Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Percentage Accurate: 82.6% → 95.0%

Time: 14.1s

Alternatives: 15

Speedup: 1.0×

• 25.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))

C

double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Python

def code(x, y, z):
	return (x * y) / ((z * z) * (z + 1.0))

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))

C

double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Python

def code(x, y, z):
	return (x * y) / ((z * z) * (z + 1.0))

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;t_0 \leq 10^{-291}:\\ \;\;\;\;\frac{1}{z \cdot \frac{\frac{z}{x}}{y}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (if (<= t_0 -2e+42)
     (/ (/ x z) (* z (/ z y)))
     (if (<= t_0 1e-291)
       (/ 1.0 (* z (/ (/ z x) y)))
       (if (<= t_0 5e+151)
         (* (/ y (* z z)) (/ x (+ z 1.0)))
         (/ (/ y z) (* z (/ z x))))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -2e+42) {
		tmp = (x / z) / (z * (z / y));
	} else if (t_0 <= 1e-291) {
		tmp = 1.0 / (z * ((z / x) / y));
	} else if (t_0 <= 5e+151) {
		tmp = (y / (z * z)) * (x / (z + 1.0));
	} else {
		tmp = (y / z) / (z * (z / x));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z + 1.0d0) * (z * z)
    if (t_0 <= (-2d+42)) then
        tmp = (x / z) / (z * (z / y))
    else if (t_0 <= 1d-291) then
        tmp = 1.0d0 / (z * ((z / x) / y))
    else if (t_0 <= 5d+151) then
        tmp = (y / (z * z)) * (x / (z + 1.0d0))
    else
        tmp = (y / z) / (z * (z / x))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -2e+42) {
		tmp = (x / z) / (z * (z / y));
	} else if (t_0 <= 1e-291) {
		tmp = 1.0 / (z * ((z / x) / y));
	} else if (t_0 <= 5e+151) {
		tmp = (y / (z * z)) * (x / (z + 1.0));
	} else {
		tmp = (y / z) / (z * (z / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = (z + 1.0) * (z * z)
	tmp = 0
	if t_0 <= -2e+42:
		tmp = (x / z) / (z * (z / y))
	elif t_0 <= 1e-291:
		tmp = 1.0 / (z * ((z / x) / y))
	elif t_0 <= 5e+151:
		tmp = (y / (z * z)) * (x / (z + 1.0))
	else:
		tmp = (y / z) / (z * (z / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -2e+42)
		tmp = Float64(Float64(x / z) / Float64(z * Float64(z / y)));
	elseif (t_0 <= 1e-291)
		tmp = Float64(1.0 / Float64(z * Float64(Float64(z / x) / y)));
	elseif (t_0 <= 5e+151)
		tmp = Float64(Float64(y / Float64(z * z)) * Float64(x / Float64(z + 1.0)));
	else
		tmp = Float64(Float64(y / z) / Float64(z * Float64(z / x)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_0 <= -2e+42)
		tmp = (x / z) / (z * (z / y));
	elseif (t_0 <= 1e-291)
		tmp = 1.0 / (z * ((z / x) / y));
	elseif (t_0 <= 5e+151)
		tmp = (y / (z * z)) * (x / (z + 1.0));
	else
		tmp = (y / z) / (z * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+42], N[(N[(x / z), $MachinePrecision] / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-291], N[(1.0 / N[(z * N[(N[(z / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+151], N[(N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < -2.00000000000000009e42

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. associate-*r/ 85.5%

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

      3. sqr-neg 85.5%

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

      4. associate-*l* 85.5%

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

      5. associate-*l* 85.5%

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

      6. sqr-neg 85.5%

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

      7. associate-*l* 85.5%

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

      8. distribute-lft-in 85.5%

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

      9. fma-def 85.5%

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

      10. *-rgt-identity 85.5%

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

   3. Simplified 85.5%

      \[\leadsto \color{blue}{y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 79.7%

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

      2. *-commutative 79.7%

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

      3. fma-udef 79.7%

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

      4. distribute-lft-in 29.7%

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

      5. *-un-lft-identity 29.7%

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

      6. distribute-rgt-in 79.7%

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

      7. associate-/r* 83.1%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}} \]

      8. frac-times 94.5%

         \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot \frac{y}{z}}}{z + 1} \]

      9. associate-/l* 92.8%

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

      10. div-inv 92.9%

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

      11. clear-num 92.9%

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

   5. Applied egg-rr 92.9%

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

   6. Taylor expanded in z around inf 91.5%

      \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\frac{{z}^{2}}{y}}} \]
   7. Step-by-step derivation

      1. unpow2 91.5%

         \[\leadsto \frac{\frac{x}{z}}{\frac{\color{blue}{z \cdot z}}{y}} \]

      2. associate-/l* 92.8%

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\frac{z}{\frac{y}{z}}}} \]

   8. Simplified 92.8%

      \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\frac{z}{\frac{y}{z}}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 92.9%

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\frac{z}{y} \cdot z}} \]

   10. Applied egg-rr 92.9%

       \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\frac{z}{y} \cdot z}} \]

   if -2.00000000000000009e42 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 9.99999999999999962e-292

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. associate-*r/ 78.4%

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

      3. sqr-neg 78.4%

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

      4. associate-*l* 78.4%

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

      5. associate-*l* 78.4%

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

      6. sqr-neg 78.4%

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

      7. associate-*l* 78.4%

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

      8. distribute-lft-in 78.4%

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

      9. fma-def 78.4%

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

      10. *-rgt-identity 78.4%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in z around 0 79.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 79.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]

      2. *-rgt-identity 79.8%

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

      3. times-frac 78.4%

         \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{1}} \]

      4. /-rgt-identity 78.4%

         \[\leadsto \frac{x}{z \cdot z} \cdot \color{blue}{y} \]

   6. Simplified 78.4%

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot y} \]
   7. Step-by-step derivation

      1. associate-*l/ 79.8%

         \[\leadsto \color{blue}{\frac{x \cdot y}{z \cdot z}} \]

      2. associate-/r* 93.3%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z}}{z}} \]

      3. clear-num 93.3%

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{z}{x \cdot y}}}}{z} \]

      4. add-sqr-sqrt 42.5%

         \[\leadsto \frac{\frac{1}{\frac{z}{x \cdot y}}}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}} \]

      5. sqrt-prod 42.5%

         \[\leadsto \frac{\frac{1}{\frac{z}{x \cdot y}}}{\color{blue}{\sqrt{z \cdot z}}} \]

      6. sqr-neg 42.5%

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

      7. sqrt-unprod 1.8%

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

      8. add-sqr-sqrt 2.0%

         \[\leadsto \frac{\frac{1}{\frac{z}{x \cdot y}}}{\color{blue}{-z}} \]

      9. associate-/l/ 2.0%

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

      10. add-sqr-sqrt 1.8%

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

      11. sqrt-unprod 42.5%

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

      12. sqr-neg 42.5%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{z \cdot z}} \cdot \frac{z}{x \cdot y}} \]

      13. sqrt-prod 42.6%

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

      14. add-sqr-sqrt 93.3%

          \[\leadsto \frac{1}{\color{blue}{z} \cdot \frac{z}{x \cdot y}} \]

      15. associate-/r* 99.8%

          \[\leadsto \frac{1}{z \cdot \color{blue}{\frac{\frac{z}{x}}{y}}} \]

   8. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{z \cdot \frac{\frac{z}{x}}{y}}} \]

   if 9.99999999999999962e-292 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 5.0000000000000002e151

   1. Initial program 93.1%

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

      1. *-commutative 93.1%

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

      2. sqr-neg 93.1%

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

      3. times-frac 92.6%

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

      4. sqr-neg 92.6%

         \[\leadsto \frac{y}{\color{blue}{z \cdot z}} \cdot \frac{x}{z + 1} \]

   3. Simplified 92.6%

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

   if 5.0000000000000002e151 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

      2. associate-*r/ 75.6%

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

      3. sqr-neg 75.6%

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

      4. associate-*l* 75.6%

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

      5. associate-*l* 75.6%

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

      6. sqr-neg 75.6%

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

      7. associate-*l* 75.6%

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

      8. distribute-lft-in 75.6%

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

      9. fma-def 75.6%

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

      10. *-rgt-identity 75.6%

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

   3. Simplified 75.6%

      \[\leadsto \color{blue}{y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 74.9%

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

      2. *-commutative 74.9%

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

      3. associate-/r* 78.5%

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

      4. associate-*r/ 82.6%

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

      5. fma-udef 82.6%

         \[\leadsto \frac{x \cdot \frac{y}{z}}{\color{blue}{z \cdot z + z}} \]

      6. distribute-lft1-in 82.6%

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

      7. frac-times 99.7%

         \[\leadsto \color{blue}{\frac{x}{z + 1} \cdot \frac{\frac{y}{z}}{z}} \]

      8. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{z + 1}{x}}} \cdot \frac{\frac{y}{z}}{z} \]

      9. frac-times 96.4%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{z}}{\frac{z + 1}{x} \cdot z}} \]

      10. *-un-lft-identity 96.4%

          \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{\frac{z + 1}{x} \cdot z} \]

   5. Applied egg-rr 96.4%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{\frac{z + 1}{x} \cdot z}} \]

   6. Taylor expanded in z around inf 84.4%

      \[\leadsto \frac{\frac{y}{z}}{\color{blue}{\frac{{z}^{2}}{x}}} \]
   7. Step-by-step derivation

      1. unpow2 84.4%

         \[\leadsto \frac{\frac{y}{z}}{\frac{\color{blue}{z \cdot z}}{x}} \]

      2. associate-*r/ 96.4%

         \[\leadsto \frac{\frac{y}{z}}{\color{blue}{z \cdot \frac{z}{x}}} \]

   8. Simplified 96.4%

      \[\leadsto \frac{\frac{y}{z}}{\color{blue}{z \cdot \frac{z}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{-291}:\\ \;\;\;\;\frac{1}{z \cdot \frac{\frac{z}{x}}{y}}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 2: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-65} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{+19}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x y) -1e-65) (not (<= (* x y) 4e+19)))
   (* y (/ x (* z z)))
   (* (/ x z) (/ y z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (((x * y) <= -1e-65) || !((x * y) <= 4e+19)) {
		tmp = y * (x / (z * z));
	} else {
		tmp = (x / z) * (y / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x * y) <= (-1d-65)) .or. (.not. ((x * y) <= 4d+19))) then
        tmp = y * (x / (z * z))
    else
        tmp = (x / z) * (y / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (((x * y) <= -1e-65) || !((x * y) <= 4e+19)) {
		tmp = y * (x / (z * z));
	} else {
		tmp = (x / z) * (y / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if ((x * y) <= -1e-65) or not ((x * y) <= 4e+19):
		tmp = y * (x / (z * z))
	else:
		tmp = (x / z) * (y / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * y) <= -1e-65) || !(Float64(x * y) <= 4e+19))
		tmp = Float64(y * Float64(x / Float64(z * z)));
	else
		tmp = Float64(Float64(x / z) * Float64(y / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * y) <= -1e-65) || ~(((x * y) <= 4e+19)))
		tmp = y * (x / (z * z));
	else
		tmp = (x / z) * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e-65], N[Not[LessEqual[N[(x * y), $MachinePrecision], 4e+19]], $MachinePrecision]], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -9.99999999999999923e-66 or 4e19 < (*.f64 x y)

   1. Initial program 82.5%

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

      1. *-commutative 82.5%

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

      2. associate-*r/ 84.9%

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

      3. sqr-neg 84.9%

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

      4. associate-*l* 84.9%

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

      5. associate-*l* 84.9%

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

      6. sqr-neg 84.9%

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

      7. associate-*l* 84.9%

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

      8. distribute-lft-in 84.9%

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

      9. fma-def 84.9%

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

      10. *-rgt-identity 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in z around 0 71.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 71.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]

      2. *-rgt-identity 71.8%

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

      3. times-frac 72.9%

         \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{1}} \]

      4. /-rgt-identity 72.9%

         \[\leadsto \frac{x}{z \cdot z} \cdot \color{blue}{y} \]

   6. Simplified 72.9%

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot y} \]

   if -9.99999999999999923e-66 < (*.f64 x y) < 4e19

   1. Initial program 84.0%

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

      1. *-commutative 84.0%

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

      2. associate-*r/ 87.6%

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

      3. sqr-neg 87.6%

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

      4. associate-*l* 87.6%

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

      5. associate-*l* 87.6%

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

      6. sqr-neg 87.6%

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

      7. associate-*l* 87.6%

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

      8. distribute-lft-in 87.6%

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

      9. fma-def 87.6%

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

      10. *-rgt-identity 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in z around 0 78.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 78.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]

      2. times-frac 94.6%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z}} \]

   6. Simplified 94.6%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-65} \lor \neg \left(x \cdot y \leq 4 \cdot 10^{+19}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 3: 93.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -44000000000 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{x}{z} - x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -44000000000.0) (not (<= z 0.75)))
   (* (/ x z) (/ y (* z z)))
   (* (/ y z) (- (/ x z) x))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -44000000000.0) || !(z <= 0.75)) {
		tmp = (x / z) * (y / (z * z));
	} else {
		tmp = (y / z) * ((x / z) - x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-44000000000.0d0)) .or. (.not. (z <= 0.75d0))) then
        tmp = (x / z) * (y / (z * z))
    else
        tmp = (y / z) * ((x / z) - x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -44000000000.0) || !(z <= 0.75)) {
		tmp = (x / z) * (y / (z * z));
	} else {
		tmp = (y / z) * ((x / z) - x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -44000000000.0) or not (z <= 0.75):
		tmp = (x / z) * (y / (z * z))
	else:
		tmp = (y / z) * ((x / z) - x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -44000000000.0) || !(z <= 0.75))
		tmp = Float64(Float64(x / z) * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(y / z) * Float64(Float64(x / z) - x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -44000000000.0) || ~((z <= 0.75)))
		tmp = (x / z) * (y / (z * z));
	else
		tmp = (y / z) * ((x / z) - x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[z, -44000000000.0], N[Not[LessEqual[z, 0.75]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4e10 or 0.75 < z

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. sqr-neg 79.4%

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

      3. times-frac 90.0%

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

      4. sqr-neg 90.0%

         \[\leadsto \frac{y}{\color{blue}{z \cdot z}} \cdot \frac{x}{z + 1} \]

   3. Simplified 90.0%

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

   4. Taylor expanded in z around inf 89.4%

      \[\leadsto \frac{y}{z \cdot z} \cdot \color{blue}{\frac{x}{z}} \]

   if -4.4e10 < z < 0.75

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

      2. associate-*r/ 89.5%

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

      3. sqr-neg 89.5%

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

      4. associate-*l* 89.5%

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

      5. associate-*l* 89.5%

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

      6. sqr-neg 89.5%

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

      7. associate-*l* 89.5%

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

      8. distribute-lft-in 89.5%

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

      9. fma-def 89.5%

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

      10. *-rgt-identity 89.5%

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

   3. Simplified 89.5%

      \[\leadsto \color{blue}{y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 86.6%

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

      2. *-commutative 86.6%

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

      3. fma-udef 86.6%

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

      4. distribute-lft-in 86.6%

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

      5. *-un-lft-identity 86.6%

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

      6. distribute-rgt-in 86.6%

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

      7. associate-/r* 86.6%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}} \]

      8. frac-times 96.3%

         \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot \frac{y}{z}}}{z + 1} \]

      9. associate-/l* 96.2%

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

      10. div-inv 96.2%

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

      11. clear-num 97.2%

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

   5. Applied egg-rr 97.2%

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

   6. Taylor expanded in z around 0 63.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z} + \frac{x \cdot y}{{z}^{2}}} \]
   7. Step-by-step derivation

      1. +-commutative 63.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}} + -1 \cdot \frac{x \cdot y}{z}} \]

      2. unpow2 63.3%

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

      3. times-frac 72.9%

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

      4. mul-1-neg 72.9%

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

      5. associate-*r/ 72.0%

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

      6. distribute-lft-neg-out 72.0%

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

      7. fma-def 72.0%

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

      8. distribute-lft-neg-out 72.0%

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

      9. *-commutative 72.0%

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

      10. associate-*l/ 72.9%

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

      11. fma-neg 72.9%

          \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z} - \frac{y \cdot x}{z}} \]

      12. associate-*l/ 72.0%

          \[\leadsto \frac{x}{z} \cdot \frac{y}{z} - \color{blue}{\frac{y}{z} \cdot x} \]

      13. *-commutative 72.0%

          \[\leadsto \frac{x}{z} \cdot \frac{y}{z} - \color{blue}{x \cdot \frac{y}{z}} \]

      14. distribute-rgt-out-- 94.7%

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

   8. Simplified 94.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -44000000000 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{x}{z} - x\right)\\ \end{array} \]

Alternative 4: 94.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -44000000000 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{x}{z} - x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -44000000000.0) (not (<= z 0.75)))
   (/ (/ x z) (* z (/ z y)))
   (* (/ y z) (- (/ x z) x))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -44000000000.0) || !(z <= 0.75)) {
		tmp = (x / z) / (z * (z / y));
	} else {
		tmp = (y / z) * ((x / z) - x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-44000000000.0d0)) .or. (.not. (z <= 0.75d0))) then
        tmp = (x / z) / (z * (z / y))
    else
        tmp = (y / z) * ((x / z) - x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -44000000000.0) || !(z <= 0.75)) {
		tmp = (x / z) / (z * (z / y));
	} else {
		tmp = (y / z) * ((x / z) - x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -44000000000.0) or not (z <= 0.75):
		tmp = (x / z) / (z * (z / y))
	else:
		tmp = (y / z) * ((x / z) - x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -44000000000.0) || !(z <= 0.75))
		tmp = Float64(Float64(x / z) / Float64(z * Float64(z / y)));
	else
		tmp = Float64(Float64(y / z) * Float64(Float64(x / z) - x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -44000000000.0) || ~((z <= 0.75)))
		tmp = (x / z) / (z * (z / y));
	else
		tmp = (y / z) * ((x / z) - x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[z, -44000000000.0], N[Not[LessEqual[z, 0.75]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4e10 or 0.75 < z

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-*r/ 82.4%

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

      3. sqr-neg 82.4%

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

      4. associate-*l* 82.4%

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

      5. associate-*l* 82.4%

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

      6. sqr-neg 82.4%

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

      7. associate-*l* 82.4%

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

      8. distribute-lft-in 82.4%

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

      9. fma-def 82.4%

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

      10. *-rgt-identity 82.4%

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

   3. Simplified 82.4%

      \[\leadsto \color{blue}{y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 79.4%

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

      2. *-commutative 79.4%

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

      3. fma-udef 79.4%

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

      4. distribute-lft-in 55.8%

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

      5. *-un-lft-identity 55.8%

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

      6. distribute-rgt-in 79.4%

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

      7. associate-/r* 82.6%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}} \]

      8. frac-times 97.2%

         \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot \frac{y}{z}}}{z + 1} \]

      9. associate-/l* 95.8%

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

      10. div-inv 95.8%

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

      11. clear-num 95.8%

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

   5. Applied egg-rr 95.8%

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

   6. Taylor expanded in z around inf 88.5%

      \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\frac{{z}^{2}}{y}}} \]
   7. Step-by-step derivation

      1. unpow2 88.5%

         \[\leadsto \frac{\frac{x}{z}}{\frac{\color{blue}{z \cdot z}}{y}} \]

      2. associate-/l* 95.1%

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\frac{z}{\frac{y}{z}}}} \]

   8. Simplified 95.1%

      \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\frac{z}{\frac{y}{z}}}} \]
   9. Step-by-step derivation

      1. associate-/r/ 95.2%

         \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\frac{z}{y} \cdot z}} \]

   10. Applied egg-rr 95.2%

       \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\frac{z}{y} \cdot z}} \]

   if -4.4e10 < z < 0.75

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

      2. associate-*r/ 89.5%

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

      3. sqr-neg 89.5%

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

      4. associate-*l* 89.5%

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

      5. associate-*l* 89.5%

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

      6. sqr-neg 89.5%

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

      7. associate-*l* 89.5%

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

      8. distribute-lft-in 89.5%

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

      9. fma-def 89.5%

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

      10. *-rgt-identity 89.5%

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

   3. Simplified 89.5%

      \[\leadsto \color{blue}{y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 86.6%

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

      2. *-commutative 86.6%

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

      3. fma-udef 86.6%

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

      4. distribute-lft-in 86.6%

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

      5. *-un-lft-identity 86.6%

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

      6. distribute-rgt-in 86.6%

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

      7. associate-/r* 86.6%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}} \]

      8. frac-times 96.3%

         \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot \frac{y}{z}}}{z + 1} \]

      9. associate-/l* 96.2%

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

      10. div-inv 96.2%

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

      11. clear-num 97.2%

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

   5. Applied egg-rr 97.2%

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

   6. Taylor expanded in z around 0 63.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z} + \frac{x \cdot y}{{z}^{2}}} \]
   7. Step-by-step derivation

      1. +-commutative 63.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}} + -1 \cdot \frac{x \cdot y}{z}} \]

      2. unpow2 63.3%

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

      3. times-frac 72.9%

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

      4. mul-1-neg 72.9%

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

      5. associate-*r/ 72.0%

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

      6. distribute-lft-neg-out 72.0%

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

      7. fma-def 72.0%

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

      8. distribute-lft-neg-out 72.0%

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

      9. *-commutative 72.0%

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

      10. associate-*l/ 72.9%

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

      11. fma-neg 72.9%

          \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z} - \frac{y \cdot x}{z}} \]

      12. associate-*l/ 72.0%

          \[\leadsto \frac{x}{z} \cdot \frac{y}{z} - \color{blue}{\frac{y}{z} \cdot x} \]

      13. *-commutative 72.0%

          \[\leadsto \frac{x}{z} \cdot \frac{y}{z} - \color{blue}{x \cdot \frac{y}{z}} \]

      14. distribute-rgt-out-- 94.7%

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

   8. Simplified 94.7%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -44000000000 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{x}{z} - x\right)\\ \end{array} \]

Alternative 5: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-144}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.8e-144)
   (* x (/ y (* z z)))
   (if (<= z 2.2e-78) (/ (* x (/ y z)) z) (* y (/ x (* z z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.8e-144) {
		tmp = x * (y / (z * z));
	} else if (z <= 2.2e-78) {
		tmp = (x * (y / z)) / z;
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.8d-144)) then
        tmp = x * (y / (z * z))
    else if (z <= 2.2d-78) then
        tmp = (x * (y / z)) / z
    else
        tmp = y * (x / (z * z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.8e-144) {
		tmp = x * (y / (z * z));
	} else if (z <= 2.2e-78) {
		tmp = (x * (y / z)) / z;
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1.8e-144:
		tmp = x * (y / (z * z))
	elif z <= 2.2e-78:
		tmp = (x * (y / z)) / z
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.8e-144)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	elseif (z <= 2.2e-78)
		tmp = Float64(Float64(x * Float64(y / z)) / z);
	else
		tmp = Float64(y * Float64(x / Float64(z * z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.8e-144)
		tmp = x * (y / (z * z));
	elseif (z <= 2.2e-78)
		tmp = (x * (y / z)) / z;
	else
		tmp = y * (x / (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, -1.8e-144], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.2e-78], N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8e-144

   1. Initial program 84.3%

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

      1. *-commutative 84.3%

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

      2. sqr-neg 84.3%

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

      3. times-frac 90.8%

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

      4. sqr-neg 90.8%

         \[\leadsto \frac{y}{\color{blue}{z \cdot z}} \cdot \frac{x}{z + 1} \]

   3. Simplified 90.8%

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

   4. Taylor expanded in z around 0 74.0%

      \[\leadsto \frac{y}{z \cdot z} \cdot \color{blue}{x} \]

   if -1.8e-144 < z < 2.1999999999999999e-78

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. sqr-neg 82.9%

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

      3. times-frac 78.4%

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

      4. sqr-neg 78.4%

         \[\leadsto \frac{y}{\color{blue}{z \cdot z}} \cdot \frac{x}{z + 1} \]

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

   4. Taylor expanded in z around 0 78.4%

      \[\leadsto \frac{y}{z \cdot z} \cdot \color{blue}{x} \]
   5. Step-by-step derivation

      1. associate-/r* 85.2%

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{z}} \cdot x \]

      2. associate-*l/ 97.2%

         \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot x}{z}} \]

   6. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot x}{z}} \]

   if 2.1999999999999999e-78 < z

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-*r/ 85.1%

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

      3. sqr-neg 85.1%

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

      4. associate-*l* 85.1%

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

      5. associate-*l* 85.1%

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

      6. sqr-neg 85.1%

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

      7. associate-*l* 85.1%

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

      8. distribute-lft-in 85.1%

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

      9. fma-def 85.1%

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

      10. *-rgt-identity 85.1%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in z around 0 70.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 70.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]

      2. *-rgt-identity 70.5%

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

      3. times-frac 73.4%

         \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{1}} \]

      4. /-rgt-identity 73.4%

         \[\leadsto \frac{x}{z \cdot z} \cdot \color{blue}{y} \]

   6. Simplified 73.4%

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-144}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 6: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -44000000000:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(-z\right)}{y}}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-79}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -44000000000.0)
   (/ x (/ (* z (- z)) y))
   (if (<= z 4.3e-79) (/ (* x (/ y z)) z) (* y (/ x (* z z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -44000000000.0) {
		tmp = x / ((z * -z) / y);
	} else if (z <= 4.3e-79) {
		tmp = (x * (y / z)) / z;
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-44000000000.0d0)) then
        tmp = x / ((z * -z) / y)
    else if (z <= 4.3d-79) then
        tmp = (x * (y / z)) / z
    else
        tmp = y * (x / (z * z))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -44000000000.0) {
		tmp = x / ((z * -z) / y);
	} else if (z <= 4.3e-79) {
		tmp = (x * (y / z)) / z;
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -44000000000.0:
		tmp = x / ((z * -z) / y)
	elif z <= 4.3e-79:
		tmp = (x * (y / z)) / z
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -44000000000.0)
		tmp = Float64(x / Float64(Float64(z * Float64(-z)) / y));
	elseif (z <= 4.3e-79)
		tmp = Float64(Float64(x * Float64(y / z)) / z);
	else
		tmp = Float64(y * Float64(x / Float64(z * z)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -44000000000.0)
		tmp = x / ((z * -z) / y);
	elseif (z <= 4.3e-79)
		tmp = (x * (y / z)) / z;
	else
		tmp = y * (x / (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, -44000000000.0], N[(x / N[(N[(z * (-z)), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.3e-79], N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.4e10

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. sqr-neg 79.7%

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

      3. times-frac 92.4%

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

      4. sqr-neg 92.4%

         \[\leadsto \frac{y}{\color{blue}{z \cdot z}} \cdot \frac{x}{z + 1} \]

   3. Simplified 92.4%

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

   4. Taylor expanded in z around 0 68.4%

      \[\leadsto \frac{y}{z \cdot z} \cdot \color{blue}{x} \]
   5. Step-by-step derivation

      1. *-commutative 68.4%

         \[\leadsto \color{blue}{x \cdot \frac{y}{z \cdot z}} \]

      2. associate-*r/ 64.4%

         \[\leadsto \color{blue}{\frac{x \cdot y}{z \cdot z}} \]

      3. frac-times 53.5%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z}} \]

      4. clear-num 53.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \cdot \frac{y}{z} \]

      5. frac-2neg 53.6%

         \[\leadsto \frac{1}{\frac{z}{x}} \cdot \color{blue}{\frac{-y}{-z}} \]

      6. frac-times 66.7%

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

      7. *-un-lft-identity 66.7%

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

      8. add-sqr-sqrt 31.7%

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

      9. sqrt-unprod 51.3%

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

      10. sqr-neg 51.3%

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

      11. sqrt-unprod 35.6%

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

      12. add-sqr-sqrt 69.6%

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

   6. Applied egg-rr 69.6%

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

      1. associate-/r* 56.4%

         \[\leadsto \color{blue}{\frac{\frac{y}{\frac{z}{x}}}{-z}} \]

      2. associate-/l* 54.7%

         \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{z}}}{-z} \]

      3. associate-/r* 67.1%

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

      4. *-commutative 67.1%

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

      5. distribute-rgt-neg-in 67.1%

         \[\leadsto \frac{x \cdot y}{\color{blue}{-z \cdot z}} \]

      6. associate-/l* 71.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{-z \cdot z}{y}}} \]

   8. Simplified 71.2%

      \[\leadsto \color{blue}{\frac{x}{\frac{-z \cdot z}{y}}} \]

   if -4.4e10 < z < 4.29999999999999982e-79

   1. Initial program 85.9%

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

      1. *-commutative 85.9%

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

      2. sqr-neg 85.9%

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

      3. times-frac 81.3%

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

      4. sqr-neg 81.3%

         \[\leadsto \frac{y}{\color{blue}{z \cdot z}} \cdot \frac{x}{z + 1} \]

   3. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

   4. Taylor expanded in z around 0 80.2%

      \[\leadsto \frac{y}{z \cdot z} \cdot \color{blue}{x} \]
   5. Step-by-step derivation

      1. associate-/r* 84.9%

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{z}} \cdot x \]

      2. associate-*l/ 93.5%

         \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot x}{z}} \]

   6. Applied egg-rr 93.5%

      \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot x}{z}} \]

   if 4.29999999999999982e-79 < z

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-*r/ 85.1%

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

      3. sqr-neg 85.1%

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

      4. associate-*l* 85.1%

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

      5. associate-*l* 85.1%

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

      6. sqr-neg 85.1%

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

      7. associate-*l* 85.1%

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

      8. distribute-lft-in 85.1%

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

      9. fma-def 85.1%

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

      10. *-rgt-identity 85.1%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in z around 0 70.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 70.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]

      2. *-rgt-identity 70.5%

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

      3. times-frac 73.4%

         \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{1}} \]

      4. /-rgt-identity 73.4%

         \[\leadsto \frac{x}{z \cdot z} \cdot \color{blue}{y} \]

   6. Simplified 73.4%

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -44000000000:\\ \;\;\;\;\frac{x}{\frac{z \cdot \left(-z\right)}{y}}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-79}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 7: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{x}{z}}{\left(z + 1\right) \cdot \frac{z}{y}} \end{array} \]

FPCore

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

C

assert(x < y);
double code(double x, double y, double z) {
	return (x / z) / ((z + 1.0) * (z / y));
}

Fortran

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

Java

assert x < y;
public static double code(double x, double y, double z) {
	return (x / z) / ((z + 1.0) * (z / y));
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return (x / z) / ((z + 1.0) * (z / y))

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(x / z) / Float64(Float64(z + 1.0) * Float64(z / y)))
end

MATLAB

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(x / z), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.1%

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

   1. *-commutative 83.1%

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

   2. associate-*r/ 86.1%

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

   3. sqr-neg 86.1%

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

   4. associate-*l* 86.1%

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

   5. associate-*l* 86.1%

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

   6. sqr-neg 86.1%

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

   7. associate-*l* 86.1%

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

   8. distribute-lft-in 86.1%

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

   9. fma-def 86.1%

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

   10. *-rgt-identity 86.1%

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

3. Simplified 86.1%

   \[\leadsto \color{blue}{y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 83.1%

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

   2. *-commutative 83.1%

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

   3. fma-udef 83.1%

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

   4. distribute-lft-in 71.8%

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

   5. *-un-lft-identity 71.8%

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

   6. distribute-rgt-in 83.1%

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

   7. associate-/r* 84.7%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}} \]

   8. frac-times 96.7%

      \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot \frac{y}{z}}}{z + 1} \]

   9. associate-/l* 96.0%

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

   10. div-inv 96.0%

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

   11. clear-num 96.5%

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

5. Applied egg-rr 96.5%

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

6. Final simplification 96.5%

   \[\leadsto \frac{\frac{x}{z}}{\left(z + 1\right) \cdot \frac{z}{y}} \]

Alternative 8: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y, double z) {
	return ((x / z) * (y / z)) / (z + 1.0);
}

Fortran

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

Java

assert x < y;
public static double code(double x, double y, double z) {
	return ((x / z) * (y / z)) / (z + 1.0);
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return ((x / z) * (y / z)) / (z + 1.0)

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(Float64(x / z) * Float64(y / z)) / Float64(z + 1.0))
end

MATLAB

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.1%

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

   1. associate-*l* 83.1%

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

   2. times-frac 93.3%

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

   3. associate-/r* 96.4%

      \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{y}{z}}{z + 1}} \]

   4. associate-*r/ 96.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot \frac{y}{z}}{z + 1}} \]

3. Simplified 96.7%

   \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot \frac{y}{z}}{z + 1}} \]

4. Final simplification 96.7%

   \[\leadsto \frac{\frac{x}{z} \cdot \frac{y}{z}}{z + 1} \]

Alternative 9: 77.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -2.1e-81) (* x (/ y (* z z))) (* (/ x z) (/ y z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e-81) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x / z) * (y / z);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.1d-81)) then
        tmp = x * (y / (z * z))
    else
        tmp = (x / z) * (y / z)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e-81) {
		tmp = x * (y / (z * z));
	} else {
		tmp = (x / z) * (y / z);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -2.1e-81:
		tmp = x * (y / (z * z))
	else:
		tmp = (x / z) * (y / z)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -2.1e-81)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(x / z) * Float64(y / z));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.1e-81)
		tmp = x * (y / (z * z));
	else
		tmp = (x / z) * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -2.1e-81], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0999999999999999e-81

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. sqr-neg 84.5%

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

      3. times-frac 95.0%

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

      4. sqr-neg 95.0%

         \[\leadsto \frac{y}{\color{blue}{z \cdot z}} \cdot \frac{x}{z + 1} \]

   3. Simplified 95.0%

      \[\leadsto \color{blue}{\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}} \]

   4. Taylor expanded in z around 0 80.1%

      \[\leadsto \frac{y}{z \cdot z} \cdot \color{blue}{x} \]

   if -2.0999999999999999e-81 < x

   1. Initial program 82.6%

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

      1. *-commutative 82.6%

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

      2. associate-*r/ 83.9%

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

      3. sqr-neg 83.9%

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

      4. associate-*l* 83.9%

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

      5. associate-*l* 83.9%

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

      6. sqr-neg 83.9%

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

      7. associate-*l* 83.9%

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

      8. distribute-lft-in 83.9%

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

      9. fma-def 83.9%

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

      10. *-rgt-identity 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in z around 0 73.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 73.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]

      2. times-frac 78.4%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z}} \]

   6. Simplified 78.4%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 10: 40.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -5e-310) (* (/ x z) (- y)) (* (/ x z) y)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e-310) {
		tmp = (x / z) * -y;
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-5d-310)) then
        tmp = (x / z) * -y
    else
        tmp = (x / z) * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e-310) {
		tmp = (x / z) * -y;
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -5e-310:
		tmp = (x / z) * -y
	else:
		tmp = (x / z) * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -5e-310)
		tmp = Float64(Float64(x / z) * Float64(-y));
	else
		tmp = Float64(Float64(x / z) * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5e-310)
		tmp = (x / z) * -y;
	else
		tmp = (x / z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, -5e-310], N[(N[(x / z), $MachinePrecision] * (-y)), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.999999999999985e-310

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-*r/ 86.9%

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

      3. sqr-neg 86.9%

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

      4. associate-*l* 86.9%

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

      5. associate-*l* 86.9%

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

      6. sqr-neg 86.9%

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

      7. associate-*l* 86.9%

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

      8. distribute-lft-in 86.9%

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

      9. fma-def 86.9%

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

      10. *-rgt-identity 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in z around 0 61.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z} + \frac{x \cdot y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. +-commutative 61.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}} + -1 \cdot \frac{x \cdot y}{z}} \]

      2. unpow2 61.0%

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

      3. *-commutative 61.0%

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

      4. times-frac 67.2%

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

      5. *-commutative 67.2%

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

      6. associate-*r/ 69.6%

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

      7. associate-*r* 69.6%

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

      8. neg-mul-1 69.6%

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

      9. distribute-rgt-out 69.6%

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

   6. Simplified 69.6%

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

   7. Taylor expanded in z around inf 38.9%

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

      1. mul-1-neg 38.9%

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

   9. Simplified 38.9%

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

   if -4.999999999999985e-310 < z

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. associate-*r/ 85.4%

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

      3. sqr-neg 85.4%

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

      4. associate-*l* 85.4%

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

      5. associate-*l* 85.4%

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

      6. sqr-neg 85.4%

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

      7. associate-*l* 85.4%

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

      8. distribute-lft-in 85.4%

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

      9. fma-def 85.4%

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

      10. *-rgt-identity 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in z around 0 40.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z} + \frac{x \cdot y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. +-commutative 40.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}} + -1 \cdot \frac{x \cdot y}{z}} \]

      2. unpow2 40.7%

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

      3. *-commutative 40.7%

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

      4. times-frac 44.8%

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

      5. *-commutative 44.8%

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

      6. associate-*r/ 46.5%

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

      7. associate-*r* 46.5%

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

      8. neg-mul-1 46.5%

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

      9. distribute-rgt-out 68.4%

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

   6. Simplified 68.4%

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

   7. Taylor expanded in z around inf 20.6%

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

      1. mul-1-neg 20.6%

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

   9. Simplified 20.6%

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

       1. expm1-log1p-u 20.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{z} \cdot \left(-y\right)\right)\right)} \]

       2. expm1-udef 27.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \left(-y\right)\right)} - 1} \]

       3. add-sqr-sqrt 14.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)}\right)} - 1 \]

       4. sqrt-unprod 33.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)} - 1 \]

       5. sqr-neg 33.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \sqrt{\color{blue}{y \cdot y}}\right)} - 1 \]

       6. sqrt-unprod 21.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} - 1 \]

       7. add-sqr-sqrt 44.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{y}\right)} - 1 \]

       8. associate-*l/ 44.3%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot y}{z}}\right)} - 1 \]

   11. Applied egg-rr 44.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 35.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)\right)} \]

       2. expm1-log1p 42.8%

          \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]

       3. associate-*l/ 46.0%

          \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

   13. Simplified 46.0%

       \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Alternative 11: 40.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -5e-310) (* x (/ (- y) z)) (* (/ x z) y)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e-310) {
		tmp = x * (-y / z);
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-5d-310)) then
        tmp = x * (-y / z)
    else
        tmp = (x / z) * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5e-310) {
		tmp = x * (-y / z);
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -5e-310:
		tmp = x * (-y / z)
	else:
		tmp = (x / z) * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -5e-310)
		tmp = Float64(x * Float64(Float64(-y) / z));
	else
		tmp = Float64(Float64(x / z) * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5e-310)
		tmp = x * (-y / z);
	else
		tmp = (x / z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[z, -5e-310], N[(x * N[((-y) / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.999999999999985e-310

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-*r/ 86.9%

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

      3. sqr-neg 86.9%

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

      4. associate-*l* 86.9%

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

      5. associate-*l* 86.9%

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

      6. sqr-neg 86.9%

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

      7. associate-*l* 86.9%

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

      8. distribute-lft-in 86.9%

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

      9. fma-def 86.9%

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

      10. *-rgt-identity 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in z around 0 61.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z} + \frac{x \cdot y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. +-commutative 61.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}} + -1 \cdot \frac{x \cdot y}{z}} \]

      2. unpow2 61.0%

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

      3. *-commutative 61.0%

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

      4. times-frac 67.2%

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

      5. *-commutative 67.2%

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

      6. associate-*r/ 69.6%

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

      7. associate-*r* 69.6%

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

      8. neg-mul-1 69.6%

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

      9. distribute-rgt-out 69.6%

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

   6. Simplified 69.6%

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

   7. Taylor expanded in z around inf 35.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z}} \]
   8. Step-by-step derivation

      1. mul-1-neg 35.0%

         \[\leadsto \color{blue}{-\frac{x \cdot y}{z}} \]

      2. associate-*r/ 40.5%

         \[\leadsto -\color{blue}{x \cdot \frac{y}{z}} \]

      3. distribute-lft-neg-out 40.5%

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

      4. *-commutative 40.5%

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

   9. Simplified 40.5%

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

   if -4.999999999999985e-310 < z

   1. Initial program 83.9%

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

      1. *-commutative 83.9%

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

      2. associate-*r/ 85.4%

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

      3. sqr-neg 85.4%

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

      4. associate-*l* 85.4%

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

      5. associate-*l* 85.4%

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

      6. sqr-neg 85.4%

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

      7. associate-*l* 85.4%

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

      8. distribute-lft-in 85.4%

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

      9. fma-def 85.4%

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

      10. *-rgt-identity 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in z around 0 40.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z} + \frac{x \cdot y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. +-commutative 40.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}} + -1 \cdot \frac{x \cdot y}{z}} \]

      2. unpow2 40.7%

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

      3. *-commutative 40.7%

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

      4. times-frac 44.8%

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

      5. *-commutative 44.8%

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

      6. associate-*r/ 46.5%

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

      7. associate-*r* 46.5%

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

      8. neg-mul-1 46.5%

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

      9. distribute-rgt-out 68.4%

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

   6. Simplified 68.4%

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

   7. Taylor expanded in z around inf 20.6%

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

      1. mul-1-neg 20.6%

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

   9. Simplified 20.6%

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

       1. expm1-log1p-u 20.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{z} \cdot \left(-y\right)\right)\right)} \]

       2. expm1-udef 27.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \left(-y\right)\right)} - 1} \]

       3. add-sqr-sqrt 14.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)}\right)} - 1 \]

       4. sqrt-unprod 33.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)} - 1 \]

       5. sqr-neg 33.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \sqrt{\color{blue}{y \cdot y}}\right)} - 1 \]

       6. sqrt-unprod 21.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} - 1 \]

       7. add-sqr-sqrt 44.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{y}\right)} - 1 \]

       8. associate-*l/ 44.3%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot y}{z}}\right)} - 1 \]

   11. Applied egg-rr 44.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 35.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)\right)} \]

       2. expm1-log1p 42.8%

          \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]

       3. associate-*l/ 46.0%

          \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

   13. Simplified 46.0%

       \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Alternative 12: 33.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -5e+89) (* x (/ y z)) (* (/ x z) y)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e+89) {
		tmp = x * (y / z);
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-5d+89)) then
        tmp = x * (y / z)
    else
        tmp = (x / z) * y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e+89) {
		tmp = x * (y / z);
	} else {
		tmp = (x / z) * y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -5e+89:
		tmp = x * (y / z)
	else:
		tmp = (x / z) * y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -5e+89)
		tmp = Float64(x * Float64(y / z));
	else
		tmp = Float64(Float64(x / z) * y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5e+89)
		tmp = x * (y / z);
	else
		tmp = (x / z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -5e+89], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999983e89

   1. Initial program 74.6%

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

      1. *-commutative 74.6%

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

      2. associate-*r/ 85.6%

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

      3. sqr-neg 85.6%

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

      4. associate-*l* 85.7%

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

      5. associate-*l* 85.6%

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

      6. sqr-neg 85.6%

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

      7. associate-*l* 85.7%

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

      8. distribute-lft-in 85.7%

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

      9. fma-def 85.7%

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

      10. *-rgt-identity 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in z around 0 30.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z} + \frac{x \cdot y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. +-commutative 30.8%

         \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}} + -1 \cdot \frac{x \cdot y}{z}} \]

      2. unpow2 30.8%

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

      3. *-commutative 30.8%

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

      4. times-frac 31.0%

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

      5. *-commutative 31.0%

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

      6. associate-*r/ 31.1%

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

      7. associate-*r* 31.1%

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

      8. neg-mul-1 31.1%

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

      9. distribute-rgt-out 54.5%

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

   6. Simplified 54.5%

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

   7. Taylor expanded in z around inf 12.1%

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

      1. mul-1-neg 12.1%

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

   9. Simplified 12.1%

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

       1. expm1-log1p-u 5.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{z} \cdot \left(-y\right)\right)\right)} \]

       2. expm1-udef 15.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \left(-y\right)\right)} - 1} \]

       3. add-sqr-sqrt 7.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)}\right)} - 1 \]

       4. sqrt-unprod 13.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)} - 1 \]

       5. sqr-neg 13.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \sqrt{\color{blue}{y \cdot y}}\right)} - 1 \]

       6. sqrt-unprod 9.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} - 1 \]

       7. add-sqr-sqrt 33.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{y}\right)} - 1 \]

       8. associate-*l/ 33.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot y}{z}}\right)} - 1 \]

   11. Applied egg-rr 33.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 23.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)\right)} \]

       2. expm1-log1p 30.6%

          \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]

       3. associate-*r/ 38.8%

          \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]

   13. Simplified 38.8%

       \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]

   if -4.99999999999999983e89 < x

   1. Initial program 85.1%

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

      1. *-commutative 85.1%

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

      2. associate-*r/ 86.2%

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

      3. sqr-neg 86.2%

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

      4. associate-*l* 86.2%

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

      5. associate-*l* 86.2%

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

      6. sqr-neg 86.2%

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

      7. associate-*l* 86.2%

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

      8. distribute-lft-in 86.2%

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

      9. fma-def 86.2%

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

      10. *-rgt-identity 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in z around 0 54.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z} + \frac{x \cdot y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. +-commutative 54.5%

         \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}} + -1 \cdot \frac{x \cdot y}{z}} \]

      2. unpow2 54.5%

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

      3. *-commutative 54.5%

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

      4. times-frac 60.6%

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

      5. *-commutative 60.6%

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

      6. associate-*r/ 63.1%

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

      7. associate-*r* 63.1%

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

      8. neg-mul-1 63.1%

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

      9. distribute-rgt-out 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in z around inf 33.0%

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

      1. mul-1-neg 33.0%

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

   9. Simplified 33.0%

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

       1. expm1-log1p-u 29.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{z} \cdot \left(-y\right)\right)\right)} \]

       2. expm1-udef 36.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \left(-y\right)\right)} - 1} \]

       3. add-sqr-sqrt 17.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)}\right)} - 1 \]

       4. sqrt-unprod 35.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)} - 1 \]

       5. sqr-neg 35.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \sqrt{\color{blue}{y \cdot y}}\right)} - 1 \]

       6. sqrt-unprod 22.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} - 1 \]

       7. add-sqr-sqrt 40.2%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{y}\right)} - 1 \]

       8. associate-*l/ 40.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot y}{z}}\right)} - 1 \]

   11. Applied egg-rr 40.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 29.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)\right)} \]

       2. expm1-log1p 34.0%

          \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]

       3. associate-*l/ 37.4%

          \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

   13. Simplified 37.4%

       \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Alternative 13: 33.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+90}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -4e+90) (* x (/ y z)) (/ y (/ z x))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -4e+90) {
		tmp = x * (y / z);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-4d+90)) then
        tmp = x * (y / z)
    else
        tmp = y / (z / x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4e+90) {
		tmp = x * (y / z);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -4e+90:
		tmp = x * (y / z)
	else:
		tmp = y / (z / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -4e+90)
		tmp = Float64(x * Float64(y / z));
	else
		tmp = Float64(y / Float64(z / x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4e+90)
		tmp = x * (y / z);
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -4e+90], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.99999999999999987e90

   1. Initial program 74.6%

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

      1. *-commutative 74.6%

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

      2. associate-*r/ 85.6%

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

      3. sqr-neg 85.6%

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

      4. associate-*l* 85.7%

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

      5. associate-*l* 85.6%

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

      6. sqr-neg 85.6%

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

      7. associate-*l* 85.7%

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

      8. distribute-lft-in 85.7%

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

      9. fma-def 85.7%

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

      10. *-rgt-identity 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in z around 0 30.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z} + \frac{x \cdot y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. +-commutative 30.8%

         \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}} + -1 \cdot \frac{x \cdot y}{z}} \]

      2. unpow2 30.8%

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

      3. *-commutative 30.8%

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

      4. times-frac 31.0%

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

      5. *-commutative 31.0%

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

      6. associate-*r/ 31.1%

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

      7. associate-*r* 31.1%

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

      8. neg-mul-1 31.1%

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

      9. distribute-rgt-out 54.5%

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

   6. Simplified 54.5%

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

   7. Taylor expanded in z around inf 12.1%

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

      1. mul-1-neg 12.1%

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

   9. Simplified 12.1%

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

       1. expm1-log1p-u 5.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{z} \cdot \left(-y\right)\right)\right)} \]

       2. expm1-udef 15.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \left(-y\right)\right)} - 1} \]

       3. add-sqr-sqrt 7.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)}\right)} - 1 \]

       4. sqrt-unprod 13.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)} - 1 \]

       5. sqr-neg 13.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \sqrt{\color{blue}{y \cdot y}}\right)} - 1 \]

       6. sqrt-unprod 9.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} - 1 \]

       7. add-sqr-sqrt 33.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{y}\right)} - 1 \]

       8. associate-*l/ 33.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot y}{z}}\right)} - 1 \]

   11. Applied egg-rr 33.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 23.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)\right)} \]

       2. expm1-log1p 30.6%

          \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]

       3. associate-*r/ 38.8%

          \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]

   13. Simplified 38.8%

       \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]

   if -3.99999999999999987e90 < x

   1. Initial program 85.1%

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

      1. *-commutative 85.1%

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

      2. associate-*r/ 86.2%

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

      3. sqr-neg 86.2%

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

      4. associate-*l* 86.2%

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

      5. associate-*l* 86.2%

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

      6. sqr-neg 86.2%

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

      7. associate-*l* 86.2%

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

      8. distribute-lft-in 86.2%

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

      9. fma-def 86.2%

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

      10. *-rgt-identity 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in z around 0 54.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z} + \frac{x \cdot y}{{z}^{2}}} \]
   5. Step-by-step derivation

      1. +-commutative 54.5%

         \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}} + -1 \cdot \frac{x \cdot y}{z}} \]

      2. unpow2 54.5%

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

      3. *-commutative 54.5%

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

      4. times-frac 60.6%

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

      5. *-commutative 60.6%

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

      6. associate-*r/ 63.1%

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

      7. associate-*r* 63.1%

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

      8. neg-mul-1 63.1%

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

      9. distribute-rgt-out 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in z around inf 33.0%

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

      1. mul-1-neg 33.0%

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

   9. Simplified 33.0%

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

       1. *-commutative 33.0%

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

       2. clear-num 33.0%

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

       3. un-div-inv 33.0%

          \[\leadsto \color{blue}{\frac{-y}{\frac{z}{x}}} \]

       4. add-sqr-sqrt 16.4%

          \[\leadsto \frac{\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}{\frac{z}{x}} \]

       5. sqrt-unprod 35.1%

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

       6. sqr-neg 35.1%

          \[\leadsto \frac{\sqrt{\color{blue}{y \cdot y}}}{\frac{z}{x}} \]

       7. sqrt-unprod 20.0%

          \[\leadsto \frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\frac{z}{x}} \]

       8. add-sqr-sqrt 37.4%

          \[\leadsto \frac{\color{blue}{y}}{\frac{z}{x}} \]

   11. Applied egg-rr 37.4%

       \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+90}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 14: 75.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y, double z) {
	return (x / z) * (y / z);
}

Fortran

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

Java

assert x < y;
public static double code(double x, double y, double z) {
	return (x / z) * (y / z);
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return (x / z) * (y / z)

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(x / z) * Float64(y / z))
end

MATLAB

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.1%

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

   1. *-commutative 83.1%

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

   2. associate-*r/ 86.1%

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

   3. sqr-neg 86.1%

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

   4. associate-*l* 86.1%

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

   5. associate-*l* 86.1%

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

   6. sqr-neg 86.1%

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

   7. associate-*l* 86.1%

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

   8. distribute-lft-in 86.1%

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

   9. fma-def 86.1%

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

   10. *-rgt-identity 86.1%

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

3. Simplified 86.1%

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

4. Taylor expanded in z around 0 74.8%

   \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 74.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]

   2. times-frac 74.5%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z}} \]

6. Simplified 74.5%

   \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z}} \]

7. Final simplification 74.5%

   \[\leadsto \frac{x}{z} \cdot \frac{y}{z} \]

Alternative 15: 31.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y, double z) {
	return x * (y / z);
}

Fortran

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

Java

assert x < y;
public static double code(double x, double y, double z) {
	return x * (y / z);
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return x * (y / z)

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(x * Float64(y / z))
end

MATLAB

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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.1%

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

   1. *-commutative 83.1%

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

   2. associate-*r/ 86.1%

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

   3. sqr-neg 86.1%

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

   4. associate-*l* 86.1%

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

   5. associate-*l* 86.1%

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

   6. sqr-neg 86.1%

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

   7. associate-*l* 86.1%

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

   8. distribute-lft-in 86.1%

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

   9. fma-def 86.1%

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

   10. *-rgt-identity 86.1%

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

3. Simplified 86.1%

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

4. Taylor expanded in z around 0 50.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{z} + \frac{x \cdot y}{{z}^{2}}} \]
5. Step-by-step derivation

   1. +-commutative 50.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}} + -1 \cdot \frac{x \cdot y}{z}} \]

   2. unpow2 50.1%

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

   3. *-commutative 50.1%

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

   4. times-frac 55.2%

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

   5. *-commutative 55.2%

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

   6. associate-*r/ 57.2%

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

   7. associate-*r* 57.2%

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

   8. neg-mul-1 57.2%

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

   9. distribute-rgt-out 69.0%

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

6. Simplified 69.0%

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

7. Taylor expanded in z around inf 29.1%

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

   1. mul-1-neg 29.1%

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

9. Simplified 29.1%

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

    1. expm1-log1p-u 24.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{z} \cdot \left(-y\right)\right)\right)} \]

    2. expm1-udef 32.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \left(-y\right)\right)} - 1} \]

    3. add-sqr-sqrt 15.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)}\right)} - 1 \]

    4. sqrt-unprod 31.4%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)} - 1 \]

    5. sqr-neg 31.4%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \sqrt{\color{blue}{y \cdot y}}\right)} - 1 \]

    6. sqrt-unprod 19.8%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}\right)} - 1 \]

    7. add-sqr-sqrt 39.0%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{z} \cdot \color{blue}{y}\right)} - 1 \]

    8. associate-*l/ 39.0%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot y}{z}}\right)} - 1 \]

11. Applied egg-rr 39.0%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 28.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot y}{z}\right)\right)} \]

    2. expm1-log1p 33.3%

       \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]

    3. associate-*r/ 36.3%

       \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]

13. Simplified 36.3%

    \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]

14. Final simplification 36.3%

    \[\leadsto x \cdot \frac{y}{z} \]

Developer target: 95.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z 249.6182814532307)
   (/ (* y (/ x z)) (+ z (* z z)))
   (/ (* (/ (/ y z) (+ 1.0 z)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < 249.6182814532307d0) then
        tmp = (y * (x / z)) / (z + (z * z))
    else
        tmp = (((y / z) / (1.0d0 + z)) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < 249.6182814532307:
		tmp = (y * (x / z)) / (z + (z * z))
	else:
		tmp = (((y / z) / (1.0 + z)) * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < 249.6182814532307)
		tmp = Float64(Float64(y * Float64(x / z)) / Float64(z + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) / Float64(1.0 + z)) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < 249.6182814532307)
		tmp = (y * (x / z)) / (z + (z * z));
	else
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, 249.6182814532307], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(z + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]

Reproduce

herbie shell --seed 2023275 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))