Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.0% → 97.1%

Time: 4.2s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+98}:\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \leq 10^{-296}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3e+98)
   (+ x (/ x (/ z y)))
   (if (<= x 1e-296) (fma (/ x z) y x) (* x (+ 1.0 (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e+98) {
		tmp = x + (x / (z / y));
	} else if (x <= 1e-296) {
		tmp = fma((x / z), y, x);
	} else {
		tmp = x * (1.0 + (y / z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3e+98)
		tmp = Float64(x + Float64(x / Float64(z / y)));
	elseif (x <= 1e-296)
		tmp = fma(Float64(x / z), y, x);
	else
		tmp = Float64(x * Float64(1.0 + Float64(y / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3e+98], N[(x + N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-296], N[(N[(x / z), $MachinePrecision] * y + x), $MachinePrecision], N[(x * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.0000000000000001e98

   1. Initial program 75.7%

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

      1. associate-*l/ 87.6%

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

      2. distribute-rgt-in 70.0%

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

      3. *-commutative 70.0%

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

      4. associate-/r/ 87.5%

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

      5. *-inverses 87.5%

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

      6. /-rgt-identity 87.5%

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

      7. associate-*r/ 89.7%

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

      8. *-commutative 89.7%

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

      9. associate-*r/ 100.0%

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

      10. fma-def 99.9%

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

   3. Simplified 99.9%

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

      1. fma-udef 100.0%

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

      2. clear-num 99.9%

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

      3. un-div-inv 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -3.0000000000000001e98 < x < 1e-296

   1. Initial program 90.9%

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

      1. associate-*l/ 84.6%

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

      2. distribute-lft-in 84.7%

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

      3. fma-def 84.7%

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

      4. associate-/r/ 96.8%

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

      5. *-inverses 96.8%

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

      6. /-rgt-identity 96.8%

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

   3. Simplified 96.8%

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

   if 1e-296 < x

   1. Initial program 85.8%

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

      1. associate-*l/ 81.4%

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

      2. distribute-rgt-in 75.9%

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

      3. *-commutative 75.9%

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

      4. associate-/r/ 89.3%

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

      5. *-inverses 89.3%

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

      6. /-rgt-identity 89.3%

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

      7. associate-*r/ 95.7%

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

      8. *-commutative 95.7%

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

      9. associate-*r/ 97.5%

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

      10. fma-def 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in x around 0 97.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+98}:\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \leq 10^{-296}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \]

Alternative 2: 71.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 14.6:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+80}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.6e+72)
   x
   (if (<= z 14.6)
     (* y (/ x z))
     (if (<= z 1.8e+80) x (if (<= z 6.4e+134) (* x (/ y z)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.6e+72) {
		tmp = x;
	} else if (z <= 14.6) {
		tmp = y * (x / z);
	} else if (z <= 1.8e+80) {
		tmp = x;
	} else if (z <= 6.4e+134) {
		tmp = x * (y / z);
	} else {
		tmp = x;
	}
	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 <= (-1.6d+72)) then
        tmp = x
    else if (z <= 14.6d0) then
        tmp = y * (x / z)
    else if (z <= 1.8d+80) then
        tmp = x
    else if (z <= 6.4d+134) then
        tmp = x * (y / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.6e+72) {
		tmp = x;
	} else if (z <= 14.6) {
		tmp = y * (x / z);
	} else if (z <= 1.8e+80) {
		tmp = x;
	} else if (z <= 6.4e+134) {
		tmp = x * (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.6e+72:
		tmp = x
	elif z <= 14.6:
		tmp = y * (x / z)
	elif z <= 1.8e+80:
		tmp = x
	elif z <= 6.4e+134:
		tmp = x * (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.6e+72)
		tmp = x;
	elseif (z <= 14.6)
		tmp = Float64(y * Float64(x / z));
	elseif (z <= 1.8e+80)
		tmp = x;
	elseif (z <= 6.4e+134)
		tmp = Float64(x * Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.6e+72)
		tmp = x;
	elseif (z <= 14.6)
		tmp = y * (x / z);
	elseif (z <= 1.8e+80)
		tmp = x;
	elseif (z <= 6.4e+134)
		tmp = x * (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.6e+72], x, If[LessEqual[z, 14.6], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.8e+80], x, If[LessEqual[z, 6.4e+134], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6000000000000001e72 or 14.5999999999999996 < z < 1.79999999999999997e80 or 6.4000000000000001e134 < z

   1. Initial program 73.6%

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

      1. associate-*l/ 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in z around inf 82.5%

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

   if -1.6000000000000001e72 < z < 14.5999999999999996

   1. Initial program 94.0%

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

      1. associate-*l/ 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in z around 0 73.9%

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

      1. associate-*r/ 73.5%

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

   6. Simplified 73.5%

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

   if 1.79999999999999997e80 < z < 6.4000000000000001e134

   1. Initial program 85.3%

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

      1. associate-*l/ 85.4%

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

      2. distribute-rgt-in 85.4%

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

      3. *-commutative 85.4%

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

      4. associate-/r/ 85.6%

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

      5. *-inverses 85.6%

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

      6. /-rgt-identity 85.6%

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

      7. associate-*r/ 85.5%

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

      8. *-commutative 85.5%

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

      9. associate-*r/ 99.8%

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

      10. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.8%

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

   5. Taylor expanded in y around inf 68.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 14.6:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+80}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 71.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 16:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+81}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.8e+71)
   x
   (if (<= z 16.0)
     (/ y (/ z x))
     (if (<= z 5.7e+81) x (if (<= z 5.9e+134) (* x (/ y z)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.8e+71) {
		tmp = x;
	} else if (z <= 16.0) {
		tmp = y / (z / x);
	} else if (z <= 5.7e+81) {
		tmp = x;
	} else if (z <= 5.9e+134) {
		tmp = x * (y / z);
	} else {
		tmp = x;
	}
	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 <= (-1.8d+71)) then
        tmp = x
    else if (z <= 16.0d0) then
        tmp = y / (z / x)
    else if (z <= 5.7d+81) then
        tmp = x
    else if (z <= 5.9d+134) then
        tmp = x * (y / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.8e+71) {
		tmp = x;
	} else if (z <= 16.0) {
		tmp = y / (z / x);
	} else if (z <= 5.7e+81) {
		tmp = x;
	} else if (z <= 5.9e+134) {
		tmp = x * (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.8e+71:
		tmp = x
	elif z <= 16.0:
		tmp = y / (z / x)
	elif z <= 5.7e+81:
		tmp = x
	elif z <= 5.9e+134:
		tmp = x * (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.8e+71)
		tmp = x;
	elseif (z <= 16.0)
		tmp = Float64(y / Float64(z / x));
	elseif (z <= 5.7e+81)
		tmp = x;
	elseif (z <= 5.9e+134)
		tmp = Float64(x * Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.8e+71)
		tmp = x;
	elseif (z <= 16.0)
		tmp = y / (z / x);
	elseif (z <= 5.7e+81)
		tmp = x;
	elseif (z <= 5.9e+134)
		tmp = x * (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.8e+71], x, If[LessEqual[z, 16.0], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.7e+81], x, If[LessEqual[z, 5.9e+134], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8e71 or 16 < z < 5.70000000000000033e81 or 5.90000000000000008e134 < z

   1. Initial program 73.6%

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

      1. associate-*l/ 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in z around inf 82.5%

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

   if -1.8e71 < z < 16

   1. Initial program 94.0%

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

      1. associate-*l/ 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in z around 0 73.9%

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

      1. associate-/l* 73.8%

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

   6. Simplified 73.8%

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

   if 5.70000000000000033e81 < z < 5.90000000000000008e134

   1. Initial program 85.3%

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

      1. associate-*l/ 85.4%

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

      2. distribute-rgt-in 85.4%

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

      3. *-commutative 85.4%

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

      4. associate-/r/ 85.6%

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

      5. *-inverses 85.6%

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

      6. /-rgt-identity 85.6%

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

      7. associate-*r/ 85.5%

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

      8. *-commutative 85.5%

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

      9. associate-*r/ 99.8%

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

      10. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.8%

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

   5. Taylor expanded in y around inf 68.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 16:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+81}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 71.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 28.5:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+80}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.6e-50)
   x
   (if (<= z 28.5)
     (/ (* x y) z)
     (if (<= z 9.6e+80) x (if (<= z 5.9e+134) (* x (/ y z)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.6e-50) {
		tmp = x;
	} else if (z <= 28.5) {
		tmp = (x * y) / z;
	} else if (z <= 9.6e+80) {
		tmp = x;
	} else if (z <= 5.9e+134) {
		tmp = x * (y / z);
	} else {
		tmp = x;
	}
	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 <= (-7.6d-50)) then
        tmp = x
    else if (z <= 28.5d0) then
        tmp = (x * y) / z
    else if (z <= 9.6d+80) then
        tmp = x
    else if (z <= 5.9d+134) then
        tmp = x * (y / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.6e-50) {
		tmp = x;
	} else if (z <= 28.5) {
		tmp = (x * y) / z;
	} else if (z <= 9.6e+80) {
		tmp = x;
	} else if (z <= 5.9e+134) {
		tmp = x * (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.6e-50:
		tmp = x
	elif z <= 28.5:
		tmp = (x * y) / z
	elif z <= 9.6e+80:
		tmp = x
	elif z <= 5.9e+134:
		tmp = x * (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.6e-50)
		tmp = x;
	elseif (z <= 28.5)
		tmp = Float64(Float64(x * y) / z);
	elseif (z <= 9.6e+80)
		tmp = x;
	elseif (z <= 5.9e+134)
		tmp = Float64(x * Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7.6e-50)
		tmp = x;
	elseif (z <= 28.5)
		tmp = (x * y) / z;
	elseif (z <= 9.6e+80)
		tmp = x;
	elseif (z <= 5.9e+134)
		tmp = x * (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.6e-50], x, If[LessEqual[z, 28.5], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 9.6e+80], x, If[LessEqual[z, 5.9e+134], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.5999999999999998e-50 or 28.5 < z < 9.59999999999999916e80 or 5.90000000000000008e134 < z

   1. Initial program 78.4%

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

      1. associate-*l/ 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in z around inf 76.0%

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

   if -7.5999999999999998e-50 < z < 28.5

   1. Initial program 93.5%

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

      1. associate-*l/ 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in z around 0 79.3%

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

   if 9.59999999999999916e80 < z < 5.90000000000000008e134

   1. Initial program 85.3%

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

      1. associate-*l/ 85.4%

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

      2. distribute-rgt-in 85.4%

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

      3. *-commutative 85.4%

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

      4. associate-/r/ 85.6%

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

      5. *-inverses 85.6%

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

      6. /-rgt-identity 85.6%

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

      7. associate-*r/ 85.5%

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

      8. *-commutative 85.5%

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

      9. associate-*r/ 99.8%

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

      10. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.8%

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

   5. Taylor expanded in y around inf 68.6%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 28.5:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+80}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.9 \cdot 10^{+134}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 96.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+23}:\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{-266}:\\ \;\;\;\;\frac{x \cdot \left(z + y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.2e+23)
   (+ x (/ x (/ z y)))
   (if (<= x 6.1e-266) (/ (* x (+ z y)) z) (* x (+ 1.0 (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.2e+23) {
		tmp = x + (x / (z / y));
	} else if (x <= 6.1e-266) {
		tmp = (x * (z + y)) / z;
	} else {
		tmp = x * (1.0 + (y / 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 (x <= (-6.2d+23)) then
        tmp = x + (x / (z / y))
    else if (x <= 6.1d-266) then
        tmp = (x * (z + y)) / z
    else
        tmp = x * (1.0d0 + (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.2e+23) {
		tmp = x + (x / (z / y));
	} else if (x <= 6.1e-266) {
		tmp = (x * (z + y)) / z;
	} else {
		tmp = x * (1.0 + (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.2e+23:
		tmp = x + (x / (z / y))
	elif x <= 6.1e-266:
		tmp = (x * (z + y)) / z
	else:
		tmp = x * (1.0 + (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.2e+23)
		tmp = Float64(x + Float64(x / Float64(z / y)));
	elseif (x <= 6.1e-266)
		tmp = Float64(Float64(x * Float64(z + y)) / z);
	else
		tmp = Float64(x * Float64(1.0 + Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6.2e+23)
		tmp = x + (x / (z / y));
	elseif (x <= 6.1e-266)
		tmp = (x * (z + y)) / z;
	else
		tmp = x * (1.0 + (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.2e+23], N[(x + N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.1e-266], N[(N[(x * N[(z + y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.19999999999999941e23

   1. Initial program 77.6%

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

      1. associate-*l/ 91.4%

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

      2. distribute-rgt-in 79.3%

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

      3. *-commutative 79.3%

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

      4. associate-/r/ 91.4%

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

      5. *-inverses 91.4%

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

      6. /-rgt-identity 91.4%

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

      7. associate-*r/ 90.1%

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

      8. *-commutative 90.1%

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

      9. associate-*r/ 99.9%

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

      10. fma-def 99.8%

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

   3. Simplified 99.8%

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

      1. fma-udef 99.9%

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

      2. clear-num 99.9%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -6.19999999999999941e23 < x < 6.1e-266

   1. Initial program 94.1%

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

   if 6.1e-266 < x

   1. Initial program 84.7%

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

      1. associate-*l/ 85.0%

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

      2. distribute-rgt-in 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-/r/ 91.0%

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

      5. *-inverses 91.0%

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

      6. /-rgt-identity 91.0%

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

      7. associate-*r/ 95.4%

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

      8. *-commutative 95.4%

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

      9. associate-*r/ 98.2%

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

      10. fma-def 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in x around 0 98.2%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+23}:\\ \;\;\;\;x + \frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{-266}:\\ \;\;\;\;\frac{x \cdot \left(z + y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{y}{z}\right)\\ \end{array} \]

Alternative 6: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 15:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.3e+71) x (if (<= z 15.0) (* y (/ x z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.3e+71) {
		tmp = x;
	} else if (z <= 15.0) {
		tmp = y * (x / z);
	} else {
		tmp = x;
	}
	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 <= (-4.3d+71)) then
        tmp = x
    else if (z <= 15.0d0) then
        tmp = y * (x / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.3e+71) {
		tmp = x;
	} else if (z <= 15.0) {
		tmp = y * (x / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.3e+71:
		tmp = x
	elif z <= 15.0:
		tmp = y * (x / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.3e+71)
		tmp = x;
	elseif (z <= 15.0)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.3e+71)
		tmp = x;
	elseif (z <= 15.0)
		tmp = y * (x / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.3e+71], x, If[LessEqual[z, 15.0], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if z < -4.29999999999999984e71 or 15 < z

   1. Initial program 75.0%

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

      1. associate-*l/ 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in z around inf 76.9%

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

   if -4.29999999999999984e71 < z < 15

   1. Initial program 94.0%

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

      1. associate-*l/ 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in z around 0 73.9%

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

      1. associate-*r/ 73.5%

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

   6. Simplified 73.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 15:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (1.0d0 + (y / z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

   1. associate-*l/ 83.7%

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

   2. distribute-rgt-in 78.0%

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

   3. *-commutative 78.0%

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

   4. associate-/r/ 91.7%

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

   5. *-inverses 91.7%

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

   6. /-rgt-identity 91.7%

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

   7. associate-*r/ 93.8%

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

   8. *-commutative 93.8%

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

   9. associate-*r/ 95.1%

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

   10. fma-def 95.1%

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

3. Simplified 95.1%

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

4. Taylor expanded in x around 0 95.1%

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

5. Final simplification 95.1%

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

Alternative 8: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{x}{\frac{z}{y}} \end{array} \]

FPCore

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

C

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

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 + (x / (z / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

   1. associate-*l/ 83.7%

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

   2. distribute-rgt-in 78.0%

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

   3. *-commutative 78.0%

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

   4. associate-/r/ 91.7%

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

   5. *-inverses 91.7%

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

   6. /-rgt-identity 91.7%

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

   7. associate-*r/ 93.8%

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

   8. *-commutative 93.8%

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

   9. associate-*r/ 95.1%

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

   10. fma-def 95.1%

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

3. Simplified 95.1%

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

   1. fma-udef 95.1%

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

   2. clear-num 95.1%

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

   3. un-div-inv 95.1%

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

5. Applied egg-rr 95.1%

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

6. Final simplification 95.1%

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

Alternative 9: 51.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

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
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 85.9%

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

   1. associate-*l/ 83.7%

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

3. Simplified 83.7%

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

4. Taylor expanded in z around inf 46.8%

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

5. Final simplification 46.8%

   \[\leadsto x \]

Developer target: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{z}{y + z}} \end{array} \]

FPCore

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

C

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

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 / (z / (y + z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023238 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))