Main:z from

Percentage Accurate: 91.9% → 98.5%

Time: 25.0s

Alternatives: 13

Speedup: 2.0×

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: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;t_1 - \sqrt{x} \leq 0.8:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \sqrt{x}\right)\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))))
   (if (<= (- t_1 (sqrt x)) 0.8)
     (/ 1.0 (+ t_1 (sqrt x)))
     (+
      (+ t_1 (- (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))) (sqrt x)))
      (+
       (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z))))
       (- (sqrt (+ 1.0 t)) (sqrt t)))))))

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 tmp;
	if ((t_1 - sqrt(x)) <= 0.8) {
		tmp = 1.0 / (t_1 + sqrt(x));
	} else {
		tmp = (t_1 + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) - sqrt(x))) + ((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + (sqrt((1.0 + t)) - sqrt(t)));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((x + 1.0d0))
    if ((t_1 - sqrt(x)) <= 0.8d0) then
        tmp = 1.0d0 / (t_1 + sqrt(x))
    else
        tmp = (t_1 + ((1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y))) - sqrt(x))) + ((1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))) + (sqrt((1.0d0 + t)) - sqrt(t)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double tmp;
	if ((t_1 - Math.sqrt(x)) <= 0.8) {
		tmp = 1.0 / (t_1 + Math.sqrt(x));
	} else {
		tmp = (t_1 + ((1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y))) - Math.sqrt(x))) + ((1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)))) + (Math.sqrt((1.0 + t)) - Math.sqrt(t)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	tmp = 0
	if (t_1 - math.sqrt(x)) <= 0.8:
		tmp = 1.0 / (t_1 + math.sqrt(x))
	else:
		tmp = (t_1 + ((1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y))) - math.sqrt(x))) + ((1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))) + (math.sqrt((1.0 + t)) - math.sqrt(t)))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_1 - sqrt(x)) <= 0.8)
		tmp = Float64(1.0 / Float64(t_1 + sqrt(x)));
	else
		tmp = Float64(Float64(t_1 + Float64(Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))) - sqrt(x))) + Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z)))) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	tmp = 0.0;
	if ((t_1 - sqrt(x)) <= 0.8)
		tmp = 1.0 / (t_1 + sqrt(x));
	else
		tmp = (t_1 + ((1.0 / (sqrt((1.0 + y)) + sqrt(y))) - sqrt(x))) + ((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + (sqrt((1.0 + t)) - sqrt(t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.8], N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + N[(N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 0.80000000000000004

   1. Initial program 85.8%

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

      1. associate-+l+ 85.8%

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

      2. +-commutative 85.8%

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

      3. associate-+r- 42.9%

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

      4. associate-+l- 11.1%

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

      5. +-commutative 11.1%

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

      6. associate--l+ 11.1%

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

      7. +-commutative 11.1%

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

   3. Simplified 7.8%

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

   4. Taylor expanded in t around inf 4.7%

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

      1. associate--l+ 24.0%

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

      2. +-commutative 24.0%

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

      3. +-commutative 24.0%

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

   6. Simplified 24.0%

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

   7. Taylor expanded in z around inf 5.8%

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

      1. +-commutative 5.8%

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

      2. associate--l+ 6.7%

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

      3. +-commutative 6.7%

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

   9. Simplified 6.7%

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

   10. Taylor expanded in y around inf 4.4%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 4.4%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 4.8%

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

       3. add-sqr-sqrt 4.4%

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

   12. Applied egg-rr 4.4%

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

       1. associate--l+ 9.5%

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

       2. +-inverses 9.5%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 9.5%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 9.5%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   14. Simplified 9.5%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]

   if 0.80000000000000004 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   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. Step-by-step derivation

      1. associate-+l+ 97.7%

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

      2. associate-+l- 97.7%

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

      3. +-commutative 97.7%

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

      4. sub-neg 97.7%

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

      5. sub-neg 97.7%

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

      6. +-commutative 97.7%

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

      7. +-commutative 97.7%

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

   3. Simplified 97.7%

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

      1. flip-- 97.7%

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

      2. add-sqr-sqrt 79.8%

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

      3. +-commutative 79.8%

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

      4. add-sqr-sqrt 98.0%

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

      5. +-commutative 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in z around 0 98.6%

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

      1. flip-- 98.8%

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

      2. add-sqr-sqrt 83.9%

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

      3. add-sqr-sqrt 99.2%

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

   8. Applied egg-rr 99.2%

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

      1. associate--l+ 99.6%

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

      2. +-inverses 99.6%

         \[\leadsto \left(\sqrt{x + 1} - \left(\sqrt{x} - \frac{1 + \color{blue}{0}}{\sqrt{1 + y} + \sqrt{y}}\right)\right) + \left(\frac{1}{\sqrt{z + 1} + \sqrt{z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) \]

      3. metadata-eval 99.6%

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

   10. Simplified 99.6%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.8:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(\frac{1}{\sqrt{1 + y} + \sqrt{y}} - \sqrt{x}\right)\right) + \left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\\ \end{array} \]

Alternative 2: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y} - \sqrt{y}\\ t_2 := \sqrt{x + 1}\\ \mathbf{if}\;t_1 + \left(t_2 - \sqrt{x}\right) \leq 1:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(1 + t_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt (+ 1.0 y)) (sqrt y))) (t_2 (sqrt (+ x 1.0))))
   (if (<= (+ t_1 (- t_2 (sqrt x))) 1.0)
     (/ 1.0 (+ t_2 (sqrt x)))
     (+
      (+ (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))) (- (sqrt (+ 1.0 t)) (sqrt t)))
      (+ 1.0 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 + y)) - sqrt(y);
	double t_2 = sqrt((x + 1.0));
	double tmp;
	if ((t_1 + (t_2 - sqrt(x))) <= 1.0) {
		tmp = 1.0 / (t_2 + sqrt(x));
	} else {
		tmp = ((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + (sqrt((1.0 + t)) - sqrt(t))) + (1.0 + t_1);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt((1.0d0 + y)) - sqrt(y)
    t_2 = sqrt((x + 1.0d0))
    if ((t_1 + (t_2 - sqrt(x))) <= 1.0d0) then
        tmp = 1.0d0 / (t_2 + sqrt(x))
    else
        tmp = ((1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))) + (sqrt((1.0d0 + t)) - sqrt(t))) + (1.0d0 + 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((1.0 + y)) - Math.sqrt(y);
	double t_2 = Math.sqrt((x + 1.0));
	double tmp;
	if ((t_1 + (t_2 - Math.sqrt(x))) <= 1.0) {
		tmp = 1.0 / (t_2 + Math.sqrt(x));
	} else {
		tmp = ((1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)))) + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + (1.0 + t_1);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y)) - math.sqrt(y)
	t_2 = math.sqrt((x + 1.0))
	tmp = 0
	if (t_1 + (t_2 - math.sqrt(x))) <= 1.0:
		tmp = 1.0 / (t_2 + math.sqrt(x))
	else:
		tmp = ((1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))) + (math.sqrt((1.0 + t)) - math.sqrt(t))) + (1.0 + 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(1.0 + y)) - sqrt(y))
	t_2 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_1 + Float64(t_2 - sqrt(x))) <= 1.0)
		tmp = Float64(1.0 / Float64(t_2 + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z)))) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + Float64(1.0 + 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((1.0 + y)) - sqrt(y);
	t_2 = sqrt((x + 1.0));
	tmp = 0.0;
	if ((t_1 + (t_2 - sqrt(x))) <= 1.0)
		tmp = 1.0 / (t_2 + sqrt(x));
	else
		tmp = ((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + (sqrt((1.0 + t)) - sqrt(t))) + (1.0 + 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[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(t$95$2 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y))) < 1

   1. Initial program 89.3%

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

      1. associate-+l+ 89.3%

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

      2. +-commutative 89.3%

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

      3. associate-+r- 58.0%

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

      4. associate-+l- 34.7%

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

      5. +-commutative 34.7%

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

      6. associate--l+ 34.7%

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

      7. +-commutative 34.7%

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

   3. Simplified 24.0%

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

   4. Taylor expanded in t around inf 5.0%

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

      1. associate--l+ 19.0%

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

      2. +-commutative 19.0%

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

      3. +-commutative 19.0%

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

   6. Simplified 19.0%

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

   7. Taylor expanded in z around inf 4.7%

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

      1. +-commutative 4.7%

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

      2. associate--l+ 14.0%

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

      3. +-commutative 14.0%

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

   9. Simplified 14.0%

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

   10. Taylor expanded in y around inf 13.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 13.0%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 13.3%

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

       3. add-sqr-sqrt 13.0%

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

   12. Applied egg-rr 13.0%

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

       1. associate--l+ 16.7%

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

       2. +-inverses 16.7%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 16.7%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 16.7%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   14. Simplified 16.7%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]

   if 1 < (+.f64 (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) (-.f64 (sqrt.f64 (+.f64 y 1)) (sqrt.f64 y)))

   1. Initial program 97.2%

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

      1. associate-+l+ 97.2%

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

      2. associate-+l- 97.2%

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

      3. +-commutative 97.2%

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

      4. sub-neg 97.2%

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

      5. sub-neg 97.2%

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

      6. +-commutative 97.2%

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

      7. +-commutative 97.2%

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

   3. Simplified 97.2%

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

      1. flip-- 97.2%

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

      2. add-sqr-sqrt 81.0%

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

      3. +-commutative 81.0%

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

      4. add-sqr-sqrt 97.6%

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

      5. +-commutative 97.6%

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

   5. Applied egg-rr 97.6%

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

   6. Taylor expanded in z around 0 98.4%

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

   7. Taylor expanded in x around 0 92.3%

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

      1. associate--l+ 91.0%

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

   9. Simplified 92.3%

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

4. Final simplification 40.9%

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

Alternative 3: 97.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;t_1 - \sqrt{x} \leq 0.8:\\ \;\;\;\;\frac{1}{t_1 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(t_1 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ x 1.0))))
   (if (<= (- t_1 (sqrt x)) 0.8)
     (/ 1.0 (+ t_1 (sqrt x)))
     (+
      (+ (/ 1.0 (+ (sqrt z) (sqrt (+ 1.0 z)))) (- (sqrt (+ 1.0 t)) (sqrt t)))
      (+ t_1 (- (- (sqrt (+ 1.0 y)) (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 tmp;
	if ((t_1 - sqrt(x)) <= 0.8) {
		tmp = 1.0 / (t_1 + sqrt(x));
	} else {
		tmp = ((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + (sqrt((1.0 + t)) - sqrt(t))) + (t_1 + ((sqrt((1.0 + y)) - sqrt(y)) - sqrt(x)));
	}
	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((x + 1.0d0))
    if ((t_1 - sqrt(x)) <= 0.8d0) then
        tmp = 1.0d0 / (t_1 + sqrt(x))
    else
        tmp = ((1.0d0 / (sqrt(z) + sqrt((1.0d0 + z)))) + (sqrt((1.0d0 + t)) - sqrt(t))) + (t_1 + ((sqrt((1.0d0 + y)) - sqrt(y)) - sqrt(x)))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0));
	double tmp;
	if ((t_1 - Math.sqrt(x)) <= 0.8) {
		tmp = 1.0 / (t_1 + Math.sqrt(x));
	} else {
		tmp = ((1.0 / (Math.sqrt(z) + Math.sqrt((1.0 + z)))) + (Math.sqrt((1.0 + t)) - Math.sqrt(t))) + (t_1 + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) - Math.sqrt(x)));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0))
	tmp = 0
	if (t_1 - math.sqrt(x)) <= 0.8:
		tmp = 1.0 / (t_1 + math.sqrt(x))
	else:
		tmp = ((1.0 / (math.sqrt(z) + math.sqrt((1.0 + z)))) + (math.sqrt((1.0 + t)) - math.sqrt(t))) + (t_1 + ((math.sqrt((1.0 + y)) - math.sqrt(y)) - math.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))
	tmp = 0.0
	if (Float64(t_1 - sqrt(x)) <= 0.8)
		tmp = Float64(1.0 / Float64(t_1 + sqrt(x)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sqrt(z) + sqrt(Float64(1.0 + z)))) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t))) + Float64(t_1 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - sqrt(x))));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0));
	tmp = 0.0;
	if ((t_1 - sqrt(x)) <= 0.8)
		tmp = 1.0 / (t_1 + sqrt(x));
	else
		tmp = ((1.0 / (sqrt(z) + sqrt((1.0 + z)))) + (sqrt((1.0 + t)) - sqrt(t))) + (t_1 + ((sqrt((1.0 + y)) - sqrt(y)) - sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.8], N[(1.0 / N[(t$95$1 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sqrt[z], $MachinePrecision] + N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 0.80000000000000004

   1. Initial program 85.8%

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

      1. associate-+l+ 85.8%

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

      2. +-commutative 85.8%

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

      3. associate-+r- 42.9%

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

      4. associate-+l- 11.1%

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

      5. +-commutative 11.1%

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

      6. associate--l+ 11.1%

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

      7. +-commutative 11.1%

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

   3. Simplified 7.8%

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

   4. Taylor expanded in t around inf 4.7%

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

      1. associate--l+ 24.0%

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

      2. +-commutative 24.0%

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

      3. +-commutative 24.0%

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

   6. Simplified 24.0%

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

   7. Taylor expanded in z around inf 5.8%

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

      1. +-commutative 5.8%

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

      2. associate--l+ 6.7%

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

      3. +-commutative 6.7%

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

   9. Simplified 6.7%

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

   10. Taylor expanded in y around inf 4.4%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 4.4%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 4.8%

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

       3. add-sqr-sqrt 4.4%

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

   12. Applied egg-rr 4.4%

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

       1. associate--l+ 9.5%

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

       2. +-inverses 9.5%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 9.5%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 9.5%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   14. Simplified 9.5%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]

   if 0.80000000000000004 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   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. Step-by-step derivation

      1. associate-+l+ 97.7%

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

      2. associate-+l- 97.7%

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

      3. +-commutative 97.7%

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

      4. sub-neg 97.7%

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

      5. sub-neg 97.7%

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

      6. +-commutative 97.7%

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

      7. +-commutative 97.7%

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

   3. Simplified 97.7%

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

      1. flip-- 97.7%

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

      2. add-sqr-sqrt 79.8%

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

      3. +-commutative 79.8%

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

      4. add-sqr-sqrt 98.0%

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

      5. +-commutative 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in z around 0 98.6%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.8:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{z} + \sqrt{1 + z}} + \left(\sqrt{1 + t} - \sqrt{t}\right)\right) + \left(\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \sqrt{x}\right)\right)\\ \end{array} \]

Alternative 4: 96.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{z} - \sqrt{1 + z}\\ t_2 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 7.5 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) - t_1\right) + 2\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+22}:\\ \;\;\;\;t_2 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_2 + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (sqrt z) (sqrt (+ 1.0 z)))) (t_2 (sqrt (+ x 1.0))))
   (if (<= y 7.5e-32)
     (+ (- (- (sqrt (+ 1.0 t)) (sqrt t)) t_1) 2.0)
     (if (<= y 2.05e+22)
       (+ t_2 (- (- (sqrt (+ 1.0 y)) (sqrt y)) (+ (sqrt x) t_1)))
       (/ 1.0 (+ t_2 (sqrt x)))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt(z) - sqrt((1.0 + z));
	double t_2 = sqrt((x + 1.0));
	double tmp;
	if (y <= 7.5e-32) {
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) - t_1) + 2.0;
	} else if (y <= 2.05e+22) {
		tmp = t_2 + ((sqrt((1.0 + y)) - sqrt(y)) - (sqrt(x) + t_1));
	} else {
		tmp = 1.0 / (t_2 + sqrt(x));
	}
	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) :: tmp
    t_1 = sqrt(z) - sqrt((1.0d0 + z))
    t_2 = sqrt((x + 1.0d0))
    if (y <= 7.5d-32) then
        tmp = ((sqrt((1.0d0 + t)) - sqrt(t)) - t_1) + 2.0d0
    else if (y <= 2.05d+22) then
        tmp = t_2 + ((sqrt((1.0d0 + y)) - sqrt(y)) - (sqrt(x) + t_1))
    else
        tmp = 1.0d0 / (t_2 + sqrt(x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt(z) - Math.sqrt((1.0 + z));
	double t_2 = Math.sqrt((x + 1.0));
	double tmp;
	if (y <= 7.5e-32) {
		tmp = ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) - t_1) + 2.0;
	} else if (y <= 2.05e+22) {
		tmp = t_2 + ((Math.sqrt((1.0 + y)) - Math.sqrt(y)) - (Math.sqrt(x) + t_1));
	} else {
		tmp = 1.0 / (t_2 + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt(z) - math.sqrt((1.0 + z))
	t_2 = math.sqrt((x + 1.0))
	tmp = 0
	if y <= 7.5e-32:
		tmp = ((math.sqrt((1.0 + t)) - math.sqrt(t)) - t_1) + 2.0
	elif y <= 2.05e+22:
		tmp = t_2 + ((math.sqrt((1.0 + y)) - math.sqrt(y)) - (math.sqrt(x) + t_1))
	else:
		tmp = 1.0 / (t_2 + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(sqrt(z) - sqrt(Float64(1.0 + z)))
	t_2 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (y <= 7.5e-32)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) - t_1) + 2.0);
	elseif (y <= 2.05e+22)
		tmp = Float64(t_2 + Float64(Float64(sqrt(Float64(1.0 + y)) - sqrt(y)) - Float64(sqrt(x) + t_1)));
	else
		tmp = Float64(1.0 / Float64(t_2 + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt(z) - sqrt((1.0 + z));
	t_2 = sqrt((x + 1.0));
	tmp = 0.0;
	if (y <= 7.5e-32)
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) - t_1) + 2.0;
	elseif (y <= 2.05e+22)
		tmp = t_2 + ((sqrt((1.0 + y)) - sqrt(y)) - (sqrt(x) + t_1));
	else
		tmp = 1.0 / (t_2 + sqrt(x));
	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[z], $MachinePrecision] - N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 7.5e-32], N[(N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 2.05e+22], N[(t$95$2 + N[(N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$2 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.49999999999999953e-32

   1. Initial program 96.9%

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

      1. associate-+l+ 96.9%

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

      2. associate-+l- 63.7%

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

      3. +-commutative 63.7%

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

      4. sub-neg 63.7%

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

      5. sub-neg 63.7%

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

      6. +-commutative 63.7%

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

      7. +-commutative 63.7%

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

   3. Simplified 63.7%

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

   4. Taylor expanded in x around 0 60.8%

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

   5. Taylor expanded in y around 0 60.8%

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

   if 7.49999999999999953e-32 < y < 2.0499999999999999e22

   1. Initial program 88.4%

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

      1. associate-+l+ 88.4%

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

      2. +-commutative 88.4%

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

      3. associate-+r- 49.9%

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

      4. associate-+l- 44.4%

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

      5. +-commutative 44.4%

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

      6. associate--l+ 44.4%

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

      7. +-commutative 44.4%

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

   3. Simplified 35.7%

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

   4. Taylor expanded in t around inf 30.4%

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

      1. +-commutative 30.4%

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

      2. +-commutative 30.4%

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

      3. associate--l+ 31.9%

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

   6. Simplified 31.9%

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

   if 2.0499999999999999e22 < y

   1. Initial program 86.4%

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

      1. associate-+l+ 86.4%

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

      2. +-commutative 86.4%

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

      3. associate-+r- 86.4%

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

      4. associate-+l- 52.8%

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

      5. +-commutative 52.8%

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

      6. associate--l+ 52.8%

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

      7. +-commutative 52.8%

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

   3. Simplified 35.6%

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

   4. Taylor expanded in t around inf 3.2%

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

      1. associate--l+ 4.5%

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

      2. +-commutative 4.5%

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

      3. +-commutative 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in z around inf 3.2%

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

      1. +-commutative 3.2%

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

      2. associate--l+ 19.7%

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

      3. +-commutative 19.7%

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

   9. Simplified 19.7%

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

   10. Taylor expanded in y around inf 20.1%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 20.1%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 20.3%

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

       3. add-sqr-sqrt 20.1%

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

   12. Applied egg-rr 20.1%

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

       1. associate--l+ 24.2%

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

       2. +-inverses 24.2%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 24.2%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 24.2%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   14. Simplified 24.2%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \left(\sqrt{z} - \sqrt{1 + z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{x + 1} + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) - \left(\sqrt{x} + \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 5: 95.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 70000000000:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 70000000000.0)
   (+
    (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))
    (- (- (sqrt (+ 1.0 t)) (sqrt t)) (- (sqrt z) (sqrt (+ 1.0 z)))))
   (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 70000000000.0) {
		tmp = (1.0 + (sqrt((1.0 + y)) - sqrt(y))) + ((sqrt((1.0 + t)) - sqrt(t)) - (sqrt(z) - sqrt((1.0 + z))));
	} else {
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	}
	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) :: tmp
    if (y <= 70000000000.0d0) then
        tmp = (1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))) + ((sqrt((1.0d0 + t)) - sqrt(t)) - (sqrt(z) - sqrt((1.0d0 + z))))
    else
        tmp = 1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))
    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 tmp;
	if (y <= 70000000000.0) {
		tmp = (1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y))) + ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) - (Math.sqrt(z) - Math.sqrt((1.0 + z))));
	} else {
		tmp = 1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 70000000000.0:
		tmp = (1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))) + ((math.sqrt((1.0 + t)) - math.sqrt(t)) - (math.sqrt(z) - math.sqrt((1.0 + z))))
	else:
		tmp = 1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 70000000000.0)
		tmp = Float64(Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))) + Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) - Float64(sqrt(z) - sqrt(Float64(1.0 + z)))));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 70000000000.0)
		tmp = (1.0 + (sqrt((1.0 + y)) - sqrt(y))) + ((sqrt((1.0 + t)) - sqrt(t)) - (sqrt(z) - sqrt((1.0 + z))));
	else
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	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_] := If[LessEqual[y, 70000000000.0], N[(N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7e10

   1. Initial program 96.1%

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

      1. associate-+l+ 96.1%

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

      2. associate-+l- 60.9%

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

      3. +-commutative 60.9%

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

      4. sub-neg 60.9%

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

      5. sub-neg 60.9%

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

      6. +-commutative 60.9%

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

      7. +-commutative 60.9%

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

   3. Simplified 60.9%

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

   4. Taylor expanded in x around 0 57.3%

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

      1. associate--l+ 57.3%

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

   6. Simplified 57.3%

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

   if 7e10 < y

   1. Initial program 85.3%

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

      1. associate-+l+ 85.3%

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

      2. +-commutative 85.3%

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

      3. associate-+r- 85.0%

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

      4. associate-+l- 53.1%

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

      5. +-commutative 53.1%

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

      6. associate--l+ 53.1%

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

      7. +-commutative 53.1%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in t around inf 4.8%

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

      1. associate--l+ 6.1%

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

      2. +-commutative 6.1%

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

      3. +-commutative 6.1%

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

   6. Simplified 6.1%

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

   7. Taylor expanded in z around inf 3.8%

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

      1. +-commutative 3.8%

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

      2. associate--l+ 19.5%

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

      3. +-commutative 19.5%

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

   9. Simplified 19.5%

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

   10. Taylor expanded in y around inf 19.9%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 19.9%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 20.1%

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

       3. add-sqr-sqrt 19.9%

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

   12. Applied egg-rr 19.9%

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

       1. associate--l+ 23.8%

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

       2. +-inverses 23.8%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 23.8%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 23.8%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   14. Simplified 23.8%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 70000000000:\\ \;\;\;\;\left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \left(\sqrt{z} - \sqrt{1 + z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 6: 86.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 5 \cdot 10^{+18}:\\ \;\;\;\;1 - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) - t_1\right) - \sqrt{1 + z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\\ \end{array} \end{array} \]

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 y))))
   (if (<= z 5e+18)
     (- 1.0 (- (- (+ (sqrt y) (sqrt z)) t_1) (sqrt (+ 1.0 z))))
     (+ (- t_1 (sqrt y)) (- (sqrt (+ x 1.0)) (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 + y));
	double tmp;
	if (z <= 5e+18) {
		tmp = 1.0 - (((sqrt(y) + sqrt(z)) - t_1) - sqrt((1.0 + z)));
	} else {
		tmp = (t_1 - sqrt(y)) + (sqrt((x + 1.0)) - sqrt(x));
	}
	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((1.0d0 + y))
    if (z <= 5d+18) then
        tmp = 1.0d0 - (((sqrt(y) + sqrt(z)) - t_1) - sqrt((1.0d0 + z)))
    else
        tmp = (t_1 - sqrt(y)) + (sqrt((x + 1.0d0)) - sqrt(x))
    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((1.0 + y));
	double tmp;
	if (z <= 5e+18) {
		tmp = 1.0 - (((Math.sqrt(y) + Math.sqrt(z)) - t_1) - Math.sqrt((1.0 + z)));
	} else {
		tmp = (t_1 - Math.sqrt(y)) + (Math.sqrt((x + 1.0)) - Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((1.0 + y))
	tmp = 0
	if z <= 5e+18:
		tmp = 1.0 - (((math.sqrt(y) + math.sqrt(z)) - t_1) - math.sqrt((1.0 + z)))
	else:
		tmp = (t_1 - math.sqrt(y)) + (math.sqrt((x + 1.0)) - math.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 + y))
	tmp = 0.0
	if (z <= 5e+18)
		tmp = Float64(1.0 - Float64(Float64(Float64(sqrt(y) + sqrt(z)) - t_1) - sqrt(Float64(1.0 + z))));
	else
		tmp = Float64(Float64(t_1 - sqrt(y)) + Float64(sqrt(Float64(x + 1.0)) - sqrt(x)));
	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((1.0 + y));
	tmp = 0.0;
	if (z <= 5e+18)
		tmp = 1.0 - (((sqrt(y) + sqrt(z)) - t_1) - sqrt((1.0 + z)));
	else
		tmp = (t_1 - sqrt(y)) + (sqrt((x + 1.0)) - sqrt(x));
	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[(1.0 + y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 5e+18], N[(1.0 - N[(N[(N[(N[Sqrt[y], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] - N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[Sqrt[y], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 5e18

   1. Initial program 95.4%

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

      1. associate-+l+ 95.4%

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

      2. +-commutative 95.4%

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

      3. associate-+r- 76.6%

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

      4. associate-+l- 52.7%

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

      5. +-commutative 52.7%

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

      6. associate--l+ 52.7%

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

      7. +-commutative 52.7%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in t around inf 21.7%

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

      1. associate--l+ 25.9%

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

      2. +-commutative 25.9%

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

      3. +-commutative 25.9%

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

   6. Simplified 25.9%

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

   7. Taylor expanded in x around 0 35.7%

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

      1. associate--l+ 47.3%

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

      2. +-commutative 47.3%

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

      3. associate--l+ 57.1%

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

   9. Simplified 57.1%

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

   if 5e18 < z

   1. Initial program 87.8%

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

      1. associate-+l+ 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-+r- 63.7%

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

      4. associate-+l- 57.0%

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

      5. +-commutative 57.0%

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

      6. associate--l+ 56.9%

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

      7. +-commutative 56.9%

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

   3. Simplified 19.0%

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

   4. Taylor expanded in t around inf 4.4%

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

      1. associate--l+ 25.2%

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

      2. +-commutative 25.2%

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

      3. +-commutative 25.2%

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

   6. Simplified 25.2%

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

   7. Taylor expanded in z around inf 27.2%

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

      1. +-commutative 27.2%

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

      2. associate--l+ 36.4%

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

      3. +-commutative 36.4%

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

   9. Simplified 36.4%

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

       1. expm1-log1p-u 33.0%

          \[\leadsto \sqrt{1 + x} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)} \]

   11. Applied egg-rr 33.0%

       \[\leadsto \sqrt{1 + x} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + y} - \left(\sqrt{y} + \sqrt{x}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 36.4%

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

       2. flip-- 36.6%

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

       3. add-sqr-sqrt 30.8%

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

       4. +-commutative 30.8%

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

       5. +-commutative 30.8%

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

       6. +-commutative 30.8%

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

       7. unpow2 30.8%

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

       8. associate-+r- 30.8%

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

       9. +-commutative 30.8%

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

       10. *-un-lft-identity 30.8%

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

   13. Applied egg-rr 37.3%

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

       1. *-lft-identity 37.3%

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

       2. +-commutative 37.3%

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

       3. sub-neg 37.3%

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

       4. associate-+l+ 53.2%

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

       5. +-commutative 53.2%

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

       6. sub-neg 53.2%

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

   15. Simplified 53.2%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+18}:\\ \;\;\;\;1 - \left(\left(\left(\sqrt{y} + \sqrt{z}\right) - \sqrt{1 + y}\right) - \sqrt{1 + z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\\ \end{array} \]

Alternative 7: 95.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-20}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \left(\sqrt{z} - \sqrt{1 + z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+26}:\\ \;\;\;\;1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.8e-20)
   (+ (- (- (sqrt (+ 1.0 t)) (sqrt t)) (- (sqrt z) (sqrt (+ 1.0 z)))) 2.0)
   (if (<= y 1.12e+26)
     (+ 1.0 (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))
     (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.8e-20) {
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) - (sqrt(z) - sqrt((1.0 + z)))) + 2.0;
	} else if (y <= 1.12e+26) {
		tmp = 1.0 + (1.0 / (sqrt((1.0 + y)) + sqrt(y)));
	} else {
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	}
	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) :: tmp
    if (y <= 3.8d-20) then
        tmp = ((sqrt((1.0d0 + t)) - sqrt(t)) - (sqrt(z) - sqrt((1.0d0 + z)))) + 2.0d0
    else if (y <= 1.12d+26) then
        tmp = 1.0d0 + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y)))
    else
        tmp = 1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))
    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 tmp;
	if (y <= 3.8e-20) {
		tmp = ((Math.sqrt((1.0 + t)) - Math.sqrt(t)) - (Math.sqrt(z) - Math.sqrt((1.0 + z)))) + 2.0;
	} else if (y <= 1.12e+26) {
		tmp = 1.0 + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 3.8e-20:
		tmp = ((math.sqrt((1.0 + t)) - math.sqrt(t)) - (math.sqrt(z) - math.sqrt((1.0 + z)))) + 2.0
	elif y <= 1.12e+26:
		tmp = 1.0 + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y)))
	else:
		tmp = 1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.8e-20)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 + t)) - sqrt(t)) - Float64(sqrt(z) - sqrt(Float64(1.0 + z)))) + 2.0);
	elseif (y <= 1.12e+26)
		tmp = Float64(1.0 + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.8e-20)
		tmp = ((sqrt((1.0 + t)) - sqrt(t)) - (sqrt(z) - sqrt((1.0 + z)))) + 2.0;
	elseif (y <= 1.12e+26)
		tmp = 1.0 + (1.0 / (sqrt((1.0 + y)) + sqrt(y)));
	else
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	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_] := If[LessEqual[y, 3.8e-20], N[(N[(N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[z], $MachinePrecision] - N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], If[LessEqual[y, 1.12e+26], N[(1.0 + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.7999999999999998e-20

   1. Initial program 96.9%

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

      1. associate-+l+ 96.9%

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

      2. associate-+l- 62.7%

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

      3. +-commutative 62.7%

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

      4. sub-neg 62.7%

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

      5. sub-neg 62.7%

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

      6. +-commutative 62.7%

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

      7. +-commutative 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in x around 0 59.1%

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

   5. Taylor expanded in y around 0 59.1%

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

   if 3.7999999999999998e-20 < y < 1.1200000000000001e26

   1. Initial program 86.1%

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

      1. associate-+l+ 86.1%

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

      2. +-commutative 86.1%

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

      3. associate-+r- 53.7%

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

      4. associate-+l- 43.8%

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

      5. +-commutative 43.8%

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

      6. associate--l+ 43.8%

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

      7. +-commutative 43.8%

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

   3. Simplified 33.8%

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

   4. Taylor expanded in t around inf 18.0%

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

      1. associate--l+ 29.7%

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

      2. +-commutative 29.7%

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

      3. +-commutative 29.7%

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

   6. Simplified 29.7%

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

   7. Taylor expanded in z around inf 20.8%

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

      1. +-commutative 20.8%

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

      2. associate--l+ 20.8%

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

      3. +-commutative 20.8%

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

   9. Simplified 20.8%

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

       1. flip-- 20.9%

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

       2. add-sqr-sqrt 20.8%

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

   11. Applied egg-rr 20.8%

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

       1. +-commutative 20.8%

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

       2. associate--l+ 20.8%

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

       3. unpow2 20.8%

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

       4. +-commutative 20.8%

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

       5. +-commutative 20.8%

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

   13. Simplified 20.8%

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

   14. Taylor expanded in x around 0 43.1%

       \[\leadsto \color{blue}{1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}} \]

   if 1.1200000000000001e26 < y

   1. Initial program 86.5%

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

      1. associate-+l+ 86.5%

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

      2. +-commutative 86.5%

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

      3. associate-+r- 86.5%

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

      4. associate-+l- 53.9%

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

      5. +-commutative 53.9%

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

      6. associate--l+ 53.9%

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

      7. +-commutative 53.9%

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

   3. Simplified 36.3%

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

   4. Taylor expanded in t around inf 3.2%

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

      1. associate--l+ 4.4%

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

      2. +-commutative 4.4%

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

      3. +-commutative 4.4%

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

   6. Simplified 4.4%

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

   7. Taylor expanded in z around inf 3.2%

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

      1. +-commutative 3.2%

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

      2. associate--l+ 20.1%

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

      3. +-commutative 20.1%

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

   9. Simplified 20.1%

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

   10. Taylor expanded in y around inf 20.5%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 20.5%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 20.6%

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

       3. add-sqr-sqrt 20.5%

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

   12. Applied egg-rr 20.5%

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

       1. associate--l+ 24.5%

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

       2. +-inverses 24.5%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 24.5%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 24.5%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   14. Simplified 24.5%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-20}:\\ \;\;\;\;\left(\left(\sqrt{1 + t} - \sqrt{t}\right) - \left(\sqrt{z} - \sqrt{1 + z}\right)\right) + 2\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+26}:\\ \;\;\;\;1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 8: 70.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.18 \cdot 10^{-20}:\\ \;\;\;\;1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.18e-20)
   (+ 1.0 (/ 1.0 (+ (sqrt (+ 1.0 y)) (sqrt y))))
   (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.18e-20) {
		tmp = 1.0 + (1.0 / (sqrt((1.0 + y)) + sqrt(y)));
	} else {
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	}
	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) :: tmp
    if (x <= 1.18d-20) then
        tmp = 1.0d0 + (1.0d0 / (sqrt((1.0d0 + y)) + sqrt(y)))
    else
        tmp = 1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))
    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 tmp;
	if (x <= 1.18e-20) {
		tmp = 1.0 + (1.0 / (Math.sqrt((1.0 + y)) + Math.sqrt(y)));
	} else {
		tmp = 1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 1.18e-20:
		tmp = 1.0 + (1.0 / (math.sqrt((1.0 + y)) + math.sqrt(y)))
	else:
		tmp = 1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.18e-20)
		tmp = Float64(1.0 + Float64(1.0 / Float64(sqrt(Float64(1.0 + y)) + sqrt(y))));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.18e-20)
		tmp = 1.0 + (1.0 / (sqrt((1.0 + y)) + sqrt(y)));
	else
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	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_] := If[LessEqual[x, 1.18e-20], N[(1.0 + N[(1.0 / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1800000000000001e-20

   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. Step-by-step derivation

      1. associate-+l+ 97.8%

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

      2. +-commutative 97.8%

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

      3. associate-+r- 97.8%

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

      4. associate-+l- 97.8%

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

      5. +-commutative 97.8%

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

      6. associate--l+ 97.8%

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

      7. +-commutative 97.8%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in t around inf 22.2%

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

      1. associate--l+ 26.9%

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

      2. +-commutative 26.9%

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

      3. +-commutative 26.9%

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

   6. Simplified 26.9%

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

   7. Taylor expanded in z around inf 29.1%

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

      1. +-commutative 29.1%

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

      2. associate--l+ 41.0%

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

      3. +-commutative 41.0%

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

   9. Simplified 41.0%

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

       1. flip-- 41.0%

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

       2. add-sqr-sqrt 34.7%

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

   11. Applied egg-rr 34.7%

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

       1. +-commutative 34.7%

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

       2. associate--l+ 34.7%

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

       3. unpow2 34.7%

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

       4. +-commutative 34.7%

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

       5. +-commutative 34.7%

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

   13. Simplified 34.7%

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

   14. Taylor expanded in x around 0 40.9%

       \[\leadsto \color{blue}{1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}} \]

   if 1.1800000000000001e-20 < x

   1. Initial program 86.6%

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

      1. associate-+l+ 86.6%

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

      2. +-commutative 86.6%

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

      3. associate-+r- 47.2%

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

      4. associate-+l- 17.9%

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

      5. +-commutative 17.9%

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

      6. associate--l+ 17.9%

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

      7. +-commutative 17.9%

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

   3. Simplified 12.1%

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

   4. Taylor expanded in t around inf 6.1%

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

      1. associate--l+ 24.4%

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

      2. +-commutative 24.4%

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

      3. +-commutative 24.4%

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

   6. Simplified 24.4%

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

   7. Taylor expanded in z around inf 7.6%

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

      1. +-commutative 7.6%

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

      2. associate--l+ 9.2%

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

      3. +-commutative 9.2%

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

   9. Simplified 9.2%

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

   10. Taylor expanded in y around inf 6.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 6.1%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 6.4%

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

       3. add-sqr-sqrt 6.1%

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

   12. Applied egg-rr 6.1%

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

       1. associate--l+ 10.8%

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

       2. +-inverses 10.8%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 10.8%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 10.8%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   14. Simplified 10.8%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.18 \cdot 10^{-20}:\\ \;\;\;\;1 + \frac{1}{\sqrt{1 + y} + \sqrt{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 9: 61.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;y \leq 0.78:\\ \;\;\;\;1 + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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 (+ x 1.0)) (sqrt x)))) (if (<= y 0.78) (+ 1.0 t_1) t_1)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (y <= 0.78) {
		tmp = 1.0 + t_1;
	} else {
		tmp = 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((x + 1.0d0)) - sqrt(x)
    if (y <= 0.78d0) then
        tmp = 1.0d0 + t_1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (y <= 0.78) {
		tmp = 1.0 + t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = math.sqrt((x + 1.0)) - math.sqrt(x)
	tmp = 0
	if y <= 0.78:
		tmp = 1.0 + t_1
	else:
		tmp = 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(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (y <= 0.78)
		tmp = Float64(1.0 + t_1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (y <= 0.78)
		tmp = 1.0 + t_1;
	else
		tmp = 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[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.78], N[(1.0 + t$95$1), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if y < 0.78000000000000003

   1. Initial program 96.8%

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

      1. associate-+l+ 96.8%

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

      2. +-commutative 96.8%

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

      3. associate-+r- 61.6%

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

      4. associate-+l- 56.5%

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

      5. +-commutative 56.5%

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

      6. associate--l+ 56.5%

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

      7. +-commutative 56.5%

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

   3. Simplified 34.8%

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

   4. Taylor expanded in t around inf 19.3%

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

      1. associate--l+ 39.0%

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

      2. +-commutative 39.0%

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

      3. +-commutative 39.0%

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

   6. Simplified 39.0%

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

   7. Taylor expanded in z around inf 27.1%

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

      1. +-commutative 27.1%

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

      2. associate--l+ 27.1%

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

      3. +-commutative 27.1%

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

   9. Simplified 27.1%

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

   10. Taylor expanded in y around 0 26.2%

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

       1. associate--l+ 40.6%

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

   12. Simplified 40.6%

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

   if 0.78000000000000003 < y

   1. Initial program 85.1%

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

      1. associate-+l+ 85.1%

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

      2. +-commutative 85.1%

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

      3. associate-+r- 82.4%

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

      4. associate-+l- 52.3%

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

      5. +-commutative 52.3%

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

      6. associate--l+ 52.3%

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

      7. +-commutative 52.3%

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

   3. Simplified 36.1%

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

   4. Taylor expanded in t around inf 5.9%

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

      1. associate--l+ 7.7%

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

      2. +-commutative 7.7%

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

      3. +-commutative 7.7%

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

   6. Simplified 7.7%

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

   7. Taylor expanded in z around inf 4.8%

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

      1. +-commutative 4.8%

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

      2. associate--l+ 19.5%

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

      3. +-commutative 19.5%

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

   9. Simplified 19.5%

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

   10. Taylor expanded in y around inf 19.2%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.78:\\ \;\;\;\;1 + \left(\sqrt{x + 1} - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]

Alternative 10: 65.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 70000000000:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y 70000000000.0)
   (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))
   (- (sqrt (+ x 1.0)) (sqrt x))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 70000000000.0) {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	} else {
		tmp = sqrt((x + 1.0)) - sqrt(x);
	}
	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) :: tmp
    if (y <= 70000000000.0d0) then
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    else
        tmp = sqrt((x + 1.0d0)) - sqrt(x)
    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 tmp;
	if (y <= 70000000000.0) {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	} else {
		tmp = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= 70000000000.0:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	else:
		tmp = math.sqrt((x + 1.0)) - math.sqrt(x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= 70000000000.0)
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) - sqrt(x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 70000000000.0)
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	else
		tmp = sqrt((x + 1.0)) - sqrt(x);
	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_] := If[LessEqual[y, 70000000000.0], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7e10

   1. Initial program 96.1%

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

      1. associate-+l+ 96.1%

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

      2. +-commutative 96.1%

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

      3. associate-+r- 60.9%

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

      4. associate-+l- 55.8%

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

      5. +-commutative 55.8%

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

      6. associate--l+ 55.8%

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

      7. +-commutative 55.8%

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

   3. Simplified 34.4%

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

   4. Taylor expanded in t around inf 19.3%

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

      1. associate--l+ 38.4%

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

      2. +-commutative 38.4%

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

      3. +-commutative 38.4%

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

   6. Simplified 38.4%

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

   7. Taylor expanded in z around inf 26.7%

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

      1. +-commutative 26.7%

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

      2. associate--l+ 26.6%

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

      3. +-commutative 26.6%

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

   9. Simplified 26.6%

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

   10. Taylor expanded in x around 0 46.9%

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

       1. associate--l+ 46.9%

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

   12. Simplified 46.9%

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

   if 7e10 < y

   1. Initial program 85.3%

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

      1. associate-+l+ 85.3%

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

      2. +-commutative 85.3%

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

      3. associate-+r- 85.0%

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

      4. associate-+l- 53.1%

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

      5. +-commutative 53.1%

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

      6. associate--l+ 53.1%

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

      7. +-commutative 53.1%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in t around inf 4.8%

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

      1. associate--l+ 6.1%

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

      2. +-commutative 6.1%

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

      3. +-commutative 6.1%

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

   6. Simplified 6.1%

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

   7. Taylor expanded in z around inf 3.8%

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

      1. +-commutative 3.8%

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

      2. associate--l+ 19.5%

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

      3. +-commutative 19.5%

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

   9. Simplified 19.5%

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

   10. Taylor expanded in y around inf 19.9%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 70000000000:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]

Alternative 11: 69.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.18 \cdot 10^{-20}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.18e-20)
   (+ 1.0 (- (sqrt (+ 1.0 y)) (sqrt y)))
   (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.18e-20) {
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	} else {
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	}
	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) :: tmp
    if (x <= 1.18d-20) then
        tmp = 1.0d0 + (sqrt((1.0d0 + y)) - sqrt(y))
    else
        tmp = 1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))
    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 tmp;
	if (x <= 1.18e-20) {
		tmp = 1.0 + (Math.sqrt((1.0 + y)) - Math.sqrt(y));
	} else {
		tmp = 1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if x <= 1.18e-20:
		tmp = 1.0 + (math.sqrt((1.0 + y)) - math.sqrt(y))
	else:
		tmp = 1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.18e-20)
		tmp = Float64(1.0 + Float64(sqrt(Float64(1.0 + y)) - sqrt(y)));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= 1.18e-20)
		tmp = 1.0 + (sqrt((1.0 + y)) - sqrt(y));
	else
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	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_] := If[LessEqual[x, 1.18e-20], N[(1.0 + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1800000000000001e-20

   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. Step-by-step derivation

      1. associate-+l+ 97.8%

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

      2. +-commutative 97.8%

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

      3. associate-+r- 97.8%

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

      4. associate-+l- 97.8%

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

      5. +-commutative 97.8%

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

      6. associate--l+ 97.8%

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

      7. +-commutative 97.8%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in t around inf 22.2%

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

      1. associate--l+ 26.9%

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

      2. +-commutative 26.9%

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

      3. +-commutative 26.9%

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

   6. Simplified 26.9%

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

   7. Taylor expanded in z around inf 29.1%

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

      1. +-commutative 29.1%

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

      2. associate--l+ 41.0%

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

      3. +-commutative 41.0%

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

   9. Simplified 41.0%

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

   10. Taylor expanded in x around 0 29.1%

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

       1. associate--l+ 41.0%

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

   12. Simplified 41.0%

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

   if 1.1800000000000001e-20 < x

   1. Initial program 86.6%

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

      1. associate-+l+ 86.6%

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

      2. +-commutative 86.6%

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

      3. associate-+r- 47.2%

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

      4. associate-+l- 17.9%

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

      5. +-commutative 17.9%

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

      6. associate--l+ 17.9%

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

      7. +-commutative 17.9%

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

   3. Simplified 12.1%

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

   4. Taylor expanded in t around inf 6.1%

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

      1. associate--l+ 24.4%

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

      2. +-commutative 24.4%

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

      3. +-commutative 24.4%

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

   6. Simplified 24.4%

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

   7. Taylor expanded in z around inf 7.6%

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

      1. +-commutative 7.6%

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

      2. associate--l+ 9.2%

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

      3. +-commutative 9.2%

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

   9. Simplified 9.2%

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

   10. Taylor expanded in y around inf 6.0%

       \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]
   11. Step-by-step derivation

       1. flip-- 6.1%

          \[\leadsto \color{blue}{\frac{\sqrt{1 + x} \cdot \sqrt{1 + x} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}}} \]

       2. add-sqr-sqrt 6.4%

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

       3. add-sqr-sqrt 6.1%

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

   12. Applied egg-rr 6.1%

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

       1. associate--l+ 10.8%

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

       2. +-inverses 10.8%

          \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{1 + x} + \sqrt{x}} \]

       3. metadata-eval 10.8%

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{1 + x} + \sqrt{x}} \]

       4. +-commutative 10.8%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{x} + \sqrt{1 + x}}} \]

   14. Simplified 10.8%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x} + \sqrt{1 + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.18 \cdot 10^{-20}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \]

Alternative 12: 35.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \sqrt{x + 1} - \sqrt{x} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt((x + 1.0d0)) - sqrt(x)
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = sqrt((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[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.8%

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

   1. associate-+l+ 91.8%

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

   2. +-commutative 91.8%

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

   3. associate-+r- 70.5%

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

   4. associate-+l- 54.7%

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

   5. +-commutative 54.7%

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

   6. associate--l+ 54.7%

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

   7. +-commutative 54.7%

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

3. Simplified 35.3%

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

4. Taylor expanded in t around inf 13.5%

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

   1. associate--l+ 25.6%

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

   2. +-commutative 25.6%

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

   3. +-commutative 25.6%

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

6. Simplified 25.6%

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

7. Taylor expanded in z around inf 17.5%

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

   1. +-commutative 17.5%

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

   2. associate--l+ 23.8%

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

   3. +-commutative 23.8%

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

9. Simplified 23.8%

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

10. Taylor expanded in y around inf 14.6%

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

11. Final simplification 14.6%

    \[\leadsto \sqrt{x + 1} - \sqrt{x} \]

Alternative 13: 34.5% accurate, 823.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ 1 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 1.0)

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return 1.0;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return 1.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return 1.0
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 91.8%

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

   1. associate-+l+ 91.8%

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

   2. +-commutative 91.8%

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

   3. associate-+r- 70.5%

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

   4. associate-+l- 54.7%

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

   5. +-commutative 54.7%

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

   6. associate--l+ 54.7%

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

   7. +-commutative 54.7%

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

3. Simplified 35.3%

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

4. Taylor expanded in t around inf 13.5%

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

   1. associate--l+ 25.6%

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

   2. +-commutative 25.6%

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

   3. +-commutative 25.6%

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

6. Simplified 25.6%

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

7. Taylor expanded in z around inf 17.5%

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

   1. +-commutative 17.5%

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

   2. associate--l+ 23.8%

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

   3. +-commutative 23.8%

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

9. Simplified 23.8%

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

10. Taylor expanded in y around inf 14.6%

    \[\leadsto \color{blue}{\sqrt{1 + x} - \sqrt{x}} \]

11. Taylor expanded in x around 0 31.5%

    \[\leadsto \color{blue}{1} \]

12. Final simplification 31.5%

    \[\leadsto 1 \]

Developer target: 99.4% accurate, 1.0× 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 2023249 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :herbie-target
  (+ (+ (+ (/ 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)))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))