Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3

Percentage Accurate: 88.0% → 97.6%

Time: 8.0s

Alternatives: 10

Speedup: 0.6×

• 21.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / 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 - x))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / 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 - x))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+31}:\\ \;\;\;\;y + x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+31) (+ y (* x (/ (- 1.0 y) z))) (- y (* y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+31) {
		tmp = y + (x * ((1.0 - y) / z));
	} else {
		tmp = y - (y * (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 (y <= 2d+31) then
        tmp = y + (x * ((1.0d0 - y) / z))
    else
        tmp = y - (y * (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+31) {
		tmp = y + (x * ((1.0 - y) / z));
	} else {
		tmp = y - (y * (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e+31:
		tmp = y + (x * ((1.0 - y) / z))
	else:
		tmp = y - (y * (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+31)
		tmp = Float64(y + Float64(x * Float64(Float64(1.0 - y) / z)));
	else
		tmp = Float64(y - Float64(y * Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e+31)
		tmp = y + (x * ((1.0 - y) / z));
	else
		tmp = y - (y * (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e+31], N[(y + N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y - N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9999999999999999e31

   1. Initial program 91.6%

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

      1. add-cube-cbrt 91.0%

         \[\leadsto \frac{x + y \cdot \color{blue}{\left(\left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right) \cdot \sqrt[3]{z - x}\right)}}{z} \]

      2. pow3 91.0%

         \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   4. Applied egg-rr 91.0%

      \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   5. Taylor expanded in x around 0 98.4%

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

      1. neg-mul-1 98.4%

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

      2. +-commutative 98.4%

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

      3. sub-neg 98.4%

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

      4. div-sub 98.4%

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

   7. Simplified 98.4%

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

   if 1.9999999999999999e31 < y

   1. Initial program 76.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 76.9%

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

      1. associate-/l* 99.9%

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

      2. div-sub 99.9%

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

      3. *-inverses 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 85.6%

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

      1. mul-1-neg 85.6%

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

      2. associate-*l/ 99.9%

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

      3. *-commutative 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

      5. distribute-neg-frac2 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+31}:\\ \;\;\;\;y + x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -78000000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -78000000000000.0) (not (<= y 1.0)))
   (* y (- 1.0 (/ x z)))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -78000000000000.0) || !(y <= 1.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (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 ((y <= (-78000000000000.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -78000000000000.0) || !(y <= 1.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -78000000000000.0) or not (y <= 1.0):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -78000000000000.0) || !(y <= 1.0))
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -78000000000000.0) || ~((y <= 1.0)))
		tmp = y * (1.0 - (x / z));
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -78000000000000.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.8e13 or 1 < y

   1. Initial program 75.5%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 73.4%

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

      1. associate-/l* 97.7%

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

      2. div-sub 97.8%

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

      3. *-inverses 97.8%

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

   5. Simplified 97.8%

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

   if -7.8e13 < y < 1

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.4%

         \[\leadsto \frac{x + y \cdot \color{blue}{\left(\left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right) \cdot \sqrt[3]{z - x}\right)}}{z} \]

      2. pow3 99.4%

         \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   5. Taylor expanded in x around 0 99.7%

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

      1. neg-mul-1 99.7%

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

      2. +-commutative 99.7%

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

      3. sub-neg 99.7%

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

      4. div-sub 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around 0 99.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -78000000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-63} \lor \neg \left(z \leq 6.2 \cdot 10^{+86}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.8e-63) (not (<= z 6.2e+86)))
   (+ y (/ x z))
   (* x (/ (- 1.0 y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.8e-63) || !(z <= 6.2e+86)) {
		tmp = y + (x / 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 ((z <= (-3.8d-63)) .or. (.not. (z <= 6.2d+86))) then
        tmp = y + (x / 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 ((z <= -3.8e-63) || !(z <= 6.2e+86)) {
		tmp = y + (x / z);
	} else {
		tmp = x * ((1.0 - y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.8e-63) or not (z <= 6.2e+86):
		tmp = y + (x / z)
	else:
		tmp = x * ((1.0 - y) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.8e-63) || !(z <= 6.2e+86))
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(x * Float64(Float64(1.0 - y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.8e-63) || ~((z <= 6.2e+86)))
		tmp = y + (x / z);
	else
		tmp = x * ((1.0 - y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.8e-63], N[Not[LessEqual[z, 6.2e+86]], $MachinePrecision]], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.80000000000000017e-63 or 6.2000000000000004e86 < z

   1. Initial program 78.2%

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

      1. add-cube-cbrt 77.4%

         \[\leadsto \frac{x + y \cdot \color{blue}{\left(\left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right) \cdot \sqrt[3]{z - x}\right)}}{z} \]

      2. pow3 77.4%

         \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   4. Applied egg-rr 77.4%

      \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   5. Taylor expanded in x around 0 99.8%

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

      1. neg-mul-1 99.8%

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

      2. +-commutative 99.8%

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

      3. sub-neg 99.8%

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

      4. div-sub 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around 0 85.6%

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

   if -3.80000000000000017e-63 < z < 6.2000000000000004e86

   1. Initial program 98.5%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 86.5%

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

      1. associate-/l* 83.4%

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

      2. mul-1-neg 83.4%

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

      3. unsub-neg 83.4%

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

   5. Simplified 83.4%

      \[\leadsto \color{blue}{x \cdot \frac{1 - y}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-63} \lor \neg \left(z \leq 6.2 \cdot 10^{+86}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -78000000000000:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -78000000000000.0)
   (* y (- 1.0 (/ x z)))
   (if (<= y 1.0) (+ y (/ x z)) (- y (* y (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -78000000000000.0) {
		tmp = y * (1.0 - (x / z));
	} else if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = y - (y * (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 (y <= (-78000000000000.0d0)) then
        tmp = y * (1.0d0 - (x / z))
    else if (y <= 1.0d0) then
        tmp = y + (x / z)
    else
        tmp = y - (y * (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -78000000000000.0) {
		tmp = y * (1.0 - (x / z));
	} else if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = y - (y * (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -78000000000000.0:
		tmp = y * (1.0 - (x / z))
	elif y <= 1.0:
		tmp = y + (x / z)
	else:
		tmp = y - (y * (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -78000000000000.0)
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	elseif (y <= 1.0)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y - Float64(y * Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -78000000000000.0)
		tmp = y * (1.0 - (x / z));
	elseif (y <= 1.0)
		tmp = y + (x / z);
	else
		tmp = y - (y * (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -78000000000000.0], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y - N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.8e13

   1. Initial program 70.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 70.9%

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

      1. associate-/l* 99.8%

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

      2. div-sub 99.8%

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

      3. *-inverses 99.8%

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

   5. Simplified 99.8%

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

   if -7.8e13 < y < 1

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.4%

         \[\leadsto \frac{x + y \cdot \color{blue}{\left(\left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right) \cdot \sqrt[3]{z - x}\right)}}{z} \]

      2. pow3 99.4%

         \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   4. Applied egg-rr 99.4%

      \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   5. Taylor expanded in x around 0 99.7%

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

      1. neg-mul-1 99.7%

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

      2. +-commutative 99.7%

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

      3. sub-neg 99.7%

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

      4. div-sub 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around 0 99.0%

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

   if 1 < y

   1. Initial program 79.8%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 75.8%

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

      1. associate-/l* 95.8%

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

      2. div-sub 95.8%

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

      3. *-inverses 95.8%

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

   5. Simplified 95.8%

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

   6. Taylor expanded in x around 0 83.4%

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

      1. mul-1-neg 83.4%

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

      2. associate-*l/ 95.8%

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

      3. *-commutative 95.8%

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

      4. distribute-rgt-neg-in 95.8%

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

      5. distribute-neg-frac2 95.8%

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

   8. Simplified 95.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -78000000000000:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+174} \lor \neg \left(y \leq 10.6\right):\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.3e+174) (not (<= y 10.6))) (* y (/ x (- z))) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.3e+174) || !(y <= 10.6)) {
		tmp = y * (x / -z);
	} else {
		tmp = y + (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 ((y <= (-1.3d+174)) .or. (.not. (y <= 10.6d0))) then
        tmp = y * (x / -z)
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.3e+174) || !(y <= 10.6)) {
		tmp = y * (x / -z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.3e+174) or not (y <= 10.6):
		tmp = y * (x / -z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.3e+174) || !(y <= 10.6))
		tmp = Float64(y * Float64(x / Float64(-z)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.3e+174) || ~((y <= 10.6)))
		tmp = y * (x / -z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.3e+174], N[Not[LessEqual[y, 10.6]], $MachinePrecision]], N[(y * N[(x / (-z)), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2999999999999999e174 or 10.5999999999999996 < y

   1. Initial program 75.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 59.0%

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

      1. associate-/l* 58.7%

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

      2. mul-1-neg 58.7%

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

      3. unsub-neg 58.7%

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

   5. Simplified 58.7%

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

   6. Taylor expanded in y around inf 56.5%

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

      1. neg-mul-1 56.5%

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

      2. distribute-neg-frac 56.5%

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

   8. Simplified 56.5%

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

   9. Taylor expanded in x around 0 56.8%

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

       1. mul-1-neg 56.8%

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

       2. distribute-neg-frac2 56.8%

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

       3. *-commutative 56.8%

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

       4. associate-*r/ 62.2%

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

   11. Simplified 62.2%

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

   if -1.2999999999999999e174 < y < 10.5999999999999996

   1. Initial program 95.4%

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

      1. add-cube-cbrt 94.8%

         \[\leadsto \frac{x + y \cdot \color{blue}{\left(\left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right) \cdot \sqrt[3]{z - x}\right)}}{z} \]

      2. pow3 94.8%

         \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   4. Applied egg-rr 94.8%

      \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   5. Taylor expanded in x around 0 99.2%

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

      1. neg-mul-1 99.2%

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

      2. +-commutative 99.2%

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

      3. sub-neg 99.2%

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

      4. div-sub 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in y around 0 91.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+174} \lor \neg \left(y \leq 10.6\right):\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-60}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6.2e-60) y (if (<= y 6e-38) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.2e-60) {
		tmp = y;
	} else if (y <= 6e-38) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	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 (y <= (-6.2d-60)) then
        tmp = y
    else if (y <= 6d-38) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.2e-60) {
		tmp = y;
	} else if (y <= 6e-38) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6.2e-60:
		tmp = y
	elif y <= 6e-38:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6.2e-60)
		tmp = y;
	elseif (y <= 6e-38)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -6.2e-60)
		tmp = y;
	elseif (y <= 6e-38)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6.2e-60], y, If[LessEqual[y, 6e-38], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -6.19999999999999976e-60 or 5.99999999999999977e-38 < y

   1. Initial program 79.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 43.0%

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

   if -6.19999999999999976e-60 < y < 5.99999999999999977e-38

   1. Initial program 99.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 78.2%

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

Alternative 7: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+107}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.15e+107) (+ y (/ x z)) (* x (/ y (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e+107) {
		tmp = y + (x / z);
	} else {
		tmp = x * (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 (y <= 1.15d+107) then
        tmp = y + (x / z)
    else
        tmp = x * (y / -z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e+107) {
		tmp = y + (x / z);
	} else {
		tmp = x * (y / -z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.15e+107:
		tmp = y + (x / z)
	else:
		tmp = x * (y / -z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.15e+107)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(x * Float64(y / Float64(-z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.15e+107)
		tmp = y + (x / z);
	else
		tmp = x * (y / -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.15e+107], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / (-z)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.15e107

   1. Initial program 90.8%

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

      1. add-cube-cbrt 90.2%

         \[\leadsto \frac{x + y \cdot \color{blue}{\left(\left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right) \cdot \sqrt[3]{z - x}\right)}}{z} \]

      2. pow3 90.2%

         \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   4. Applied egg-rr 90.2%

      \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   5. Taylor expanded in x around 0 97.2%

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

      1. neg-mul-1 97.2%

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

      2. +-commutative 97.2%

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

      3. sub-neg 97.2%

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

      4. div-sub 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in y around 0 81.2%

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

   if 1.15e107 < y

   1. Initial program 76.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 63.7%

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

      1. associate-/l* 63.4%

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

      2. mul-1-neg 63.4%

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

      3. unsub-neg 63.4%

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

   5. Simplified 63.4%

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

   6. Taylor expanded in y around inf 63.4%

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

      1. neg-mul-1 63.4%

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

      2. distribute-neg-frac 63.4%

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

   8. Simplified 63.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+107}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.0) (+ y (/ x z)) (- y (/ x z))))

C

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

Java

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

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.0:
		tmp = y + (x / z)
	else:
		tmp = y - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = y + (x / z);
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 91.3%

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

      1. add-cube-cbrt 90.7%

         \[\leadsto \frac{x + y \cdot \color{blue}{\left(\left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right) \cdot \sqrt[3]{z - x}\right)}}{z} \]

      2. pow3 90.7%

         \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   4. Applied egg-rr 90.7%

      \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   5. Taylor expanded in x around 0 98.3%

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

      1. neg-mul-1 98.3%

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

      2. +-commutative 98.3%

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

      3. sub-neg 98.3%

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

      4. div-sub 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in y around 0 85.9%

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

   if 1 < y

   1. Initial program 79.8%

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

      1. add-cube-cbrt 79.2%

         \[\leadsto \frac{x + y \cdot \color{blue}{\left(\left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right) \cdot \sqrt[3]{z - x}\right)}}{z} \]

      2. pow3 79.2%

         \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   4. Applied egg-rr 79.2%

      \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

   5. Taylor expanded in x around 0 90.7%

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

      1. neg-mul-1 90.7%

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

      2. +-commutative 90.7%

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

      3. sub-neg 90.7%

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

      4. div-sub 90.7%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in y around 0 34.0%

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

      1. add-sqr-sqrt 14.3%

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

      2. sqrt-unprod 40.4%

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

      3. sqr-neg 40.4%

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

      4. sqrt-unprod 22.2%

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

      5. add-sqr-sqrt 47.3%

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

      6. distribute-frac-neg2 47.3%

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

      7. sub-neg 47.3%

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

   10. Applied egg-rr 47.3%

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

Alternative 9: 77.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return y + (x / 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 = y + (x / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. add-cube-cbrt 88.0%

      \[\leadsto \frac{x + y \cdot \color{blue}{\left(\left(\sqrt[3]{z - x} \cdot \sqrt[3]{z - x}\right) \cdot \sqrt[3]{z - x}\right)}}{z} \]

   2. pow3 88.0%

      \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

4. Applied egg-rr 88.0%

   \[\leadsto \frac{x + y \cdot \color{blue}{{\left(\sqrt[3]{z - x}\right)}^{3}}}{z} \]

5. Taylor expanded in x around 0 96.5%

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

   1. neg-mul-1 96.5%

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

   2. +-commutative 96.5%

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

   3. sub-neg 96.5%

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

   4. div-sub 96.5%

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

7. Simplified 96.5%

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

8. Taylor expanded in y around 0 73.5%

   \[\leadsto y + \color{blue}{\frac{x}{z}} \]
9. Add Preprocessing

Alternative 10: 40.3% accurate, 9.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 88.6%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 34.2%

   \[\leadsto \color{blue}{y} \]
4. Add Preprocessing

Developer Target 1: 93.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (y + (x / z)) - (y / (z / 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 = (y + (x / z)) - (y / (z / x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024181 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- (+ y (/ x z)) (/ y (/ z x))))

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