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

Percentage Accurate: 87.9% → 98.3%

Time: 7.4s

Alternatives: 10

Speedup: 0.8×

• 20.9% 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: 87.9% 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: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+30}:\\ \;\;\;\;y - \frac{x}{\frac{z}{y + -1}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e+30) (- y (/ x (/ z (+ y -1.0)))) (* y (- 1.0 (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+30) {
		tmp = y - (x / (z / (y + -1.0)));
	} else {
		tmp = y * (1.0 - (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 <= 4d+30) then
        tmp = y - (x / (z / (y + (-1.0d0))))
    else
        tmp = y * (1.0d0 - (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+30) {
		tmp = y - (x / (z / (y + -1.0)));
	} else {
		tmp = y * (1.0 - (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4e+30:
		tmp = y - (x / (z / (y + -1.0)))
	else:
		tmp = y * (1.0 - (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e+30)
		tmp = Float64(y - Float64(x / Float64(z / Float64(y + -1.0))));
	else
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4e+30)
		tmp = y - (x / (z / (y + -1.0)));
	else
		tmp = y * (1.0 - (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e+30], N[(y - N[(x / N[(z / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.0000000000000001e30

   1. Initial program 91.6%

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

   2. Taylor expanded in y around 0 95.0%

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

   3. Taylor expanded in z around -inf 96.7%

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

      1. mul-1-neg 96.7%

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

      2. unsub-neg 96.7%

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

      3. distribute-rgt-out 96.7%

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

      4. metadata-eval 96.7%

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

      5. sub-neg 96.7%

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

      6. associate-/l* 99.3%

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

      7. sub-neg 99.3%

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

      8. metadata-eval 99.3%

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

   5. Simplified 99.3%

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

   if 4.0000000000000001e30 < y

   1. Initial program 77.1%

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

   2. Taylor expanded in y around 0 88.6%

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

   3. Taylor expanded in y around inf 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+30}:\\ \;\;\;\;y - \frac{x}{\frac{z}{y + -1}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \]

Alternative 2: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+79}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+187} \lor \neg \left(y \leq 8.5 \cdot 10^{+281}\right):\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.2e+79)
   (+ y (/ x z))
   (if (or (<= y 1.6e+187) (not (<= y 8.5e+281)))
     (* y (/ (- x) z))
     (- y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e+79) {
		tmp = y + (x / z);
	} else if ((y <= 1.6e+187) || !(y <= 8.5e+281)) {
		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 <= 5.2d+79) then
        tmp = y + (x / z)
    else if ((y <= 1.6d+187) .or. (.not. (y <= 8.5d+281))) 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 <= 5.2e+79) {
		tmp = y + (x / z);
	} else if ((y <= 1.6e+187) || !(y <= 8.5e+281)) {
		tmp = y * (-x / z);
	} else {
		tmp = y - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.2e+79:
		tmp = y + (x / z)
	elif (y <= 1.6e+187) or not (y <= 8.5e+281):
		tmp = y * (-x / z)
	else:
		tmp = y - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.2e+79)
		tmp = Float64(y + Float64(x / z));
	elseif ((y <= 1.6e+187) || !(y <= 8.5e+281))
		tmp = Float64(y * Float64(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 <= 5.2e+79)
		tmp = y + (x / z);
	elseif ((y <= 1.6e+187) || ~((y <= 8.5e+281)))
		tmp = y * (-x / z);
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.2e+79], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 1.6e+187], N[Not[LessEqual[y, 8.5e+281]], $MachinePrecision]], N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision], N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.20000000000000029e79

   1. Initial program 90.7%

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

   2. Taylor expanded in y around 0 95.3%

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

   3. Taylor expanded in z around -inf 96.9%

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

      1. mul-1-neg 96.9%

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

      2. unsub-neg 96.9%

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

      3. distribute-rgt-out 96.9%

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

      4. metadata-eval 96.9%

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

      5. sub-neg 96.9%

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

      6. associate-/l* 99.2%

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

      7. sub-neg 99.2%

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

      8. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around 0 88.1%

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

      1. neg-mul-1 88.1%

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

      2. distribute-frac-neg 88.1%

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

   8. Simplified 88.1%

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

   9. Taylor expanded in y around 0 88.1%

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

       1. +-commutative 88.1%

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

   11. Simplified 88.1%

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

   if 5.20000000000000029e79 < y < 1.59999999999999997e187 or 8.5000000000000006e281 < y

   1. Initial program 82.9%

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

   2. Taylor expanded in y around 0 85.6%

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

   3. Taylor expanded in y around inf 99.9%

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

   4. Taylor expanded in x around inf 73.0%

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

      1. associate-*r/ 73.0%

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

      2. mul-1-neg 73.0%

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

      3. distribute-rgt-neg-in 73.0%

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

      4. associate-*r/ 81.5%

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

   6. Simplified 81.5%

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

   if 1.59999999999999997e187 < y < 8.5000000000000006e281

   1. Initial program 73.2%

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

   2. Taylor expanded in y around 0 85.7%

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

   3. Taylor expanded in z around -inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. unsub-neg 95.2%

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

      3. distribute-rgt-out 95.2%

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

      4. metadata-eval 95.2%

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

      5. sub-neg 95.2%

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

      6. associate-/l* 86.1%

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

      7. sub-neg 86.1%

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

      8. metadata-eval 86.1%

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

   5. Simplified 86.1%

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

   6. Taylor expanded in y around 0 52.7%

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

      1. neg-mul-1 52.7%

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

      2. distribute-frac-neg 52.7%

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

   8. Simplified 52.7%

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

      1. *-un-lft-identity 52.7%

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

      2. *-commutative 52.7%

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

      3. div-inv 52.7%

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

      4. add-cube-cbrt 52.7%

         \[\leadsto y \cdot 1 - \color{blue}{\left(\left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right) \cdot \sqrt[3]{-x}\right)} \cdot \frac{1}{z} \]

      5. associate-*l* 52.7%

         \[\leadsto y \cdot 1 - \color{blue}{\left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right) \cdot \left(\sqrt[3]{-x} \cdot \frac{1}{z}\right)} \]

      6. prod-diff 52.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1, -\left(\sqrt[3]{-x} \cdot \frac{1}{z}\right) \cdot \left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{-x} \cdot \frac{1}{z}, \sqrt[3]{-x} \cdot \sqrt[3]{-x}, \left(\sqrt[3]{-x} \cdot \frac{1}{z}\right) \cdot \left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right)\right)} \]

   10. Applied egg-rr 53.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1, -\left(\sqrt[3]{x} \cdot \frac{1}{z}\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \mathsf{fma}\left(-\sqrt[3]{x} \cdot \frac{1}{z}, {\left(\sqrt[3]{x}\right)}^{2}, \left(\sqrt[3]{x} \cdot \frac{1}{z}\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}\right)} \]

   12. Simplified 68.1%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+79}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+187} \lor \neg \left(y \leq 8.5 \cdot 10^{+281}\right):\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \]

Alternative 3: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+79}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+187}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+282}:\\ \;\;\;\;y - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3e+79)
   (+ y (/ x z))
   (if (<= y 2.9e+187)
     (* y (/ (- x) z))
     (if (<= y 6e+282) (- y (/ x z)) (* x (/ (- y) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e+79) {
		tmp = y + (x / z);
	} else if (y <= 2.9e+187) {
		tmp = y * (-x / z);
	} else if (y <= 6e+282) {
		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 <= 3d+79) then
        tmp = y + (x / z)
    else if (y <= 2.9d+187) then
        tmp = y * (-x / z)
    else if (y <= 6d+282) 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 <= 3e+79) {
		tmp = y + (x / z);
	} else if (y <= 2.9e+187) {
		tmp = y * (-x / z);
	} else if (y <= 6e+282) {
		tmp = y - (x / z);
	} else {
		tmp = x * (-y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3e+79:
		tmp = y + (x / z)
	elif y <= 2.9e+187:
		tmp = y * (-x / z)
	elif y <= 6e+282:
		tmp = y - (x / z)
	else:
		tmp = x * (-y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3e+79)
		tmp = Float64(y + Float64(x / z));
	elseif (y <= 2.9e+187)
		tmp = Float64(y * Float64(Float64(-x) / z));
	elseif (y <= 6e+282)
		tmp = Float64(y - Float64(x / z));
	else
		tmp = Float64(x * Float64(Float64(-y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3e+79)
		tmp = y + (x / z);
	elseif (y <= 2.9e+187)
		tmp = y * (-x / z);
	elseif (y <= 6e+282)
		tmp = y - (x / z);
	else
		tmp = x * (-y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3e+79], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+187], N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+282], N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[((-y) / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 2.99999999999999974e79

   1. Initial program 90.7%

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

   2. Taylor expanded in y around 0 95.3%

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

   3. Taylor expanded in z around -inf 96.9%

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

      1. mul-1-neg 96.9%

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

      2. unsub-neg 96.9%

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

      3. distribute-rgt-out 96.9%

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

      4. metadata-eval 96.9%

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

      5. sub-neg 96.9%

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

      6. associate-/l* 99.2%

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

      7. sub-neg 99.2%

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

      8. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around 0 88.1%

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

      1. neg-mul-1 88.1%

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

      2. distribute-frac-neg 88.1%

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

   8. Simplified 88.1%

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

   9. Taylor expanded in y around 0 88.1%

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

       1. +-commutative 88.1%

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

   11. Simplified 88.1%

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

   if 2.99999999999999974e79 < y < 2.9000000000000001e187

   1. Initial program 87.2%

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

   2. Taylor expanded in y around 0 86.6%

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

   3. Taylor expanded in y around inf 99.9%

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

   4. Taylor expanded in x around inf 74.4%

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

      1. associate-*r/ 74.4%

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

      2. mul-1-neg 74.4%

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

      3. distribute-rgt-neg-in 74.4%

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

      4. associate-*r/ 87.1%

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

   6. Simplified 87.1%

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

   if 2.9000000000000001e187 < y < 5.99999999999999994e282

   1. Initial program 73.2%

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

   2. Taylor expanded in y around 0 85.7%

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

   3. Taylor expanded in z around -inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. unsub-neg 95.2%

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

      3. distribute-rgt-out 95.2%

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

      4. metadata-eval 95.2%

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

      5. sub-neg 95.2%

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

      6. associate-/l* 86.1%

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

      7. sub-neg 86.1%

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

      8. metadata-eval 86.1%

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

   5. Simplified 86.1%

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

   6. Taylor expanded in y around 0 52.7%

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

      1. neg-mul-1 52.7%

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

      2. distribute-frac-neg 52.7%

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

   8. Simplified 52.7%

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

      1. *-un-lft-identity 52.7%

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

      2. *-commutative 52.7%

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

      3. div-inv 52.7%

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

      4. add-cube-cbrt 52.7%

         \[\leadsto y \cdot 1 - \color{blue}{\left(\left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right) \cdot \sqrt[3]{-x}\right)} \cdot \frac{1}{z} \]

      5. associate-*l* 52.7%

         \[\leadsto y \cdot 1 - \color{blue}{\left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right) \cdot \left(\sqrt[3]{-x} \cdot \frac{1}{z}\right)} \]

      6. prod-diff 52.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1, -\left(\sqrt[3]{-x} \cdot \frac{1}{z}\right) \cdot \left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{-x} \cdot \frac{1}{z}, \sqrt[3]{-x} \cdot \sqrt[3]{-x}, \left(\sqrt[3]{-x} \cdot \frac{1}{z}\right) \cdot \left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right)\right)} \]

   10. Applied egg-rr 53.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1, -\left(\sqrt[3]{x} \cdot \frac{1}{z}\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \mathsf{fma}\left(-\sqrt[3]{x} \cdot \frac{1}{z}, {\left(\sqrt[3]{x}\right)}^{2}, \left(\sqrt[3]{x} \cdot \frac{1}{z}\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}\right)} \]

   12. Simplified 68.1%

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

   if 5.99999999999999994e282 < y

   1. Initial program 72.1%

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

   2. Taylor expanded in x around inf 69.5%

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

      1. associate-/l* 67.7%

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

      2. associate-/r/ 67.7%

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

      3. mul-1-neg 67.7%

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

      4. unsub-neg 67.7%

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

   4. Simplified 67.7%

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

   5. Taylor expanded in y around inf 67.7%

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

      1. neg-mul-1 67.7%

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

      2. distribute-neg-frac 67.7%

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

   7. Simplified 67.7%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+79}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+187}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+282}:\\ \;\;\;\;y - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-y}{z}\\ \end{array} \]

Alternative 4: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+81}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+187}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+273}:\\ \;\;\;\;y - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.5e+81)
   (+ y (/ x z))
   (if (<= y 3.4e+187)
     (* y (/ (- x) z))
     (if (<= y 1.25e+273) (- y (/ x z)) (/ (* y (- x)) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.5e+81) {
		tmp = y + (x / z);
	} else if (y <= 3.4e+187) {
		tmp = y * (-x / z);
	} else if (y <= 1.25e+273) {
		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 <= 1.5d+81) then
        tmp = y + (x / z)
    else if (y <= 3.4d+187) then
        tmp = y * (-x / z)
    else if (y <= 1.25d+273) 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 <= 1.5e+81) {
		tmp = y + (x / z);
	} else if (y <= 3.4e+187) {
		tmp = y * (-x / z);
	} else if (y <= 1.25e+273) {
		tmp = y - (x / z);
	} else {
		tmp = (y * -x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.5e+81:
		tmp = y + (x / z)
	elif y <= 3.4e+187:
		tmp = y * (-x / z)
	elif y <= 1.25e+273:
		tmp = y - (x / z)
	else:
		tmp = (y * -x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.5e+81)
		tmp = Float64(y + Float64(x / z));
	elseif (y <= 3.4e+187)
		tmp = Float64(y * Float64(Float64(-x) / z));
	elseif (y <= 1.25e+273)
		tmp = Float64(y - Float64(x / z));
	else
		tmp = Float64(Float64(y * Float64(-x)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.5e+81)
		tmp = y + (x / z);
	elseif (y <= 3.4e+187)
		tmp = y * (-x / z);
	elseif (y <= 1.25e+273)
		tmp = y - (x / z);
	else
		tmp = (y * -x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.5e+81], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e+187], N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+273], N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(y * (-x)), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.49999999999999999e81

   1. Initial program 90.7%

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

   2. Taylor expanded in y around 0 95.3%

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

   3. Taylor expanded in z around -inf 96.9%

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

      1. mul-1-neg 96.9%

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

      2. unsub-neg 96.9%

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

      3. distribute-rgt-out 96.9%

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

      4. metadata-eval 96.9%

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

      5. sub-neg 96.9%

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

      6. associate-/l* 99.2%

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

      7. sub-neg 99.2%

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

      8. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around 0 88.1%

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

      1. neg-mul-1 88.1%

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

      2. distribute-frac-neg 88.1%

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

   8. Simplified 88.1%

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

   9. Taylor expanded in y around 0 88.1%

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

       1. +-commutative 88.1%

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

   11. Simplified 88.1%

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

   if 1.49999999999999999e81 < y < 3.4e187

   1. Initial program 87.2%

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

   2. Taylor expanded in y around 0 86.6%

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

   3. Taylor expanded in y around inf 99.9%

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

   4. Taylor expanded in x around inf 74.4%

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

      1. associate-*r/ 74.4%

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

      2. mul-1-neg 74.4%

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

      3. distribute-rgt-neg-in 74.4%

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

      4. associate-*r/ 87.1%

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

   6. Simplified 87.1%

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

   if 3.4e187 < y < 1.2499999999999999e273

   1. Initial program 73.2%

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

   2. Taylor expanded in y around 0 85.7%

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

   3. Taylor expanded in z around -inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. unsub-neg 95.2%

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

      3. distribute-rgt-out 95.2%

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

      4. metadata-eval 95.2%

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

      5. sub-neg 95.2%

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

      6. associate-/l* 86.1%

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

      7. sub-neg 86.1%

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

      8. metadata-eval 86.1%

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

   5. Simplified 86.1%

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

   6. Taylor expanded in y around 0 52.7%

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

      1. neg-mul-1 52.7%

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

      2. distribute-frac-neg 52.7%

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

   8. Simplified 52.7%

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

      1. *-un-lft-identity 52.7%

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

      2. *-commutative 52.7%

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

      3. div-inv 52.7%

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

      4. add-cube-cbrt 52.7%

         \[\leadsto y \cdot 1 - \color{blue}{\left(\left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right) \cdot \sqrt[3]{-x}\right)} \cdot \frac{1}{z} \]

      5. associate-*l* 52.7%

         \[\leadsto y \cdot 1 - \color{blue}{\left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right) \cdot \left(\sqrt[3]{-x} \cdot \frac{1}{z}\right)} \]

      6. prod-diff 52.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1, -\left(\sqrt[3]{-x} \cdot \frac{1}{z}\right) \cdot \left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{-x} \cdot \frac{1}{z}, \sqrt[3]{-x} \cdot \sqrt[3]{-x}, \left(\sqrt[3]{-x} \cdot \frac{1}{z}\right) \cdot \left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right)\right)} \]

   10. Applied egg-rr 53.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1, -\left(\sqrt[3]{x} \cdot \frac{1}{z}\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \mathsf{fma}\left(-\sqrt[3]{x} \cdot \frac{1}{z}, {\left(\sqrt[3]{x}\right)}^{2}, \left(\sqrt[3]{x} \cdot \frac{1}{z}\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}\right)} \]

   12. Simplified 68.1%

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

   if 1.2499999999999999e273 < y

   1. Initial program 72.1%

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

   2. Taylor expanded in x around inf 69.5%

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

      1. associate-/l* 67.7%

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

      2. associate-/r/ 67.7%

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

      3. mul-1-neg 67.7%

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

      4. unsub-neg 67.7%

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

   4. Simplified 67.7%

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

   5. Taylor expanded in y around inf 67.7%

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

      1. neg-mul-1 67.7%

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

      2. distribute-neg-frac 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in y around 0 69.5%

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

      1. mul-1-neg 69.5%

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

      2. associate-*l/ 67.7%

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

      3. distribute-rgt-neg-out 67.7%

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

      4. associate-*l/ 69.5%

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

   10. Simplified 69.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+81}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+187}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+273}:\\ \;\;\;\;y - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(-x\right)}{z}\\ \end{array} \]

Alternative 5: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -24000000 \lor \neg \left(y \leq 4.8 \cdot 10^{-27}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -24000000.0) (not (<= y 4.8e-27)))
   (* y (- 1.0 (/ x z)))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -24000000.0) || !(y <= 4.8e-27)) {
		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 <= (-24000000.0d0)) .or. (.not. (y <= 4.8d-27))) 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 <= -24000000.0) || !(y <= 4.8e-27)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -24000000.0) or not (y <= 4.8e-27):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -24000000.0) || !(y <= 4.8e-27))
		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 <= -24000000.0) || ~((y <= 4.8e-27)))
		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, -24000000.0], N[Not[LessEqual[y, 4.8e-27]], $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 < -2.4e7 or 4.80000000000000004e-27 < y

   1. Initial program 76.4%

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

   2. Taylor expanded in y around 0 94.2%

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

   3. Taylor expanded in y around inf 99.0%

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

   if -2.4e7 < y < 4.80000000000000004e-27

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 93.2%

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

   3. Taylor expanded in z around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. distribute-rgt-out 100.0%

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

      4. metadata-eval 100.0%

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

      5. sub-neg 100.0%

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

      6. associate-/l* 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 99.6%

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

      1. neg-mul-1 99.6%

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

      2. distribute-frac-neg 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around 0 99.6%

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

       1. +-commutative 99.6%

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

   11. Simplified 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -24000000 \lor \neg \left(y \leq 4.8 \cdot 10^{-27}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 6: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -24000000:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -24000000.0)
   (/ y (/ z (- z x)))
   (if (<= y 4.8e-27) (+ y (/ x z)) (* y (- 1.0 (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -24000000.0) {
		tmp = y / (z / (z - x));
	} else if (y <= 4.8e-27) {
		tmp = y + (x / z);
	} else {
		tmp = y * (1.0 - (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 <= (-24000000.0d0)) then
        tmp = y / (z / (z - x))
    else if (y <= 4.8d-27) then
        tmp = y + (x / z)
    else
        tmp = y * (1.0d0 - (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -24000000.0) {
		tmp = y / (z / (z - x));
	} else if (y <= 4.8e-27) {
		tmp = y + (x / z);
	} else {
		tmp = y * (1.0 - (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -24000000.0:
		tmp = y / (z / (z - x))
	elif y <= 4.8e-27:
		tmp = y + (x / z)
	else:
		tmp = y * (1.0 - (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -24000000.0)
		tmp = Float64(y / Float64(z / Float64(z - x)));
	elseif (y <= 4.8e-27)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -24000000.0)
		tmp = y / (z / (z - x));
	elseif (y <= 4.8e-27)
		tmp = y + (x / z);
	else
		tmp = y * (1.0 - (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -24000000.0], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e-27], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e7

   1. Initial program 71.5%

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

   2. Taylor expanded in y around inf 70.9%

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

      1. associate-/l* 99.4%

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

   4. Simplified 99.4%

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

   if -2.4e7 < y < 4.80000000000000004e-27

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 93.2%

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

   3. Taylor expanded in z around -inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

      3. distribute-rgt-out 100.0%

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

      4. metadata-eval 100.0%

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

      5. sub-neg 100.0%

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

      6. associate-/l* 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 99.6%

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

      1. neg-mul-1 99.6%

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

      2. distribute-frac-neg 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around 0 99.6%

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

       1. +-commutative 99.6%

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

   11. Simplified 99.6%

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

   if 4.80000000000000004e-27 < y

   1. Initial program 81.0%

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

   2. Taylor expanded in y around 0 89.0%

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

   3. Taylor expanded in y around inf 98.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -24000000:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \]

Alternative 7: 56.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+83}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.75e+83) y (if (<= z 9.5e+55) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.75e+83) {
		tmp = y;
	} else if (z <= 9.5e+55) {
		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 (z <= (-1.75d+83)) then
        tmp = y
    else if (z <= 9.5d+55) 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 (z <= -1.75e+83) {
		tmp = y;
	} else if (z <= 9.5e+55) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.75e+83:
		tmp = y
	elif z <= 9.5e+55:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.75e+83)
		tmp = y;
	elseif (z <= 9.5e+55)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.75e+83)
		tmp = y;
	elseif (z <= 9.5e+55)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.75e+83], y, If[LessEqual[z, 9.5e+55], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if z < -1.74999999999999989e83 or 9.49999999999999989e55 < z

   1. Initial program 72.6%

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

   2. Taylor expanded in x around 0 76.4%

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

   if -1.74999999999999989e83 < z < 9.49999999999999989e55

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0 55.5%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+83}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 8: 78.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.8e-27) (+ y (/ x z)) (* z (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.8e-27) {
		tmp = y + (x / z);
	} else {
		tmp = z * (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 <= 4.8d-27) then
        tmp = y + (x / z)
    else
        tmp = z * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.8e-27) {
		tmp = y + (x / z);
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.8e-27:
		tmp = y + (x / z)
	else:
		tmp = z * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.8e-27)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(z * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.8e-27)
		tmp = y + (x / z);
	else
		tmp = z * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.8e-27], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.80000000000000004e-27

   1. Initial program 91.2%

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

   2. Taylor expanded in y around 0 95.3%

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

   3. Taylor expanded in z around -inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. unsub-neg 96.6%

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

      3. distribute-rgt-out 96.6%

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

      4. metadata-eval 96.6%

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

      5. sub-neg 96.6%

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

      6. associate-/l* 99.3%

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

      7. sub-neg 99.3%

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

      8. metadata-eval 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in y around 0 89.8%

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

      1. neg-mul-1 89.8%

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

      2. distribute-frac-neg 89.8%

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

   8. Simplified 89.8%

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

   9. Taylor expanded in y around 0 89.8%

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

       1. +-commutative 89.8%

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

   11. Simplified 89.8%

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

   if 4.80000000000000004e-27 < y

   1. Initial program 81.0%

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

   2. Taylor expanded in x around 0 34.1%

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

      1. associate-/l* 50.0%

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

      2. associate-/r/ 55.7%

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

   4. Applied egg-rr 55.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]

Alternative 9: 80.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.8e-27) (+ y (/ x z)) (- y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.8e-27) {
		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 <= 4.8d-27) 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 <= 4.8e-27) {
		tmp = y + (x / z);
	} else {
		tmp = y - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.8e-27:
		tmp = y + (x / z)
	else:
		tmp = y - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.8e-27)
		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 <= 4.8e-27)
		tmp = y + (x / z);
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.8e-27], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.80000000000000004e-27

   1. Initial program 91.2%

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

   2. Taylor expanded in y around 0 95.3%

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

   3. Taylor expanded in z around -inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. unsub-neg 96.6%

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

      3. distribute-rgt-out 96.6%

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

      4. metadata-eval 96.6%

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

      5. sub-neg 96.6%

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

      6. associate-/l* 99.3%

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

      7. sub-neg 99.3%

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

      8. metadata-eval 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in y around 0 89.8%

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

      1. neg-mul-1 89.8%

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

      2. distribute-frac-neg 89.8%

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

   8. Simplified 89.8%

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

   9. Taylor expanded in y around 0 89.8%

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

       1. +-commutative 89.8%

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

   11. Simplified 89.8%

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

   if 4.80000000000000004e-27 < y

   1. Initial program 81.0%

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

   2. Taylor expanded in y around 0 89.0%

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

   3. Taylor expanded in z around -inf 94.0%

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

      1. mul-1-neg 94.0%

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

      2. unsub-neg 94.0%

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

      3. distribute-rgt-out 94.0%

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

      4. metadata-eval 94.0%

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

      5. sub-neg 94.0%

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

      6. associate-/l* 92.0%

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

      7. sub-neg 92.0%

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

      8. metadata-eval 92.0%

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

   5. Simplified 92.0%

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

   6. Taylor expanded in y around 0 49.3%

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

      1. neg-mul-1 49.3%

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

      2. distribute-frac-neg 49.3%

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

   8. Simplified 49.3%

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

      1. *-un-lft-identity 49.3%

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

      2. *-commutative 49.3%

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

      3. div-inv 49.3%

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

      4. add-cube-cbrt 49.3%

         \[\leadsto y \cdot 1 - \color{blue}{\left(\left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right) \cdot \sqrt[3]{-x}\right)} \cdot \frac{1}{z} \]

      5. associate-*l* 49.3%

         \[\leadsto y \cdot 1 - \color{blue}{\left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right) \cdot \left(\sqrt[3]{-x} \cdot \frac{1}{z}\right)} \]

      6. prod-diff 49.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1, -\left(\sqrt[3]{-x} \cdot \frac{1}{z}\right) \cdot \left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{-x} \cdot \frac{1}{z}, \sqrt[3]{-x} \cdot \sqrt[3]{-x}, \left(\sqrt[3]{-x} \cdot \frac{1}{z}\right) \cdot \left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right)\right)} \]

   10. Applied egg-rr 50.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1, -\left(\sqrt[3]{x} \cdot \frac{1}{z}\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}\right) + \mathsf{fma}\left(-\sqrt[3]{x} \cdot \frac{1}{z}, {\left(\sqrt[3]{x}\right)}^{2}, \left(\sqrt[3]{x} \cdot \frac{1}{z}\right) \cdot {\left(\sqrt[3]{x}\right)}^{2}\right)} \]

   12. Simplified 61.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \]

Alternative 10: 41.0% 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. Taylor expanded in x around 0 40.8%

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

3. Final simplification 40.8%

   \[\leadsto y \]

Developer target: 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 2023242 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

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

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