((x + 1.0))) - sqrt(x);
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(fma(sqrt(Float64(1.0 / z)), 0.5, sqrt(Float64(x + 1.0))) - 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[(N[(N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision] * 0.5 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
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. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   16. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

5. Applied rewrites 12.7%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

6. Taylor expanded in z around inf

   \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 12.0%

   \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

9. Taylor expanded in y around inf

   \[\leadsto \left(\sqrt{1 + x} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right) - \sqrt{x} \]
10. Step-by-step derivation

11. Applied rewrites 12.4%

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{x + 1}\right) - \sqrt{x} \]
12. Add Preprocessing

Alternative 22: 7.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 0.5 \cdot \sqrt{\frac{1}{z}} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (* 0.5 (sqrt (/ 1.0 z))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 0.5 * sqrt((1.0 / z));
}

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 = 0.5d0 * sqrt((1.0d0 / z))
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 0.5 * Math.sqrt((1.0 / z));
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 0.5 * math.sqrt((1.0 / z))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(0.5 * sqrt(Float64(1.0 / z)))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 0.5 * sqrt((1.0 / z));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(0.5 * N[Sqrt[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.5%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
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. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   16. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

5. Applied rewrites 12.7%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]

6. Taylor expanded in z around inf

   \[\leadsto \left(\sqrt{1 + x} + \left(\sqrt{1 + y} + \frac{1}{2} \cdot \sqrt{\frac{1}{z}}\right)\right) - \color{blue}{\left(\sqrt{x} + \sqrt{y}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 12.0%

   \[\leadsto \left(\mathsf{fma}\left(\sqrt{\frac{1}{z}}, 0.5, \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\sqrt{y} + \sqrt{x}\right)} \]

9. Taylor expanded in z around 0

   \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{1}{z}} \]
10. Step-by-step derivation

11. Applied rewrites 8.4%

    \[\leadsto \sqrt{\frac{1}{z}} \cdot 0.5 \]

12. Final simplification 8.4%

    \[\leadsto 0.5 \cdot \sqrt{\frac{1}{z}} \]
13. Add Preprocessing

Alternative 23: 1.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ -\sqrt{y} \end{array} \]

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 y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return -sqrt(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(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(y);
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return -math.sqrt(y)

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(-sqrt(y))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = -sqrt(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[Sqrt[y], $MachinePrecision])

Derivation

1. Initial program 91.5%

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
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. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{1 + y} + \sqrt{1 + z}\right) + \sqrt{1 + x}\right)} - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\sqrt{1 + z} + \sqrt{1 + y}\right)} + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\color{blue}{\sqrt{1 + z}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\left(\sqrt{\color{blue}{z + 1}} + \sqrt{1 + y}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(\sqrt{z + 1} + \color{blue}{\sqrt{1 + y}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{\color{blue}{y + 1}}\right) + \sqrt{1 + x}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   12. lower-sqrt.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \color{blue}{\sqrt{1 + x}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   14. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{\color{blue}{x + 1}}\right) - \left(\sqrt{x} + \left(\sqrt{y} + \sqrt{z}\right)\right) \]

   15. +-commutative N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

   16. lower-+.f64 N/A

       \[\leadsto \left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \color{blue}{\left(\left(\sqrt{y} + \sqrt{z}\right) + \sqrt{x}\right)} \]

5. Applied rewrites 12.7%

   \[\leadsto \color{blue}{\left(\left(\sqrt{z + 1} + \sqrt{y + 1}\right) + \sqrt{x + 1}\right) - \left(\left(\sqrt{z} + \sqrt{y}\right) + \sqrt{x}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 3.8%

   \[\leadsto \left(\frac{\left(z + 1\right) - \left(1 + y\right)}{\sqrt{z + 1} - \sqrt{1 + y}} + \sqrt{x + 1}\right) - \left(\left(\color{blue}{\sqrt{z}} + \sqrt{y}\right) + \sqrt{x}\right) \]

8. Taylor expanded in y around inf

   \[\leadsto -1 \cdot \color{blue}{\sqrt{y}} \]
9. Step-by-step derivation

10. Applied rewrites 1.6%

    \[\leadsto -\sqrt{y} \]
11. 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 2024255 
(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))))