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

Percentage Accurate: 83.3% → 95.5%

Time: 14.3s

Alternatives: 19

Speedup: 1.0×

• 26.2% 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: 83.3% 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: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ t_1 := \frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-238}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\frac{x \cdot y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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))) (t_1 (/ (/ y z) (* z (/ z x)))))
   (if (<= t_0 -2e+21)
     t_1
     (if (<= t_0 2e-238)
       (/ (* (/ x z) y) z)
       (if (<= t_0 5e+300) (/ (* x y) t_0) t_1)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double t_1 = (y / z) / (z * (z / x));
	double tmp;
	if (t_0 <= -2e+21) {
		tmp = t_1;
	} else if (t_0 <= 2e-238) {
		tmp = ((x / z) * y) / z;
	} else if (t_0 <= 5e+300) {
		tmp = (x * y) / t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (z + 1.0d0) * (z * z)
    t_1 = (y / z) / (z * (z / x))
    if (t_0 <= (-2d+21)) then
        tmp = t_1
    else if (t_0 <= 2d-238) then
        tmp = ((x / z) * y) / z
    else if (t_0 <= 5d+300) then
        tmp = (x * y) / t_0
    else
        tmp = t_1
    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 t_1 = (y / z) / (z * (z / x));
	double tmp;
	if (t_0 <= -2e+21) {
		tmp = t_1;
	} else if (t_0 <= 2e-238) {
		tmp = ((x / z) * y) / z;
	} else if (t_0 <= 5e+300) {
		tmp = (x * y) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = (z + 1.0) * (z * z)
	t_1 = (y / z) / (z * (z / x))
	tmp = 0
	if t_0 <= -2e+21:
		tmp = t_1
	elif t_0 <= 2e-238:
		tmp = ((x / z) * y) / z
	elif t_0 <= 5e+300:
		tmp = (x * y) / t_0
	else:
		tmp = t_1
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	t_1 = Float64(Float64(y / z) / Float64(z * Float64(z / x)))
	tmp = 0.0
	if (t_0 <= -2e+21)
		tmp = t_1;
	elseif (t_0 <= 2e-238)
		tmp = Float64(Float64(Float64(x / z) * y) / z);
	elseif (t_0 <= 5e+300)
		tmp = Float64(Float64(x * y) / t_0);
	else
		tmp = t_1;
	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);
	t_1 = (y / z) / (z * (z / x));
	tmp = 0.0;
	if (t_0 <= -2e+21)
		tmp = t_1;
	elseif (t_0 <= 2e-238)
		tmp = ((x / z) * y) / z;
	elseif (t_0 <= 5e+300)
		tmp = (x * y) / t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+21], t$95$1, If[LessEqual[t$95$0, 2e-238], N[(N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 5e+300], N[(N[(x * y), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < -2e21 or 5.00000000000000026e300 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 87.9%

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

      1. *-commutative 87.9%

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

      2. associate-*r/ 89.1%

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

      3. sqr-neg 89.1%

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

      4. associate-*l* 89.1%

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

      5. associate-*l* 89.1%

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

      6. sqr-neg 89.1%

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

      7. associate-*l* 89.1%

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

      8. distribute-lft-in 89.1%

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

      9. fma-def 89.1%

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

      10. *-rgt-identity 89.1%

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

   3. Simplified 89.1%

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

      1. associate-*r/ 87.9%

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

      2. *-commutative 87.9%

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

      3. associate-/r* 93.2%

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

      4. associate-*r/ 95.9%

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

      5. fma-udef 95.9%

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

      6. distribute-lft1-in 95.9%

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

      7. frac-times 99.4%

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

      8. clear-num 99.3%

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

      9. frac-times 96.6%

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

      10. *-un-lft-identity 96.6%

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

   5. Applied egg-rr 96.6%

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

   6. Taylor expanded in z around inf 94.7%

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

      1. unpow2 94.7%

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

      2. associate-*r/ 96.0%

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

   8. Simplified 96.0%

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

   if -2e21 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 2e-238

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-*r/ 77.3%

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

      3. sqr-neg 77.3%

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

      4. associate-*l* 77.3%

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

      5. associate-*l* 77.3%

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

      6. sqr-neg 77.3%

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

      7. associate-*l* 77.3%

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

      8. distribute-lft-in 77.3%

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

      9. fma-def 77.3%

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

      10. *-rgt-identity 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in z around 0 77.3%

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

      1. unpow2 77.3%

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

   6. Simplified 77.3%

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

      1. *-commutative 77.3%

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

      2. associate-/r* 91.5%

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

      3. associate-/r/ 98.0%

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

      4. div-inv 97.5%

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

      5. clear-num 97.5%

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

      6. associate-*r/ 97.8%

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

   8. Applied egg-rr 97.8%

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

   if 2e-238 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 5.00000000000000026e300

   1. Initial program 98.3%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{-238}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 2: 94.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (/ (/ y z) (* z (/ z x)))
   (if (<= z 3.4e-109)
     (/ (* y (- (/ x z) x)) z)
     (* (/ y (* z z)) (/ x (+ z 1.0))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y / z) / (z * (z / x));
	} else if (z <= 3.4e-109) {
		tmp = (y * ((x / z) - x)) / z;
	} else {
		tmp = (y / (z * z)) * (x / (z + 1.0));
	}
	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 <= (-1.0d0)) then
        tmp = (y / z) / (z * (z / x))
    else if (z <= 3.4d-109) then
        tmp = (y * ((x / z) - x)) / z
    else
        tmp = (y / (z * z)) * (x / (z + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y / z) / (z * (z / x));
	} else if (z <= 3.4e-109) {
		tmp = (y * ((x / z) - x)) / z;
	} else {
		tmp = (y / (z * z)) * (x / (z + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (y / z) / (z * (z / x))
	elif z <= 3.4e-109:
		tmp = (y * ((x / z) - x)) / z
	else:
		tmp = (y / (z * z)) * (x / (z + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(y / z) / Float64(z * Float64(z / x)));
	elseif (z <= 3.4e-109)
		tmp = Float64(Float64(y * Float64(Float64(x / z) - x)) / z);
	else
		tmp = Float64(Float64(y / Float64(z * z)) * Float64(x / Float64(z + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = (y / z) / (z * (z / x));
	elseif (z <= 3.4e-109)
		tmp = (y * ((x / z) - x)) / z;
	else
		tmp = (y / (z * z)) * (x / (z + 1.0));
	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, -1.0], N[(N[(y / z), $MachinePrecision] / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.4e-109], N[(N[(y * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision] * N[(x / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*r/ 91.6%

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

      3. sqr-neg 91.6%

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

      4. associate-*l* 91.6%

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

      5. associate-*l* 91.6%

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

      6. sqr-neg 91.6%

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

      7. associate-*l* 91.6%

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

      8. distribute-lft-in 91.6%

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

      9. fma-def 91.6%

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

      10. *-rgt-identity 91.6%

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

   3. Simplified 91.6%

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

      1. associate-*r/ 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-/r* 92.6%

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

      4. associate-*r/ 97.4%

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

      5. fma-udef 97.4%

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

      6. distribute-lft1-in 97.4%

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

      7. frac-times 99.0%

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

      8. clear-num 98.9%

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

      9. frac-times 97.1%

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

      10. *-un-lft-identity 97.1%

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

   5. Applied egg-rr 97.1%

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

   6. Taylor expanded in z around inf 95.1%

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

      1. unpow2 95.1%

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

      2. associate-*r/ 96.1%

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

   8. Simplified 96.1%

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

   if -1 < z < 3.40000000000000012e-109

   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/ 81.4%

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

      3. sqr-neg 81.4%

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

      4. associate-*l* 81.4%

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

      5. associate-*l* 81.4%

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

      6. sqr-neg 81.4%

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

      7. associate-*l* 81.4%

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

      8. distribute-lft-in 81.4%

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

      9. fma-def 81.4%

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

      10. *-rgt-identity 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in z around 0 67.3%

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

      1. unpow2 67.3%

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

      2. *-commutative 67.3%

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

      3. times-frac 80.8%

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

      4. mul-1-neg 80.8%

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

      5. *-commutative 80.8%

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

      6. associate-*r/ 77.1%

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

      7. distribute-lft-neg-in 77.1%

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

      8. distribute-rgt-out 95.1%

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

   6. Simplified 95.1%

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

   7. Taylor expanded in y around 0 96.8%

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

   if 3.40000000000000012e-109 < z

   1. Initial program 90.7%

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

      1. *-commutative 90.7%

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

      2. sqr-neg 90.7%

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

      3. times-frac 96.7%

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

      4. sqr-neg 96.7%

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

   3. Simplified 96.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot z} \cdot \frac{x}{z + 1}\\ \end{array} \]

Alternative 3: 93.1% accurate, 0.8× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0)))
   (* (/ x z) (/ y (* z z)))
   (/ (* (/ x z) y) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (x / z) * (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 ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = (x / z) * (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 ((z <= -1.0) || !(z <= 1.0)) {
		tmp = (x / z) * (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 (z <= -1.0) or not (z <= 1.0):
		tmp = (x / z) * (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 ((z <= -1.0) || !(z <= 1.0))
		tmp = Float64(Float64(x / z) * Float64(y / Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(x / z) * y) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.0)))
		tmp = (x / z) * (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[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 89.5%

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

      1. *-commutative 89.5%

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

      2. sqr-neg 89.5%

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

      3. times-frac 97.3%

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

      4. sqr-neg 97.3%

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

   3. Simplified 97.3%

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

   4. Taylor expanded in z around inf 96.2%

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

   if -1 < z < 1

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. associate-*r/ 83.4%

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

      3. sqr-neg 83.4%

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

      4. associate-*l* 83.4%

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

      5. associate-*l* 83.4%

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

      6. sqr-neg 83.4%

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

      7. associate-*l* 83.4%

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

      8. distribute-lft-in 83.4%

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

      9. fma-def 83.4%

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

      10. *-rgt-identity 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around 0 81.9%

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

      1. unpow2 81.9%

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

   6. Simplified 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-/r* 90.5%

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

      3. associate-/r/ 94.5%

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

      4. div-inv 94.1%

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

      5. clear-num 94.1%

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

      6. associate-*r/ 94.9%

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

   8. Applied egg-rr 94.9%

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

4. Final simplification 95.5%

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

Alternative 4: 95.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 0.75)))
   (/ (/ y z) (* z (/ z x)))
   (/ (* y (- (/ x z) x)) z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.75)) {
		tmp = (y / z) / (z * (z / x));
	} else {
		tmp = (y * ((x / z) - x)) / 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 <= (-1.0d0)) .or. (.not. (z <= 0.75d0))) then
        tmp = (y / z) / (z * (z / x))
    else
        tmp = (y * ((x / z) - x)) / 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 <= -1.0) || !(z <= 0.75)) {
		tmp = (y / z) / (z * (z / x));
	} else {
		tmp = (y * ((x / z) - x)) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.75):
		tmp = (y / z) / (z * (z / x))
	else:
		tmp = (y * ((x / z) - x)) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.75))
		tmp = Float64(Float64(y / z) / Float64(z * Float64(z / x)));
	else
		tmp = Float64(Float64(y * Float64(Float64(x / z) - x)) / z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 0.75)))
		tmp = (y / z) / (z * (z / x));
	else
		tmp = (y * ((x / z) - x)) / 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[z, -1.0], N[Not[LessEqual[z, 0.75]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.75 < z

   1. Initial program 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-*r/ 88.8%

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

      3. sqr-neg 88.8%

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

      4. associate-*l* 88.8%

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

      5. associate-*l* 88.8%

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

      6. sqr-neg 88.8%

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

      7. associate-*l* 88.8%

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

      8. distribute-lft-in 88.8%

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

      9. fma-def 88.8%

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

      10. *-rgt-identity 88.8%

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

   3. Simplified 88.8%

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

      1. associate-*r/ 89.5%

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

      2. *-commutative 89.5%

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

      3. associate-/r* 94.0%

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

      4. associate-*r/ 95.6%

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

      5. fma-udef 95.6%

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

      6. distribute-lft1-in 95.6%

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

      7. frac-times 98.0%

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

      8. clear-num 98.0%

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

      9. frac-times 95.1%

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

      10. *-un-lft-identity 95.1%

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

   5. Applied egg-rr 95.1%

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

   6. Taylor expanded in z around inf 92.9%

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

      1. unpow2 92.9%

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

      2. associate-*r/ 94.0%

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

   8. Simplified 94.0%

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

   if -1 < z < 0.75

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. associate-*r/ 83.4%

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

      3. sqr-neg 83.4%

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

      4. associate-*l* 83.4%

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

      5. associate-*l* 83.4%

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

      6. sqr-neg 83.4%

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

      7. associate-*l* 83.4%

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

      8. distribute-lft-in 83.4%

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

      9. fma-def 83.4%

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

      10. *-rgt-identity 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around 0 66.9%

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

      1. unpow2 66.9%

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

      2. *-commutative 66.9%

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

      3. times-frac 78.3%

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

      4. mul-1-neg 78.3%

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

      5. *-commutative 78.3%

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

      6. associate-*r/ 75.1%

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

      7. distribute-lft-neg-in 75.1%

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

      8. distribute-rgt-out 94.9%

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

   6. Simplified 94.9%

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

   7. Taylor expanded in y around 0 95.7%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \end{array} \]

Alternative 5: 92.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (x / z) * (y / (z * z));
	} else if (z <= 1.0) {
		tmp = ((x / z) * y) / z;
	} 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) :: tmp
    if (z <= (-1.0d0)) then
        tmp = (x / z) * (y / (z * z))
    else if (z <= 1.0d0) then
        tmp = ((x / z) * y) / z
    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 tmp;
	if (z <= -1.0) {
		tmp = (x / z) * (y / (z * z));
	} else if (z <= 1.0) {
		tmp = ((x / z) * y) / z;
	} else {
		tmp = y / ((z * z) * (z / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (x / z) * (y / (z * z))
	elif z <= 1.0:
		tmp = ((x / z) * y) / z
	else:
		tmp = y / ((z * z) * (z / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(x / z) * Float64(y / Float64(z * z)));
	elseif (z <= 1.0)
		tmp = Float64(Float64(Float64(x / z) * y) / z);
	else
		tmp = Float64(y / Float64(Float64(z * z) * 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 (z <= -1.0)
		tmp = (x / z) * (y / (z * z));
	elseif (z <= 1.0)
		tmp = ((x / z) * y) / z;
	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_] := If[LessEqual[z, -1.0], N[(N[(x / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(N[(z * z), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

      2. sqr-neg 89.7%

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

      3. times-frac 99.0%

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

      4. sqr-neg 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in z around inf 97.9%

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

   if -1 < z < 1

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. associate-*r/ 83.4%

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

      3. sqr-neg 83.4%

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

      4. associate-*l* 83.4%

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

      5. associate-*l* 83.4%

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

      6. sqr-neg 83.4%

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

      7. associate-*l* 83.4%

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

      8. distribute-lft-in 83.4%

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

      9. fma-def 83.4%

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

      10. *-rgt-identity 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around 0 81.9%

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

      1. unpow2 81.9%

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

   6. Simplified 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-/r* 90.5%

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

      3. associate-/r/ 94.5%

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

      4. div-inv 94.1%

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

      5. clear-num 94.1%

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

      6. associate-*r/ 94.9%

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

   8. Applied egg-rr 94.9%

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

   if 1 < z

   1. Initial program 89.3%

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

      1. *-commutative 89.3%

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

      2. sqr-neg 89.3%

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

      3. times-frac 95.7%

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

      4. sqr-neg 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in z around inf 94.6%

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

      1. *-commutative 94.6%

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

      2. clear-num 94.5%

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

      3. frac-times 90.6%

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

      4. *-un-lft-identity 90.6%

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

   6. Applied egg-rr 90.6%

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

4. Final simplification 94.6%

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

Alternative 6: 93.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(z \cdot z\right) \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* (/ x z) (/ y (* z z)))
   (if (<= z 0.75) (/ (* y (- (/ x z) x)) z) (/ y (* (* z z) (/ z x))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (x / z) * (y / (z * z));
	} else if (z <= 0.75) {
		tmp = (y * ((x / z) - x)) / z;
	} 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) :: tmp
    if (z <= (-1.0d0)) then
        tmp = (x / z) * (y / (z * z))
    else if (z <= 0.75d0) then
        tmp = (y * ((x / z) - x)) / z
    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 tmp;
	if (z <= -1.0) {
		tmp = (x / z) * (y / (z * z));
	} else if (z <= 0.75) {
		tmp = (y * ((x / z) - x)) / z;
	} else {
		tmp = y / ((z * z) * (z / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (x / z) * (y / (z * z))
	elif z <= 0.75:
		tmp = (y * ((x / z) - x)) / z
	else:
		tmp = y / ((z * z) * (z / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(x / z) * Float64(y / Float64(z * z)));
	elseif (z <= 0.75)
		tmp = Float64(Float64(y * Float64(Float64(x / z) - x)) / z);
	else
		tmp = Float64(y / Float64(Float64(z * z) * 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 (z <= -1.0)
		tmp = (x / z) * (y / (z * z));
	elseif (z <= 0.75)
		tmp = (y * ((x / z) - x)) / z;
	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_] := If[LessEqual[z, -1.0], N[(N[(x / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.75], N[(N[(y * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(N[(z * z), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

      2. sqr-neg 89.7%

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

      3. times-frac 99.0%

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

      4. sqr-neg 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in z around inf 97.9%

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

   if -1 < z < 0.75

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. associate-*r/ 83.4%

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

      3. sqr-neg 83.4%

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

      4. associate-*l* 83.4%

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

      5. associate-*l* 83.4%

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

      6. sqr-neg 83.4%

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

      7. associate-*l* 83.4%

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

      8. distribute-lft-in 83.4%

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

      9. fma-def 83.4%

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

      10. *-rgt-identity 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around 0 66.9%

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

      1. unpow2 66.9%

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

      2. *-commutative 66.9%

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

      3. times-frac 78.3%

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

      4. mul-1-neg 78.3%

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

      5. *-commutative 78.3%

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

      6. associate-*r/ 75.1%

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

      7. distribute-lft-neg-in 75.1%

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

      8. distribute-rgt-out 94.9%

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

   6. Simplified 94.9%

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

   7. Taylor expanded in y around 0 95.7%

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

   if 0.75 < z

   1. Initial program 89.3%

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

      1. *-commutative 89.3%

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

      2. sqr-neg 89.3%

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

      3. times-frac 95.7%

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

      4. sqr-neg 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in z around inf 94.6%

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

      1. *-commutative 94.6%

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

      2. clear-num 94.5%

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

      3. frac-times 90.6%

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

      4. *-un-lft-identity 90.6%

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

   6. Applied egg-rr 90.6%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(z \cdot z\right) \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 7: 93.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x \cdot y}{z}}{z}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (/ (/ y z) (* z (/ z x)))
   (if (<= z 0.75) (/ (* y (- (/ x z) x)) z) (/ (/ (/ (* x y) z) z) z))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = (y / z) / (z * (z / x));
	} else if (z <= 0.75) {
		tmp = (y * ((x / z) - x)) / z;
	} else {
		tmp = (((x * y) / z) / 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 <= (-1.0d0)) then
        tmp = (y / z) / (z * (z / x))
    else if (z <= 0.75d0) then
        tmp = (y * ((x / z) - x)) / z
    else
        tmp = (((x * y) / z) / 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 <= -1.0) {
		tmp = (y / z) / (z * (z / x));
	} else if (z <= 0.75) {
		tmp = (y * ((x / z) - x)) / z;
	} else {
		tmp = (((x * y) / z) / z) / z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = (y / z) / (z * (z / x))
	elif z <= 0.75:
		tmp = (y * ((x / z) - x)) / z
	else:
		tmp = (((x * y) / z) / z) / z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(Float64(y / z) / Float64(z * Float64(z / x)));
	elseif (z <= 0.75)
		tmp = Float64(Float64(y * Float64(Float64(x / z) - x)) / z);
	else
		tmp = Float64(Float64(Float64(Float64(x * y) / z) / 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 <= -1.0)
		tmp = (y / z) / (z * (z / x));
	elseif (z <= 0.75)
		tmp = (y * ((x / z) - x)) / z;
	else
		tmp = (((x * y) / z) / 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, -1.0], N[(N[(y / z), $MachinePrecision] / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.75], N[(N[(y * N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*r/ 91.6%

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

      3. sqr-neg 91.6%

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

      4. associate-*l* 91.6%

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

      5. associate-*l* 91.6%

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

      6. sqr-neg 91.6%

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

      7. associate-*l* 91.6%

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

      8. distribute-lft-in 91.6%

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

      9. fma-def 91.6%

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

      10. *-rgt-identity 91.6%

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

   3. Simplified 91.6%

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

      1. associate-*r/ 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-/r* 92.6%

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

      4. associate-*r/ 97.4%

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

      5. fma-udef 97.4%

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

      6. distribute-lft1-in 97.4%

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

      7. frac-times 99.0%

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

      8. clear-num 98.9%

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

      9. frac-times 97.1%

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

      10. *-un-lft-identity 97.1%

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

   5. Applied egg-rr 97.1%

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

   6. Taylor expanded in z around inf 95.1%

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

      1. unpow2 95.1%

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

      2. associate-*r/ 96.1%

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

   8. Simplified 96.1%

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

   if -1 < z < 0.75

   1. Initial program 84.2%

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

      1. *-commutative 84.2%

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

      2. associate-*r/ 83.4%

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

      3. sqr-neg 83.4%

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

      4. associate-*l* 83.4%

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

      5. associate-*l* 83.4%

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

      6. sqr-neg 83.4%

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

      7. associate-*l* 83.4%

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

      8. distribute-lft-in 83.4%

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

      9. fma-def 83.4%

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

      10. *-rgt-identity 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in z around 0 66.9%

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

      1. unpow2 66.9%

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

      2. *-commutative 66.9%

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

      3. times-frac 78.3%

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

      4. mul-1-neg 78.3%

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

      5. *-commutative 78.3%

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

      6. associate-*r/ 75.1%

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

      7. distribute-lft-neg-in 75.1%

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

      8. distribute-rgt-out 94.9%

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

   6. Simplified 94.9%

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

   7. Taylor expanded in y around 0 95.7%

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

   if 0.75 < z

   1. Initial program 89.3%

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

      1. *-commutative 89.3%

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

      2. sqr-neg 89.3%

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

      3. times-frac 95.7%

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

      4. sqr-neg 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in z around inf 94.6%

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

      1. associate-/r* 95.9%

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

      2. associate-*l/ 97.1%

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

      3. associate-*l/ 95.6%

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

      4. associate-*r/ 95.6%

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

   6. Applied egg-rr 95.6%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{y \cdot \left(\frac{x}{z} - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x \cdot y}{z}}{z}}{z}\\ \end{array} \]

Alternative 8: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9000000000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -9000000000.0)
   (* x (/ y (* z z)))
   (if (<= x 5.5e-178) (* (/ x z) (/ y z)) (* y (/ x (* z z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -9000000000.0) {
		tmp = x * (y / (z * z));
	} else if (x <= 5.5e-178) {
		tmp = (x / z) * (y / 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 (x <= (-9000000000.0d0)) then
        tmp = x * (y / (z * z))
    else if (x <= 5.5d-178) then
        tmp = (x / z) * (y / 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 (x <= -9000000000.0) {
		tmp = x * (y / (z * z));
	} else if (x <= 5.5e-178) {
		tmp = (x / z) * (y / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -9000000000.0:
		tmp = x * (y / (z * z))
	elif x <= 5.5e-178:
		tmp = (x / z) * (y / z)
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -9000000000.0)
		tmp = Float64(x * Float64(y / Float64(z * z)));
	elseif (x <= 5.5e-178)
		tmp = Float64(Float64(x / z) * Float64(y / 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 (x <= -9000000000.0)
		tmp = x * (y / (z * z));
	elseif (x <= 5.5e-178)
		tmp = (x / z) * (y / 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[x, -9000000000.0], N[(x * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e-178], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9e9

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. associate-*r/ 89.9%

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

      3. sqr-neg 89.9%

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

      4. associate-*l* 89.9%

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

      5. associate-*l* 89.9%

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

      6. sqr-neg 89.9%

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

      7. associate-*l* 89.9%

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

      8. distribute-lft-in 90.0%

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

      9. fma-def 90.0%

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

      10. *-rgt-identity 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in z around 0 75.8%

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

      1. unpow2 75.8%

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

      2. associate-*l/ 81.4%

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

   6. Simplified 81.4%

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

   if -9e9 < x < 5.50000000000000028e-178

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-*r/ 76.4%

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

      3. sqr-neg 76.4%

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

      4. associate-*l* 76.4%

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

      5. associate-*l* 76.4%

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

      6. sqr-neg 76.4%

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

      7. associate-*l* 76.4%

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

      8. distribute-lft-in 76.4%

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

      9. fma-def 76.4%

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

      10. *-rgt-identity 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in z around 0 50.5%

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

      1. unpow2 50.5%

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

      2. *-commutative 50.5%

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

      3. times-frac 69.8%

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

      4. mul-1-neg 69.8%

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

      5. *-commutative 69.8%

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

      6. associate-*r/ 67.3%

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

      7. distribute-lft-neg-in 67.3%

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

      8. distribute-rgt-out 71.0%

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

   6. Simplified 71.0%

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

   7. Taylor expanded in z around 0 81.2%

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

   if 5.50000000000000028e-178 < x

   1. Initial program 92.8%

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

      1. *-commutative 92.8%

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

      2. associate-*r/ 90.9%

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

      3. sqr-neg 90.9%

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

      4. associate-*l* 90.9%

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

      5. associate-*l* 90.9%

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

      6. sqr-neg 90.9%

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

      7. associate-*l* 90.9%

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

      8. distribute-lft-in 90.9%

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

      9. fma-def 90.9%

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

      10. *-rgt-identity 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in z around 0 83.6%

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

      1. unpow2 83.6%

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

   6. Simplified 83.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9000000000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 9: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1600000000:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -1600000000.0)
   (/ x (/ (* z z) y))
   (if (<= x 4.2e-178) (* (/ x z) (/ y z)) (* y (/ x (* z z))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1600000000.0) {
		tmp = x / ((z * z) / y);
	} else if (x <= 4.2e-178) {
		tmp = (x / z) * (y / 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 (x <= (-1600000000.0d0)) then
        tmp = x / ((z * z) / y)
    else if (x <= 4.2d-178) then
        tmp = (x / z) * (y / 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 (x <= -1600000000.0) {
		tmp = x / ((z * z) / y);
	} else if (x <= 4.2e-178) {
		tmp = (x / z) * (y / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -1600000000.0:
		tmp = x / ((z * z) / y)
	elif x <= 4.2e-178:
		tmp = (x / z) * (y / z)
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -1600000000.0)
		tmp = Float64(x / Float64(Float64(z * z) / y));
	elseif (x <= 4.2e-178)
		tmp = Float64(Float64(x / z) * Float64(y / 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 (x <= -1600000000.0)
		tmp = x / ((z * z) / y);
	elseif (x <= 4.2e-178)
		tmp = (x / z) * (y / 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[x, -1600000000.0], N[(x / N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e-178], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6e9

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

      2. associate-*r/ 89.9%

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

      3. sqr-neg 89.9%

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

      4. associate-*l* 89.9%

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

      5. associate-*l* 89.9%

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

      6. sqr-neg 89.9%

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

      7. associate-*l* 89.9%

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

      8. distribute-lft-in 90.0%

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

      9. fma-def 90.0%

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

      10. *-rgt-identity 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in z around 0 75.8%

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

      1. unpow2 75.8%

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

      2. *-commutative 75.8%

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

      3. associate-/l* 81.5%

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

   6. Simplified 81.5%

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

   if -1.6e9 < x < 4.2e-178

   1. Initial program 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-*r/ 76.4%

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

      3. sqr-neg 76.4%

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

      4. associate-*l* 76.4%

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

      5. associate-*l* 76.4%

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

      6. sqr-neg 76.4%

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

      7. associate-*l* 76.4%

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

      8. distribute-lft-in 76.4%

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

      9. fma-def 76.4%

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

      10. *-rgt-identity 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in z around 0 50.5%

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

      1. unpow2 50.5%

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

      2. *-commutative 50.5%

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

      3. times-frac 69.8%

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

      4. mul-1-neg 69.8%

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

      5. *-commutative 69.8%

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

      6. associate-*r/ 67.3%

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

      7. distribute-lft-neg-in 67.3%

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

      8. distribute-rgt-out 71.0%

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

   6. Simplified 71.0%

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

   7. Taylor expanded in z around 0 81.2%

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

   if 4.2e-178 < x

   1. Initial program 92.8%

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

      1. *-commutative 92.8%

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

      2. associate-*r/ 90.9%

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

      3. sqr-neg 90.9%

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

      4. associate-*l* 90.9%

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

      5. associate-*l* 90.9%

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

      6. sqr-neg 90.9%

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

      7. associate-*l* 90.9%

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

      8. distribute-lft-in 90.9%

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

      9. fma-def 90.9%

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

      10. *-rgt-identity 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in z around 0 83.6%

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

      1. unpow2 83.6%

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

   6. Simplified 83.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1600000000:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 10: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y, double z) {
	return ((x / z) / (z / y)) / (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) / (z / y)) / (z + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 86.7%

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

   1. associate-*l* 86.7%

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

   2. times-frac 96.4%

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

   3. associate-/r* 96.8%

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

   4. associate-*r/ 97.3%

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

3. Simplified 97.3%

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

   1. clear-num 97.3%

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

   2. un-div-inv 97.5%

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

5. Applied egg-rr 97.5%

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

6. Final simplification 97.5%

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

Alternative 11: 97.0% 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 86.7%

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

   1. associate-*l* 86.7%

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

   2. times-frac 96.4%

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

   3. associate-/r* 96.8%

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

   4. associate-*r/ 97.3%

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

3. Simplified 97.3%

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

4. Final simplification 97.3%

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

Alternative 12: 78.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+16}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 1e+16) (* (/ x z) (/ y z)) (* y (/ x (* z z)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+16) {
		tmp = (x / z) * (y / 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 (y <= 1d+16) then
        tmp = (x / z) * (y / 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 (y <= 1e+16) {
		tmp = (x / z) * (y / z);
	} else {
		tmp = y * (x / (z * z));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= 1e+16:
		tmp = (x / z) * (y / z)
	else:
		tmp = y * (x / (z * z))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= 1e+16)
		tmp = Float64(Float64(x / z) * Float64(y / 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 (y <= 1e+16)
		tmp = (x / z) * (y / 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[y, 1e+16], N[(N[(x / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e16

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      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 53.7%

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

      1. unpow2 53.7%

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

      2. *-commutative 53.7%

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

      3. times-frac 61.8%

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

      4. mul-1-neg 61.8%

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

      5. *-commutative 61.8%

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

      6. associate-*r/ 64.4%

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

      7. distribute-lft-neg-in 64.4%

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

      8. distribute-rgt-out 71.3%

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

   6. Simplified 71.3%

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

   7. Taylor expanded in z around 0 77.9%

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

   if 1e16 < y

   1. Initial program 91.5%

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

      1. *-commutative 91.5%

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

      2. associate-*r/ 94.4%

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

      3. sqr-neg 94.4%

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

      4. associate-*l* 94.4%

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

      5. associate-*l* 94.4%

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

      6. sqr-neg 94.4%

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

      7. associate-*l* 94.4%

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

      8. distribute-lft-in 94.4%

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

      9. fma-def 94.4%

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

      10. *-rgt-identity 94.4%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in z around 0 91.0%

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

      1. unpow2 91.0%

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

   6. Simplified 91.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+16}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 13: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -5.5e-85) (/ x (* z (/ z y))) (/ y (* z (/ z x)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.5e-85) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y / (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) :: tmp
    if (x <= (-5.5d-85)) then
        tmp = x / (z * (z / y))
    else
        tmp = y / (z * (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 <= -5.5e-85) {
		tmp = x / (z * (z / y));
	} else {
		tmp = y / (z * (z / x));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -5.5e-85:
		tmp = x / (z * (z / y))
	else:
		tmp = y / (z * (z / x))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -5.5e-85)
		tmp = Float64(x / Float64(z * Float64(z / y)));
	else
		tmp = Float64(y / Float64(z * 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 <= -5.5e-85)
		tmp = x / (z * (z / y));
	else
		tmp = y / (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_] := If[LessEqual[x, -5.5e-85], N[(x / N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4999999999999997e-85

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

      2. associate-*r/ 87.9%

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

      3. sqr-neg 87.9%

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

      4. associate-*l* 87.9%

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

      5. associate-*l* 87.9%

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

      6. sqr-neg 87.9%

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

      7. associate-*l* 87.9%

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

      8. distribute-lft-in 87.9%

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

      9. fma-def 87.9%

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

      10. *-rgt-identity 87.9%

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

   3. Simplified 87.9%

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

   4. Taylor expanded in z around 0 72.0%

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

      1. unpow2 72.0%

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

   6. Simplified 72.0%

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

      1. *-commutative 72.0%

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

      2. associate-/r* 70.2%

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

      3. associate-/r/ 72.5%

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

      4. associate-/l/ 78.1%

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

   8. Applied egg-rr 78.1%

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

   if -5.4999999999999997e-85 < x

   1. Initial program 87.0%

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

      1. *-commutative 87.0%

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

      2. associate-*r/ 85.3%

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

      3. sqr-neg 85.3%

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

      4. associate-*l* 85.3%

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

      5. associate-*l* 85.3%

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

      6. sqr-neg 85.3%

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

      7. associate-*l* 85.3%

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

      8. distribute-lft-in 85.3%

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

      9. fma-def 85.3%

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

      10. *-rgt-identity 85.3%

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

   3. Simplified 85.3%

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

      1. associate-*r/ 87.0%

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

      2. *-commutative 87.0%

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

      3. associate-/r* 93.5%

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

      4. associate-*r/ 95.3%

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

      5. fma-udef 95.3%

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

      6. distribute-lft1-in 95.3%

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

      7. frac-times 90.7%

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

      8. clear-num 90.6%

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

      9. frac-times 94.1%

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

      10. *-un-lft-identity 94.1%

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

   5. Applied egg-rr 94.1%

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

   6. Taylor expanded in z around 0 77.7%

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

      1. unpow2 77.7%

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

      2. associate-/l* 79.6%

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

      3. associate-*r/ 83.2%

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

   8. Simplified 83.2%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 14: 40.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-310}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= z -2e-310) (* y (/ x (- z))) (/ x (/ z y))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e-310) {
		tmp = y * (x / -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 <= (-2d-310)) then
        tmp = y * (x / -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 <= -2e-310) {
		tmp = y * (x / -z);
	} else {
		tmp = x / (z / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if z <= -2e-310:
		tmp = y * (x / -z)
	else:
		tmp = x / (z / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (z <= -2e-310)
		tmp = Float64(y * Float64(x / Float64(-z)));
	else
		tmp = Float64(x / Float64(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 <= -2e-310)
		tmp = y * (x / -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, -2e-310], N[(y * N[(x / (-z)), $MachinePrecision]), $MachinePrecision], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.999999999999994e-310

   1. Initial program 88.1%

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

      1. *-commutative 88.1%

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

      2. associate-*r/ 88.1%

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

      3. sqr-neg 88.1%

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

      4. associate-*l* 88.1%

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

      5. associate-*l* 88.1%

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

      6. sqr-neg 88.1%

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

      7. associate-*l* 88.1%

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

      8. distribute-lft-in 88.1%

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

      9. fma-def 88.1%

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

      10. *-rgt-identity 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in z around 0 61.6%

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

      1. unpow2 61.6%

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

      2. *-commutative 61.6%

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

      3. times-frac 67.0%

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

      4. mul-1-neg 67.0%

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

      5. *-commutative 67.0%

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

      6. associate-*r/ 68.4%

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

      7. distribute-lft-neg-in 68.4%

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

      8. distribute-rgt-out 68.4%

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

   6. Simplified 68.4%

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

   7. Taylor expanded in z around inf 36.2%

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

      1. associate-*r/ 41.4%

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

      2. neg-mul-1 41.4%

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

      3. distribute-rgt-neg-in 41.4%

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

      4. neg-mul-1 41.4%

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

      5. metadata-eval 41.4%

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

      6. times-frac 41.4%

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

      7. *-lft-identity 41.4%

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

      8. neg-mul-1 41.4%

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

   9. Simplified 41.4%

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

   if -1.999999999999994e-310 < z

   1. Initial program 85.4%

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

      1. *-commutative 85.4%

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

      2. associate-*r/ 84.0%

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

      3. sqr-neg 84.0%

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

      4. associate-*l* 84.0%

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

      5. associate-*l* 84.0%

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

      6. sqr-neg 84.0%

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

      7. associate-*l* 84.0%

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

      8. distribute-lft-in 84.0%

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

      9. fma-def 84.0%

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

      10. *-rgt-identity 84.0%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in z around 0 38.5%

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

      1. unpow2 38.5%

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

      2. *-commutative 38.5%

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

      3. times-frac 44.8%

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

      4. mul-1-neg 44.8%

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

      5. *-commutative 44.8%

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

      6. associate-*r/ 46.0%

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

      7. distribute-lft-neg-in 46.0%

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

      8. distribute-rgt-out 66.4%

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

   6. Simplified 66.4%

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

   7. Taylor expanded in z around inf 14.1%

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

      1. associate-*r/ 18.5%

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

      2. neg-mul-1 18.5%

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

      3. distribute-rgt-neg-in 18.5%

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

      4. neg-mul-1 18.5%

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

      5. metadata-eval 18.5%

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

      6. times-frac 18.5%

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

      7. *-lft-identity 18.5%

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

      8. neg-mul-1 18.5%

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

   9. Simplified 18.5%

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

       1. expm1-log1p-u 18.4%

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

       2. expm1-udef 33.3%

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

       3. add-sqr-sqrt 22.5%

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

       4. sqrt-unprod 39.8%

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

       5. frac-times 34.2%

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

       6. sqr-neg 34.2%

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

       7. frac-times 39.8%

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

       8. sqrt-unprod 29.1%

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

       9. add-sqr-sqrt 43.2%

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

       10. associate-*r/ 43.3%

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

   11. Applied egg-rr 43.3%

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

       1. expm1-def 24.3%

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

       2. expm1-log1p 34.0%

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

       3. *-commutative 34.0%

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

       4. associate-/l* 41.4%

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

   13. Simplified 41.4%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-310}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 15: 32.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-269}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -3.2e-269) (* x (/ y z)) (* (/ x z) y)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.2e-269) {
		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 <= (-3.2d-269)) 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 <= -3.2e-269) {
		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 <= -3.2e-269:
		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 <= -3.2e-269)
		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 <= -3.2e-269)
		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, -3.2e-269], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000001e-269

   1. Initial program 82.3%

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

      1. *-commutative 82.3%

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

      2. associate-*r/ 82.9%

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

      3. sqr-neg 82.9%

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

      4. associate-*l* 82.9%

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

      5. associate-*l* 82.9%

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

      6. sqr-neg 82.9%

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

      7. associate-*l* 82.9%

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

      8. distribute-lft-in 82.9%

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

      9. fma-def 82.9%

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

      10. *-rgt-identity 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in z around 0 46.6%

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

      1. unpow2 46.6%

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

      2. *-commutative 46.6%

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

      3. times-frac 54.8%

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

      4. mul-1-neg 54.8%

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

      5. *-commutative 54.8%

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

      6. associate-*r/ 58.0%

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

      7. distribute-lft-neg-in 58.0%

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

      8. distribute-rgt-out 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in z around inf 21.3%

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

      1. associate-*r/ 23.9%

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

      2. neg-mul-1 23.9%

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

      3. distribute-rgt-neg-in 23.9%

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

      4. neg-mul-1 23.9%

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

      5. metadata-eval 23.9%

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

      6. times-frac 23.9%

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

      7. *-lft-identity 23.9%

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

      8. neg-mul-1 23.9%

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

   9. Simplified 23.9%

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

       1. expm1-log1p-u 18.8%

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

       2. expm1-udef 34.4%

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

       3. add-sqr-sqrt 16.2%

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

       4. sqrt-unprod 29.8%

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

       5. frac-times 25.0%

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

       6. sqr-neg 25.0%

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

       7. frac-times 29.8%

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

       8. sqrt-unprod 18.2%

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

       9. add-sqr-sqrt 34.7%

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

       10. associate-*r/ 34.8%

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

   11. Applied egg-rr 34.8%

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

       1. expm1-def 17.4%

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

       2. expm1-log1p 21.8%

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

       3. *-commutative 21.8%

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

       4. associate-/l* 29.6%

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

   13. Simplified 29.6%

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

       1. *-un-lft-identity 29.6%

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

       2. associate-*l/ 29.6%

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

       3. clear-num 28.7%

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

   15. Applied egg-rr 28.7%

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

   if -3.2000000000000001e-269 < x

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-*r/ 88.4%

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

      3. sqr-neg 88.4%

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

      4. associate-*l* 88.4%

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

      5. associate-*l* 88.4%

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

      6. sqr-neg 88.4%

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

      7. associate-*l* 88.4%

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

      8. distribute-lft-in 88.4%

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

      9. fma-def 88.4%

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

      10. *-rgt-identity 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in z around 0 52.6%

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

      1. unpow2 52.6%

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

      2. *-commutative 52.6%

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

      3. times-frac 56.7%

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

      4. mul-1-neg 56.7%

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

      5. *-commutative 56.7%

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

      6. associate-*r/ 56.6%

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

      7. distribute-lft-neg-in 56.6%

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

      8. distribute-rgt-out 66.8%

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

   6. Simplified 66.8%

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

   7. Taylor expanded in z around inf 28.0%

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

      1. associate-*r/ 34.4%

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

      2. neg-mul-1 34.4%

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

      3. distribute-rgt-neg-in 34.4%

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

      4. neg-mul-1 34.4%

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

      5. metadata-eval 34.4%

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

      6. times-frac 34.4%

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

      7. *-lft-identity 34.4%

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

      8. neg-mul-1 34.4%

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

   9. Simplified 34.4%

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

       1. expm1-log1p-u 28.4%

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

       2. expm1-udef 39.9%

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

       3. add-sqr-sqrt 30.4%

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

       4. sqrt-unprod 46.2%

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

       5. frac-times 44.6%

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

       6. sqr-neg 44.6%

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

       7. frac-times 46.2%

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

       8. sqrt-unprod 31.4%

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

       9. add-sqr-sqrt 43.2%

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

       10. associate-*r/ 43.2%

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

   11. Applied egg-rr 43.2%

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

       1. expm1-def 25.3%

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

       2. expm1-log1p 30.8%

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

       3. *-commutative 30.8%

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

       4. associate-/l* 32.6%

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

   13. Simplified 32.6%

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

       1. associate-/r/ 38.5%

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

   15. Applied egg-rr 38.5%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-269}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Alternative 16: 32.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-268}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -1e-268) (/ x (/ z y)) (* (/ x z) y)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e-268) {
		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 (x <= (-1d-268)) 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 (x <= -1e-268) {
		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 x <= -1e-268:
		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 (x <= -1e-268)
		tmp = Float64(x / Float64(z / 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 (x <= -1e-268)
		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[x, -1e-268], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999958e-269

   1. Initial program 82.3%

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

      1. *-commutative 82.3%

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

      2. associate-*r/ 82.9%

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

      3. sqr-neg 82.9%

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

      4. associate-*l* 82.9%

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

      5. associate-*l* 82.9%

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

      6. sqr-neg 82.9%

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

      7. associate-*l* 82.9%

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

      8. distribute-lft-in 82.9%

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

      9. fma-def 82.9%

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

      10. *-rgt-identity 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in z around 0 46.6%

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

      1. unpow2 46.6%

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

      2. *-commutative 46.6%

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

      3. times-frac 54.8%

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

      4. mul-1-neg 54.8%

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

      5. *-commutative 54.8%

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

      6. associate-*r/ 58.0%

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

      7. distribute-lft-neg-in 58.0%

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

      8. distribute-rgt-out 68.2%

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

   6. Simplified 68.2%

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

   7. Taylor expanded in z around inf 21.3%

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

      1. associate-*r/ 23.9%

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

      2. neg-mul-1 23.9%

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

      3. distribute-rgt-neg-in 23.9%

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

      4. neg-mul-1 23.9%

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

      5. metadata-eval 23.9%

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

      6. times-frac 23.9%

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

      7. *-lft-identity 23.9%

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

      8. neg-mul-1 23.9%

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

   9. Simplified 23.9%

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

       1. expm1-log1p-u 18.8%

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

       2. expm1-udef 34.4%

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

       3. add-sqr-sqrt 16.2%

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

       4. sqrt-unprod 29.8%

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

       5. frac-times 25.0%

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

       6. sqr-neg 25.0%

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

       7. frac-times 29.8%

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

       8. sqrt-unprod 18.2%

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

       9. add-sqr-sqrt 34.7%

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

       10. associate-*r/ 34.8%

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

   11. Applied egg-rr 34.8%

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

       1. expm1-def 17.4%

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

       2. expm1-log1p 21.8%

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

       3. *-commutative 21.8%

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

       4. associate-/l* 29.6%

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

   13. Simplified 29.6%

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

   if -9.99999999999999958e-269 < x

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-*r/ 88.4%

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

      3. sqr-neg 88.4%

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

      4. associate-*l* 88.4%

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

      5. associate-*l* 88.4%

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

      6. sqr-neg 88.4%

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

      7. associate-*l* 88.4%

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

      8. distribute-lft-in 88.4%

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

      9. fma-def 88.4%

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

      10. *-rgt-identity 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in z around 0 52.6%

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

      1. unpow2 52.6%

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

      2. *-commutative 52.6%

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

      3. times-frac 56.7%

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

      4. mul-1-neg 56.7%

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

      5. *-commutative 56.7%

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

      6. associate-*r/ 56.6%

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

      7. distribute-lft-neg-in 56.6%

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

      8. distribute-rgt-out 66.8%

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

   6. Simplified 66.8%

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

   7. Taylor expanded in z around inf 28.0%

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

      1. associate-*r/ 34.4%

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

      2. neg-mul-1 34.4%

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

      3. distribute-rgt-neg-in 34.4%

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

      4. neg-mul-1 34.4%

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

      5. metadata-eval 34.4%

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

      6. times-frac 34.4%

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

      7. *-lft-identity 34.4%

         \[\leadsto y \cdot \frac{\color{blue}{x}}{-1 \cdot z} \]

      8. neg-mul-1 34.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{-z}} \]

   9. Simplified 34.4%

      \[\leadsto \color{blue}{y \cdot \frac{x}{-z}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 28.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{x}{-z}\right)\right)} \]

       2. expm1-udef 39.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y \cdot \frac{x}{-z}\right)} - 1} \]

       3. add-sqr-sqrt 30.4%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\left(\sqrt{\frac{x}{-z}} \cdot \sqrt{\frac{x}{-z}}\right)}\right)} - 1 \]

       4. sqrt-unprod 46.2%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\sqrt{\frac{x}{-z} \cdot \frac{x}{-z}}}\right)} - 1 \]

       5. frac-times 44.6%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \sqrt{\color{blue}{\frac{x \cdot x}{\left(-z\right) \cdot \left(-z\right)}}}\right)} - 1 \]

       6. sqr-neg 44.6%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \sqrt{\frac{x \cdot x}{\color{blue}{z \cdot z}}}\right)} - 1 \]

       7. frac-times 46.2%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \sqrt{\color{blue}{\frac{x}{z} \cdot \frac{x}{z}}}\right)} - 1 \]

       8. sqrt-unprod 31.4%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\left(\sqrt{\frac{x}{z}} \cdot \sqrt{\frac{x}{z}}\right)}\right)} - 1 \]

       9. add-sqr-sqrt 43.2%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\frac{x}{z}}\right)} - 1 \]

       10. associate-*r/ 43.2%

           \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{y \cdot x}{z}}\right)} - 1 \]

   11. Applied egg-rr 43.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y \cdot x}{z}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 25.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot x}{z}\right)\right)} \]

       2. expm1-log1p 30.8%

          \[\leadsto \color{blue}{\frac{y \cdot x}{z}} \]

       3. *-commutative 30.8%

          \[\leadsto \frac{\color{blue}{x \cdot y}}{z} \]

       4. associate-/l* 32.6%

          \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]

   13. Simplified 32.6%

       \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]
   14. Step-by-step derivation

       1. associate-/r/ 38.5%

          \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

   15. Applied egg-rr 38.5%

       \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-268}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Alternative 17: 32.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-270}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -2.5e-270) (/ x (/ z y)) (/ y (/ z x))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.5e-270) {
		tmp = x / (z / y);
	} 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 <= (-2.5d-270)) then
        tmp = x / (z / y)
    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 <= -2.5e-270) {
		tmp = x / (z / y);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -2.5e-270:
		tmp = x / (z / y)
	else:
		tmp = y / (z / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -2.5e-270)
		tmp = Float64(x / Float64(z / y));
	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 <= -2.5e-270)
		tmp = x / (z / y);
	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, -2.5e-270], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4999999999999999e-270

   1. Initial program 82.3%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 82.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

      2. associate-*r/ 82.9%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

      3. sqr-neg 82.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(-z\right) \cdot \left(-z\right)\right)} \cdot \left(z + 1\right)} \]

      4. associate-*l* 82.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(-z\right) \cdot \left(\left(-z\right) \cdot \left(z + 1\right)\right)}} \]

      5. associate-*l* 82.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(-z\right) \cdot \left(-z\right)\right) \cdot \left(z + 1\right)}} \]

      6. sqr-neg 82.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)} \]

      7. associate-*l* 82.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      8. distribute-lft-in 82.9%

         \[\leadsto y \cdot \frac{x}{z \cdot \color{blue}{\left(z \cdot z + z \cdot 1\right)}} \]

      9. fma-def 82.9%

         \[\leadsto y \cdot \frac{x}{z \cdot \color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      10. *-rgt-identity 82.9%

          \[\leadsto y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 82.9%

      \[\leadsto \color{blue}{y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 46.6%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}} + -1 \cdot \frac{y \cdot x}{z}} \]
   5. Step-by-step derivation

      1. unpow2 46.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{z \cdot z}} + -1 \cdot \frac{y \cdot x}{z} \]

      2. *-commutative 46.6%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{z \cdot z} + -1 \cdot \frac{y \cdot x}{z} \]

      3. times-frac 54.8%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z}} + -1 \cdot \frac{y \cdot x}{z} \]

      4. mul-1-neg 54.8%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{z} + \color{blue}{\left(-\frac{y \cdot x}{z}\right)} \]

      5. *-commutative 54.8%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{z} + \left(-\frac{\color{blue}{x \cdot y}}{z}\right) \]

      6. associate-*r/ 58.0%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{z} + \left(-\color{blue}{x \cdot \frac{y}{z}}\right) \]

      7. distribute-lft-neg-in 58.0%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{z} + \color{blue}{\left(-x\right) \cdot \frac{y}{z}} \]

      8. distribute-rgt-out 68.2%

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(\frac{x}{z} + \left(-x\right)\right)} \]

   6. Simplified 68.2%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(\frac{x}{z} + \left(-x\right)\right)} \]

   7. Taylor expanded in z around inf 21.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot x}{z}} \]
   8. Step-by-step derivation

      1. associate-*r/ 23.9%

         \[\leadsto -1 \cdot \color{blue}{\left(y \cdot \frac{x}{z}\right)} \]

      2. neg-mul-1 23.9%

         \[\leadsto \color{blue}{-y \cdot \frac{x}{z}} \]

      3. distribute-rgt-neg-in 23.9%

         \[\leadsto \color{blue}{y \cdot \left(-\frac{x}{z}\right)} \]

      4. neg-mul-1 23.9%

         \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{x}{z}\right)} \]

      5. metadata-eval 23.9%

         \[\leadsto y \cdot \left(\color{blue}{\frac{1}{-1}} \cdot \frac{x}{z}\right) \]

      6. times-frac 23.9%

         \[\leadsto y \cdot \color{blue}{\frac{1 \cdot x}{-1 \cdot z}} \]

      7. *-lft-identity 23.9%

         \[\leadsto y \cdot \frac{\color{blue}{x}}{-1 \cdot z} \]

      8. neg-mul-1 23.9%

         \[\leadsto y \cdot \frac{x}{\color{blue}{-z}} \]

   9. Simplified 23.9%

      \[\leadsto \color{blue}{y \cdot \frac{x}{-z}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 18.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{x}{-z}\right)\right)} \]

       2. expm1-udef 34.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y \cdot \frac{x}{-z}\right)} - 1} \]

       3. add-sqr-sqrt 16.2%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\left(\sqrt{\frac{x}{-z}} \cdot \sqrt{\frac{x}{-z}}\right)}\right)} - 1 \]

       4. sqrt-unprod 29.8%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\sqrt{\frac{x}{-z} \cdot \frac{x}{-z}}}\right)} - 1 \]

       5. frac-times 25.0%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \sqrt{\color{blue}{\frac{x \cdot x}{\left(-z\right) \cdot \left(-z\right)}}}\right)} - 1 \]

       6. sqr-neg 25.0%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \sqrt{\frac{x \cdot x}{\color{blue}{z \cdot z}}}\right)} - 1 \]

       7. frac-times 29.8%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \sqrt{\color{blue}{\frac{x}{z} \cdot \frac{x}{z}}}\right)} - 1 \]

       8. sqrt-unprod 18.2%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\left(\sqrt{\frac{x}{z}} \cdot \sqrt{\frac{x}{z}}\right)}\right)} - 1 \]

       9. add-sqr-sqrt 34.7%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\frac{x}{z}}\right)} - 1 \]

       10. associate-*r/ 34.8%

           \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{y \cdot x}{z}}\right)} - 1 \]

   11. Applied egg-rr 34.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y \cdot x}{z}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 17.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot x}{z}\right)\right)} \]

       2. expm1-log1p 21.8%

          \[\leadsto \color{blue}{\frac{y \cdot x}{z}} \]

       3. *-commutative 21.8%

          \[\leadsto \frac{\color{blue}{x \cdot y}}{z} \]

       4. associate-/l* 29.6%

          \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]

   13. Simplified 29.6%

       \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]

   if -2.4999999999999999e-270 < x

   1. Initial program 90.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 90.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

      2. associate-*r/ 88.4%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

      3. sqr-neg 88.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(-z\right) \cdot \left(-z\right)\right)} \cdot \left(z + 1\right)} \]

      4. associate-*l* 88.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(-z\right) \cdot \left(\left(-z\right) \cdot \left(z + 1\right)\right)}} \]

      5. associate-*l* 88.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(-z\right) \cdot \left(-z\right)\right) \cdot \left(z + 1\right)}} \]

      6. sqr-neg 88.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)} \]

      7. associate-*l* 88.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      8. distribute-lft-in 88.4%

         \[\leadsto y \cdot \frac{x}{z \cdot \color{blue}{\left(z \cdot z + z \cdot 1\right)}} \]

      9. fma-def 88.4%

         \[\leadsto y \cdot \frac{x}{z \cdot \color{blue}{\mathsf{fma}\left(z, z, z \cdot 1\right)}} \]

      10. *-rgt-identity 88.4%

          \[\leadsto y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, \color{blue}{z}\right)} \]

   3. Simplified 88.4%

      \[\leadsto \color{blue}{y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}} \]

   4. Taylor expanded in z around 0 52.6%

      \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}} + -1 \cdot \frac{y \cdot x}{z}} \]
   5. Step-by-step derivation

      1. unpow2 52.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{z \cdot z}} + -1 \cdot \frac{y \cdot x}{z} \]

      2. *-commutative 52.6%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{z \cdot z} + -1 \cdot \frac{y \cdot x}{z} \]

      3. times-frac 56.7%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z}} + -1 \cdot \frac{y \cdot x}{z} \]

      4. mul-1-neg 56.7%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{z} + \color{blue}{\left(-\frac{y \cdot x}{z}\right)} \]

      5. *-commutative 56.7%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{z} + \left(-\frac{\color{blue}{x \cdot y}}{z}\right) \]

      6. associate-*r/ 56.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{z} + \left(-\color{blue}{x \cdot \frac{y}{z}}\right) \]

      7. distribute-lft-neg-in 56.6%

         \[\leadsto \frac{x}{z} \cdot \frac{y}{z} + \color{blue}{\left(-x\right) \cdot \frac{y}{z}} \]

      8. distribute-rgt-out 66.8%

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(\frac{x}{z} + \left(-x\right)\right)} \]

   6. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(\frac{x}{z} + \left(-x\right)\right)} \]

   7. Taylor expanded in z around inf 28.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot x}{z}} \]
   8. Step-by-step derivation

      1. associate-*r/ 34.4%

         \[\leadsto -1 \cdot \color{blue}{\left(y \cdot \frac{x}{z}\right)} \]

      2. neg-mul-1 34.4%

         \[\leadsto \color{blue}{-y \cdot \frac{x}{z}} \]

      3. distribute-rgt-neg-in 34.4%

         \[\leadsto \color{blue}{y \cdot \left(-\frac{x}{z}\right)} \]

      4. neg-mul-1 34.4%

         \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{x}{z}\right)} \]

      5. metadata-eval 34.4%

         \[\leadsto y \cdot \left(\color{blue}{\frac{1}{-1}} \cdot \frac{x}{z}\right) \]

      6. times-frac 34.4%

         \[\leadsto y \cdot \color{blue}{\frac{1 \cdot x}{-1 \cdot z}} \]

      7. *-lft-identity 34.4%

         \[\leadsto y \cdot \frac{\color{blue}{x}}{-1 \cdot z} \]

      8. neg-mul-1 34.4%

         \[\leadsto y \cdot \frac{x}{\color{blue}{-z}} \]

   9. Simplified 34.4%

      \[\leadsto \color{blue}{y \cdot \frac{x}{-z}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 28.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{x}{-z}\right)\right)} \]

       2. expm1-udef 39.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y \cdot \frac{x}{-z}\right)} - 1} \]

       3. add-sqr-sqrt 30.4%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\left(\sqrt{\frac{x}{-z}} \cdot \sqrt{\frac{x}{-z}}\right)}\right)} - 1 \]

       4. sqrt-unprod 46.2%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\sqrt{\frac{x}{-z} \cdot \frac{x}{-z}}}\right)} - 1 \]

       5. frac-times 44.6%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \sqrt{\color{blue}{\frac{x \cdot x}{\left(-z\right) \cdot \left(-z\right)}}}\right)} - 1 \]

       6. sqr-neg 44.6%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \sqrt{\frac{x \cdot x}{\color{blue}{z \cdot z}}}\right)} - 1 \]

       7. frac-times 46.2%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \sqrt{\color{blue}{\frac{x}{z} \cdot \frac{x}{z}}}\right)} - 1 \]

       8. sqrt-unprod 31.4%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\left(\sqrt{\frac{x}{z}} \cdot \sqrt{\frac{x}{z}}\right)}\right)} - 1 \]

       9. add-sqr-sqrt 43.2%

          \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\frac{x}{z}}\right)} - 1 \]

       10. associate-*r/ 43.2%

           \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{y \cdot x}{z}}\right)} - 1 \]

   11. Applied egg-rr 43.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y \cdot x}{z}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 25.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot x}{z}\right)\right)} \]

       2. expm1-log1p 30.8%

          \[\leadsto \color{blue}{\frac{y \cdot x}{z}} \]

       3. associate-/l* 39.1%

          \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}} \]

   13. Simplified 39.1%

       \[\leadsto \color{blue}{\frac{y}{\frac{z}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-270}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 18: 72.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ y \cdot \frac{x}{z \cdot z} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* y (/ x (* z z))))

C

assert(x < y);
double code(double x, double y, double z) {
	return y * (x / (z * 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 = y * (x / (z * z))
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	return y * (x / (z * z));
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return y * (x / (z * z))

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(y * Float64(x / Float64(z * z)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = y * (x / (z * z));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(y * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.7%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 86.7%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

   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* 86.0%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(-z\right) \cdot \left(\left(-z\right) \cdot \left(z + 1\right)\right)}} \]

   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* 86.0%

      \[\leadsto y \cdot \frac{x}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

   8. distribute-lft-in 86.0%

      \[\leadsto y \cdot \frac{x}{z \cdot \color{blue}{\left(z \cdot z + z \cdot 1\right)}} \]

   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 77.4%

   \[\leadsto y \cdot \color{blue}{\frac{x}{{z}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 77.4%

      \[\leadsto y \cdot \frac{x}{\color{blue}{z \cdot z}} \]

6. Simplified 77.4%

   \[\leadsto y \cdot \color{blue}{\frac{x}{z \cdot z}} \]

7. Final simplification 77.4%

   \[\leadsto y \cdot \frac{x}{z \cdot z} \]

Alternative 19: 31.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{z} \cdot y \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* (/ x z) y))

C

assert(x < y);
double code(double x, double y, double z) {
	return (x / 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) * y
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	return (x / z) * y;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return (x / z) * y

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(x / z) * y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = (x / 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] * y), $MachinePrecision]

Derivation

1. Initial program 86.7%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 86.7%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

   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* 86.0%

      \[\leadsto y \cdot \frac{x}{\color{blue}{\left(-z\right) \cdot \left(\left(-z\right) \cdot \left(z + 1\right)\right)}} \]

   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* 86.0%

      \[\leadsto y \cdot \frac{x}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

   8. distribute-lft-in 86.0%

      \[\leadsto y \cdot \frac{x}{z \cdot \color{blue}{\left(z \cdot z + z \cdot 1\right)}} \]

   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 50.0%

   \[\leadsto \color{blue}{\frac{y \cdot x}{{z}^{2}} + -1 \cdot \frac{y \cdot x}{z}} \]
5. Step-by-step derivation

   1. unpow2 50.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{z \cdot z}} + -1 \cdot \frac{y \cdot x}{z} \]

   2. *-commutative 50.0%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{z \cdot z} + -1 \cdot \frac{y \cdot x}{z} \]

   3. times-frac 55.9%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{z}} + -1 \cdot \frac{y \cdot x}{z} \]

   4. mul-1-neg 55.9%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{z} + \color{blue}{\left(-\frac{y \cdot x}{z}\right)} \]

   5. *-commutative 55.9%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{z} + \left(-\frac{\color{blue}{x \cdot y}}{z}\right) \]

   6. associate-*r/ 57.2%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{z} + \left(-\color{blue}{x \cdot \frac{y}{z}}\right) \]

   7. distribute-lft-neg-in 57.2%

      \[\leadsto \frac{x}{z} \cdot \frac{y}{z} + \color{blue}{\left(-x\right) \cdot \frac{y}{z}} \]

   8. distribute-rgt-out 67.4%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(\frac{x}{z} + \left(-x\right)\right)} \]

6. Simplified 67.4%

   \[\leadsto \color{blue}{\frac{y}{z} \cdot \left(\frac{x}{z} + \left(-x\right)\right)} \]

7. Taylor expanded in z around inf 25.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot x}{z}} \]
8. Step-by-step derivation

   1. associate-*r/ 29.9%

      \[\leadsto -1 \cdot \color{blue}{\left(y \cdot \frac{x}{z}\right)} \]

   2. neg-mul-1 29.9%

      \[\leadsto \color{blue}{-y \cdot \frac{x}{z}} \]

   3. distribute-rgt-neg-in 29.9%

      \[\leadsto \color{blue}{y \cdot \left(-\frac{x}{z}\right)} \]

   4. neg-mul-1 29.9%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot \frac{x}{z}\right)} \]

   5. metadata-eval 29.9%

      \[\leadsto y \cdot \left(\color{blue}{\frac{1}{-1}} \cdot \frac{x}{z}\right) \]

   6. times-frac 29.9%

      \[\leadsto y \cdot \color{blue}{\frac{1 \cdot x}{-1 \cdot z}} \]

   7. *-lft-identity 29.9%

      \[\leadsto y \cdot \frac{\color{blue}{x}}{-1 \cdot z} \]

   8. neg-mul-1 29.9%

      \[\leadsto y \cdot \frac{x}{\color{blue}{-z}} \]

9. Simplified 29.9%

   \[\leadsto \color{blue}{y \cdot \frac{x}{-z}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 24.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{x}{-z}\right)\right)} \]

    2. expm1-udef 37.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y \cdot \frac{x}{-z}\right)} - 1} \]

    3. add-sqr-sqrt 24.4%

       \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\left(\sqrt{\frac{x}{-z}} \cdot \sqrt{\frac{x}{-z}}\right)}\right)} - 1 \]

    4. sqrt-unprod 39.2%

       \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\sqrt{\frac{x}{-z} \cdot \frac{x}{-z}}}\right)} - 1 \]

    5. frac-times 36.2%

       \[\leadsto e^{\mathsf{log1p}\left(y \cdot \sqrt{\color{blue}{\frac{x \cdot x}{\left(-z\right) \cdot \left(-z\right)}}}\right)} - 1 \]

    6. sqr-neg 36.2%

       \[\leadsto e^{\mathsf{log1p}\left(y \cdot \sqrt{\frac{x \cdot x}{\color{blue}{z \cdot z}}}\right)} - 1 \]

    7. frac-times 39.2%

       \[\leadsto e^{\mathsf{log1p}\left(y \cdot \sqrt{\color{blue}{\frac{x}{z} \cdot \frac{x}{z}}}\right)} - 1 \]

    8. sqrt-unprod 25.8%

       \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\left(\sqrt{\frac{x}{z}} \cdot \sqrt{\frac{x}{z}}\right)}\right)} - 1 \]

    9. add-sqr-sqrt 39.6%

       \[\leadsto e^{\mathsf{log1p}\left(y \cdot \color{blue}{\frac{x}{z}}\right)} - 1 \]

    10. associate-*r/ 39.6%

        \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{y \cdot x}{z}}\right)} - 1 \]

11. Applied egg-rr 39.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y \cdot x}{z}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 21.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot x}{z}\right)\right)} \]

    2. expm1-log1p 27.0%

       \[\leadsto \color{blue}{\frac{y \cdot x}{z}} \]

    3. *-commutative 27.0%

       \[\leadsto \frac{\color{blue}{x \cdot y}}{z} \]

    4. associate-/l* 31.4%

       \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]

13. Simplified 31.4%

    \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]
14. Step-by-step derivation

    1. associate-/r/ 32.8%

       \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

15. Applied egg-rr 32.8%

    \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

16. Final simplification 32.8%

    \[\leadsto \frac{x}{z} \cdot y \]

Developer target: 95.7% 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 2023274 
(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))))