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

Percentage Accurate: 88.7% → 99.7%

Time: 8.0s

Alternatives: 11

Speedup: 0.5×

• 20.8% 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: 88.7% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+66} \lor \neg \left(y \leq 30000\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x \cdot \left(1 - y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5e+66) (not (<= y 30000.0)))
   (* y (/ (- z x) z))
   (+ y (/ (* x (- 1.0 y)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e+66) || !(y <= 30000.0)) {
		tmp = y * ((z - x) / z);
	} else {
		tmp = y + ((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 ((y <= (-5d+66)) .or. (.not. (y <= 30000.0d0))) then
        tmp = y * ((z - x) / z)
    else
        tmp = y + ((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 ((y <= -5e+66) || !(y <= 30000.0)) {
		tmp = y * ((z - x) / z);
	} else {
		tmp = y + ((x * (1.0 - y)) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5e+66) or not (y <= 30000.0):
		tmp = y * ((z - x) / z)
	else:
		tmp = y + ((x * (1.0 - y)) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5e+66) || !(y <= 30000.0))
		tmp = Float64(y * Float64(Float64(z - x) / z));
	else
		tmp = Float64(y + Float64(Float64(x * Float64(1.0 - y)) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5e+66) || ~((y <= 30000.0)))
		tmp = y * ((z - x) / z);
	else
		tmp = y + ((x * (1.0 - y)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5e+66], N[Not[LessEqual[y, 30000.0]], $MachinePrecision]], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999991e66 or 3e4 < y

   1. Initial program 75.3%

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

   3. Taylor expanded in y around inf 75.3%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

   if -4.99999999999999991e66 < y < 3e4

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. associate-+r+ 99.7%

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

      2. +-commutative 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. div-sub 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in z around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+66} \lor \neg \left(y \leq 30000\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x \cdot \left(1 - y\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+36} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.2e+36) (not (<= y 1.0))) (* y (/ (- z x) z)) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.2e+36) || !(y <= 1.0)) {
		tmp = y * ((z - 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 <= (-4.2d+36)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * ((z - 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 <= -4.2e+36) || !(y <= 1.0)) {
		tmp = y * ((z - x) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.2e+36) or not (y <= 1.0):
		tmp = y * ((z - x) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.2e+36) || !(y <= 1.0))
		tmp = Float64(y * Float64(Float64(z - 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 <= -4.2e+36) || ~((y <= 1.0)))
		tmp = y * ((z - x) / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.2e+36], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.20000000000000009e36 or 1 < y

   1. Initial program 76.4%

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

   3. Taylor expanded in y around inf 75.4%

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

      1. associate-/l* 98.9%

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

   5. Simplified 98.9%

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

   if -4.20000000000000009e36 < y < 1

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. associate-+r+ 99.7%

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

      2. +-commutative 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. div-sub 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in z around inf 100.0%

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

   7. Taylor expanded in y around 0 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+36} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-17} \lor \neg \left(x \leq 9.6 \cdot 10^{+120}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.05e-17) (not (<= x 9.6e+120)))
   (* x (/ (- 1.0 y) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.05e-17) || !(x <= 9.6e+120)) {
		tmp = x * ((1.0 - y) / 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 ((x <= (-1.05d-17)) .or. (.not. (x <= 9.6d+120))) then
        tmp = x * ((1.0d0 - y) / 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 ((x <= -1.05e-17) || !(x <= 9.6e+120)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.05e-17) or not (x <= 9.6e+120):
		tmp = x * ((1.0 - y) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.05e-17) || !(x <= 9.6e+120))
		tmp = Float64(x * Float64(Float64(1.0 - y) / z));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.05e-17) || ~((x <= 9.6e+120)))
		tmp = x * ((1.0 - y) / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.05e-17], N[Not[LessEqual[x, 9.6e+120]], $MachinePrecision]], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.04999999999999996e-17 or 9.60000000000000004e120 < x

   1. Initial program 89.2%

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

   3. Taylor expanded in x around inf 81.2%

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

      1. associate-/l* 85.8%

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

      2. mul-1-neg 85.8%

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

      3. unsub-neg 85.8%

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

   5. Simplified 85.8%

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

   if -1.04999999999999996e-17 < x < 9.60000000000000004e120

   1. Initial program 89.5%

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

   3. Taylor expanded in x around inf 81.5%

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

      1. associate-+r+ 81.5%

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

      2. +-commutative 81.5%

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

      3. mul-1-neg 81.5%

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

      4. unsub-neg 81.5%

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

      5. div-sub 81.5%

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

   5. Simplified 81.5%

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

   6. Taylor expanded in z around inf 97.8%

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

   7. Taylor expanded in y around 0 88.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-17} \lor \neg \left(x \leq 9.6 \cdot 10^{+120}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.2e+36)
   (/ y (/ z (- z x)))
   (if (<= y 1.0) (+ y (/ x z)) (* y (/ (- z x) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.2e+36) {
		tmp = y / (z / (z - x));
	} else if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = y * ((z - x) / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.2d+36)) then
        tmp = y / (z / (z - x))
    else if (y <= 1.0d0) then
        tmp = y + (x / z)
    else
        tmp = y * ((z - x) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.2e+36) {
		tmp = y / (z / (z - x));
	} else if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = y * ((z - x) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.2e+36:
		tmp = y / (z / (z - x))
	elif y <= 1.0:
		tmp = y + (x / z)
	else:
		tmp = y * ((z - x) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.2e+36)
		tmp = Float64(y / Float64(z / Float64(z - x)));
	elseif (y <= 1.0)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y * Float64(Float64(z - x) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.2e+36)
		tmp = y / (z / (z - x));
	elseif (y <= 1.0)
		tmp = y + (x / z);
	else
		tmp = y * ((z - x) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.2e+36], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.20000000000000009e36

   1. Initial program 75.7%

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

   3. Taylor expanded in y around inf 75.7%

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

      1. associate-/l* 99.8%

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

   5. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.9%

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

   7. Applied egg-rr 99.9%

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

   if -4.20000000000000009e36 < y < 1

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf 99.7%

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

      1. associate-+r+ 99.7%

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

      2. +-commutative 99.7%

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

      3. mul-1-neg 99.7%

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

      4. unsub-neg 99.7%

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

      5. div-sub 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in z around inf 100.0%

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

   7. Taylor expanded in y around 0 98.9%

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

   if 1 < 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 75.3%

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

      1. associate-/l* 98.3%

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

   5. Simplified 98.3%

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

Alternative 5: 58.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+82}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.8e+82) y (if (<= y 2.15e-39) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.8e+82) {
		tmp = y;
	} else if (y <= 2.15e-39) {
		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 <= (-1.8d+82)) then
        tmp = y
    else if (y <= 2.15d-39) 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 <= -1.8e+82) {
		tmp = y;
	} else if (y <= 2.15e-39) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.8e+82:
		tmp = y
	elif y <= 2.15e-39:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.8e+82)
		tmp = y;
	elseif (y <= 2.15e-39)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.8e+82)
		tmp = y;
	elseif (y <= 2.15e-39)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.8e+82], y, If[LessEqual[y, 2.15e-39], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000007e82 or 2.15e-39 < y

   1. Initial program 77.4%

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

   3. Taylor expanded in x around 0 54.1%

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

   if -1.80000000000000007e82 < y < 2.15e-39

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0 70.3%

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

Alternative 6: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+239}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{-x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.7e+239) (+ y (/ x z)) (/ y (/ z (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.7e+239) {
		tmp = y + (x / z);
	} else {
		tmp = y / (z / -x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 3.7d+239) then
        tmp = y + (x / z)
    else
        tmp = y / (z / -x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.7e+239) {
		tmp = y + (x / z);
	} else {
		tmp = y / (z / -x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 3.7e+239:
		tmp = y + (x / z)
	else:
		tmp = y / (z / -x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.7e+239)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y / Float64(z / Float64(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 3.7e+239)
		tmp = y + (x / z);
	else
		tmp = y / (z / -x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.7e+239], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / (-x)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.69999999999999998e239

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf 88.8%

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

      1. associate-+r+ 88.8%

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

      2. +-commutative 88.8%

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

      3. mul-1-neg 88.8%

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

      4. unsub-neg 88.8%

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

      5. div-sub 88.8%

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

   5. Simplified 88.8%

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

   6. Taylor expanded in z around inf 95.9%

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

   7. Taylor expanded in y around 0 82.7%

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

   if 3.69999999999999998e239 < x

   1. Initial program 77.9%

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

   3. Taylor expanded in y around inf 60.6%

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

      1. associate-/l* 88.1%

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

   5. Simplified 88.1%

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

      1. clear-num 88.1%

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

      2. un-div-inv 88.1%

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

   7. Applied egg-rr 88.1%

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

   8. Taylor expanded in z around 0 86.3%

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

      1. associate-*r/ 86.3%

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

      2. neg-mul-1 86.3%

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

   10. Simplified 86.3%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+239}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{-x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.6 \cdot 10^{+239}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8.6e+239) (+ y (/ x z)) (* y (/ x (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8.6e+239) {
		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 (x <= 8.6d+239) 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 (x <= 8.6e+239) {
		tmp = y + (x / z);
	} else {
		tmp = y * (x / -z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 8.6e+239:
		tmp = y + (x / z)
	else:
		tmp = y * (x / -z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8.6e+239)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y * Float64(x / Float64(-z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 8.6e+239)
		tmp = y + (x / z);
	else
		tmp = y * (x / -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8.6e+239], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / (-z)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.6000000000000008e239

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf 88.8%

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

      1. associate-+r+ 88.8%

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

      2. +-commutative 88.8%

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

      3. mul-1-neg 88.8%

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

      4. unsub-neg 88.8%

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

      5. div-sub 88.8%

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

   5. Simplified 88.8%

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

   6. Taylor expanded in z around inf 95.9%

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

   7. Taylor expanded in y around 0 82.7%

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

   if 8.6000000000000008e239 < x

   1. Initial program 77.9%

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

   3. Taylor expanded in y around inf 60.6%

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

      1. associate-/l* 88.1%

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

   5. Simplified 88.1%

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

   6. Taylor expanded in z around 0 86.2%

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

      1. neg-mul-1 86.2%

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

   8. Simplified 86.2%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.6 \cdot 10^{+239}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{+239}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.1e+239) (+ y (/ x z)) (* x (/ y (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.1e+239) {
		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 (x <= 3.1d+239) 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 (x <= 3.1e+239) {
		tmp = y + (x / z);
	} else {
		tmp = x * (y / -z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 3.1e+239:
		tmp = y + (x / z)
	else:
		tmp = x * (y / -z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.1e+239)
		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 (x <= 3.1e+239)
		tmp = y + (x / z);
	else
		tmp = x * (y / -z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.1e+239], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / (-z)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000001e239

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf 88.8%

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

      1. associate-+r+ 88.8%

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

      2. +-commutative 88.8%

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

      3. mul-1-neg 88.8%

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

      4. unsub-neg 88.8%

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

      5. div-sub 88.8%

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

   5. Simplified 88.8%

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

   6. Taylor expanded in z around inf 95.9%

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

   7. Taylor expanded in y around 0 82.7%

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

   if 3.10000000000000001e239 < x

   1. Initial program 77.9%

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

   3. Taylor expanded in x around inf 77.9%

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

      1. associate-/l* 98.0%

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

      2. mul-1-neg 98.0%

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

      3. unsub-neg 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in y around inf 80.8%

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

      1. neg-mul-1 80.8%

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

      2. distribute-neg-frac 80.8%

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

   8. Simplified 80.8%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{+239}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.6% 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 93.6%

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

   3. Taylor expanded in x around inf 91.4%

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

      1. associate-+r+ 91.4%

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

      2. +-commutative 91.4%

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

      3. mul-1-neg 91.4%

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

      4. unsub-neg 91.4%

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

      5. div-sub 91.4%

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

   5. Simplified 91.4%

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

   6. Taylor expanded in z around inf 96.5%

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

   7. Taylor expanded in y around 0 88.2%

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

   if 1 < y

   1. Initial program 76.9%

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

   3. Taylor expanded in x around inf 84.0%

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

      1. associate-+r+ 84.1%

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

      2. +-commutative 84.1%

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

      3. mul-1-neg 84.1%

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

      4. unsub-neg 84.1%

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

      5. div-sub 84.1%

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

   5. Simplified 84.1%

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

   6. Taylor expanded in z around inf 89.4%

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

   7. Taylor expanded in y around 0 54.6%

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

      1. frac-2neg 54.6%

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

      2. div-inv 54.6%

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

      3. add-sqr-sqrt 25.5%

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

      4. sqrt-unprod 60.2%

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

      5. sqr-neg 60.2%

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

      6. sqrt-unprod 36.2%

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

      7. add-sqr-sqrt 65.3%

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

      8. cancel-sign-sub-inv 65.3%

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

      9. div-inv 65.3%

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

   9. Applied egg-rr 65.3%

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

Alternative 10: 78.1% 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 89.4%

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

3. Taylor expanded in x around inf 89.6%

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

   1. associate-+r+ 89.6%

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

   2. +-commutative 89.6%

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

   3. mul-1-neg 89.6%

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

   4. unsub-neg 89.6%

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

   5. div-sub 89.6%

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

5. Simplified 89.6%

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

6. Taylor expanded in z around inf 94.7%

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

7. Taylor expanded in y around 0 79.7%

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

Alternative 11: 40.5% 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 89.4%

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

3. Taylor expanded in x around 0 39.5%

   \[\leadsto \color{blue}{y} \]
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 2024182 
(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))