Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x - y) / (z - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x - y) / (z - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x - y) / (z - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{z - y} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{x - y}{z - y} \]
4. Add Preprocessing

Alternative 2: 60.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -44000000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{-104}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+35}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -44000000000000.0)
   1.0
   (if (<= y 1.4e-134)
     (/ x z)
     (if (<= y 1.28e-104)
       (/ y (- z))
       (if (<= y 2.85e-68) (/ x z) (if (<= y 3.3e+35) (/ x (- y)) 1.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -44000000000000.0) {
		tmp = 1.0;
	} else if (y <= 1.4e-134) {
		tmp = x / z;
	} else if (y <= 1.28e-104) {
		tmp = y / -z;
	} else if (y <= 2.85e-68) {
		tmp = x / z;
	} else if (y <= 3.3e+35) {
		tmp = x / -y;
	} else {
		tmp = 1.0;
	}
	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 <= (-44000000000000.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.4d-134) then
        tmp = x / z
    else if (y <= 1.28d-104) then
        tmp = y / -z
    else if (y <= 2.85d-68) then
        tmp = x / z
    else if (y <= 3.3d+35) then
        tmp = x / -y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -44000000000000.0) {
		tmp = 1.0;
	} else if (y <= 1.4e-134) {
		tmp = x / z;
	} else if (y <= 1.28e-104) {
		tmp = y / -z;
	} else if (y <= 2.85e-68) {
		tmp = x / z;
	} else if (y <= 3.3e+35) {
		tmp = x / -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -44000000000000.0:
		tmp = 1.0
	elif y <= 1.4e-134:
		tmp = x / z
	elif y <= 1.28e-104:
		tmp = y / -z
	elif y <= 2.85e-68:
		tmp = x / z
	elif y <= 3.3e+35:
		tmp = x / -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -44000000000000.0)
		tmp = 1.0;
	elseif (y <= 1.4e-134)
		tmp = Float64(x / z);
	elseif (y <= 1.28e-104)
		tmp = Float64(y / Float64(-z));
	elseif (y <= 2.85e-68)
		tmp = Float64(x / z);
	elseif (y <= 3.3e+35)
		tmp = Float64(x / Float64(-y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -44000000000000.0)
		tmp = 1.0;
	elseif (y <= 1.4e-134)
		tmp = x / z;
	elseif (y <= 1.28e-104)
		tmp = y / -z;
	elseif (y <= 2.85e-68)
		tmp = x / z;
	elseif (y <= 3.3e+35)
		tmp = x / -y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -44000000000000.0], 1.0, If[LessEqual[y, 1.4e-134], N[(x / z), $MachinePrecision], If[LessEqual[y, 1.28e-104], N[(y / (-z)), $MachinePrecision], If[LessEqual[y, 2.85e-68], N[(x / z), $MachinePrecision], If[LessEqual[y, 3.3e+35], N[(x / (-y)), $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.4e13 or 3.3000000000000002e35 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 66.7%

      \[\leadsto \color{blue}{1} \]

   if -4.4e13 < y < 1.3999999999999999e-134 or 1.27999999999999992e-104 < y < 2.8500000000000001e-68

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 76.9%

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

   if 1.3999999999999999e-134 < y < 1.27999999999999992e-104

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac2 100.0%

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

   6. Simplified 100.0%

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

   if 2.8500000000000001e-68 < y < 3.3000000000000002e35

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 62.7%

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

      1. div-sub 62.7%

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

      2. sub-neg 62.7%

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

      3. *-inverses 62.7%

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

      4. metadata-eval 62.7%

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

      5. distribute-lft-in 62.7%

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

      6. metadata-eval 62.7%

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

      7. +-commutative 62.7%

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

      8. mul-1-neg 62.7%

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

      9. unsub-neg 62.7%

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

   5. Simplified 62.7%

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

   6. Taylor expanded in x around inf 51.0%

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

      1. mul-1-neg 51.0%

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

      2. distribute-frac-neg2 51.0%

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

   8. Simplified 51.0%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -44000000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{-104}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+35}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0008 \lor \neg \left(y \leq 1.8 \cdot 10^{-142}\right) \land \left(y \leq 1.9 \cdot 10^{-77} \lor \neg \left(y \leq 3.4 \cdot 10^{+33}\right)\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0008)
         (and (not (<= y 1.8e-142)) (or (<= y 1.9e-77) (not (<= y 3.4e+33)))))
   (/ y (- y z))
   (/ x (- z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0008) || (!(y <= 1.8e-142) && ((y <= 1.9e-77) || !(y <= 3.4e+33)))) {
		tmp = y / (y - z);
	} else {
		tmp = x / (z - 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 <= (-0.0008d0)) .or. (.not. (y <= 1.8d-142)) .and. (y <= 1.9d-77) .or. (.not. (y <= 3.4d+33))) then
        tmp = y / (y - z)
    else
        tmp = x / (z - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0008) || (!(y <= 1.8e-142) && ((y <= 1.9e-77) || !(y <= 3.4e+33)))) {
		tmp = y / (y - z);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0008) or (not (y <= 1.8e-142) and ((y <= 1.9e-77) or not (y <= 3.4e+33))):
		tmp = y / (y - z)
	else:
		tmp = x / (z - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0008) || (!(y <= 1.8e-142) && ((y <= 1.9e-77) || !(y <= 3.4e+33))))
		tmp = Float64(y / Float64(y - z));
	else
		tmp = Float64(x / Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.0008) || (~((y <= 1.8e-142)) && ((y <= 1.9e-77) || ~((y <= 3.4e+33)))))
		tmp = y / (y - z);
	else
		tmp = x / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0008], And[N[Not[LessEqual[y, 1.8e-142]], $MachinePrecision], Or[LessEqual[y, 1.9e-77], N[Not[LessEqual[y, 3.4e+33]], $MachinePrecision]]]], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000038e-4 or 1.8e-142 < y < 1.8999999999999999e-77 or 3.3999999999999999e33 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 85.1%

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

      1. neg-mul-1 85.1%

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

      2. distribute-neg-frac 85.1%

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

   5. Simplified 85.1%

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

      1. frac-2neg 85.1%

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

      2. div-inv 84.9%

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

      3. remove-double-neg 84.9%

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

      4. sub-neg 84.9%

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

      5. distribute-neg-in 84.9%

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

      6. remove-double-neg 84.9%

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

   7. Applied egg-rr 84.9%

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

      1. associate-*r/ 85.1%

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

      2. *-rgt-identity 85.1%

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

      3. +-commutative 85.1%

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

      4. unsub-neg 85.1%

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

   9. Simplified 85.1%

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

   if -8.00000000000000038e-4 < y < 1.8e-142 or 1.8999999999999999e-77 < y < 3.3999999999999999e33

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 85.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0008 \lor \neg \left(y \leq 1.8 \cdot 10^{-142}\right) \land \left(y \leq 1.9 \cdot 10^{-77} \lor \neg \left(y \leq 3.4 \cdot 10^{+33}\right)\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -1.46 \cdot 10^{-67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -1.46e-67)
     t_0
     (if (<= y 1.4e-134)
       (/ x z)
       (if (<= y 1.3e-104) (/ y (- z)) (if (<= y 1.25e-69) (/ x z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -1.46e-67) {
		tmp = t_0;
	} else if (y <= 1.4e-134) {
		tmp = x / z;
	} else if (y <= 1.3e-104) {
		tmp = y / -z;
	} else if (y <= 1.25e-69) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    if (y <= (-1.46d-67)) then
        tmp = t_0
    else if (y <= 1.4d-134) then
        tmp = x / z
    else if (y <= 1.3d-104) then
        tmp = y / -z
    else if (y <= 1.25d-69) then
        tmp = x / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -1.46e-67) {
		tmp = t_0;
	} else if (y <= 1.4e-134) {
		tmp = x / z;
	} else if (y <= 1.3e-104) {
		tmp = y / -z;
	} else if (y <= 1.25e-69) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -1.46e-67:
		tmp = t_0
	elif y <= 1.4e-134:
		tmp = x / z
	elif y <= 1.3e-104:
		tmp = y / -z
	elif y <= 1.25e-69:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -1.46e-67)
		tmp = t_0;
	elseif (y <= 1.4e-134)
		tmp = Float64(x / z);
	elseif (y <= 1.3e-104)
		tmp = Float64(y / Float64(-z));
	elseif (y <= 1.25e-69)
		tmp = Float64(x / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -1.46e-67)
		tmp = t_0;
	elseif (y <= 1.4e-134)
		tmp = x / z;
	elseif (y <= 1.3e-104)
		tmp = y / -z;
	elseif (y <= 1.25e-69)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.46e-67], t$95$0, If[LessEqual[y, 1.4e-134], N[(x / z), $MachinePrecision], If[LessEqual[y, 1.3e-104], N[(y / (-z)), $MachinePrecision], If[LessEqual[y, 1.25e-69], N[(x / z), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.46000000000000002e-67 or 1.25000000000000008e-69 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 71.2%

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

      1. div-sub 71.2%

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

      2. sub-neg 71.2%

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

      3. *-inverses 71.2%

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

      4. metadata-eval 71.2%

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

      5. distribute-lft-in 71.2%

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

      6. metadata-eval 71.2%

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

      7. +-commutative 71.2%

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

      8. mul-1-neg 71.2%

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

      9. unsub-neg 71.2%

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

   5. Simplified 71.2%

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

   if -1.46000000000000002e-67 < y < 1.3999999999999999e-134 or 1.30000000000000001e-104 < y < 1.25000000000000008e-69

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 79.9%

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

   if 1.3999999999999999e-134 < y < 1.30000000000000001e-104

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac2 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.46 \cdot 10^{-67}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-69}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-104}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))) (t_1 (/ x (- z y))))
   (if (<= y -1.9e+17)
     t_0
     (if (<= y 1.3e-134)
       t_1
       (if (<= y 1.75e-104) (/ y (- z)) (if (<= y 2.6e-8) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double t_1 = x / (z - y);
	double tmp;
	if (y <= -1.9e+17) {
		tmp = t_0;
	} else if (y <= 1.3e-134) {
		tmp = t_1;
	} else if (y <= 1.75e-104) {
		tmp = y / -z;
	} else if (y <= 2.6e-8) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    t_1 = x / (z - y)
    if (y <= (-1.9d+17)) then
        tmp = t_0
    else if (y <= 1.3d-134) then
        tmp = t_1
    else if (y <= 1.75d-104) then
        tmp = y / -z
    else if (y <= 2.6d-8) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double t_1 = x / (z - y);
	double tmp;
	if (y <= -1.9e+17) {
		tmp = t_0;
	} else if (y <= 1.3e-134) {
		tmp = t_1;
	} else if (y <= 1.75e-104) {
		tmp = y / -z;
	} else if (y <= 2.6e-8) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	t_1 = x / (z - y)
	tmp = 0
	if y <= -1.9e+17:
		tmp = t_0
	elif y <= 1.3e-134:
		tmp = t_1
	elif y <= 1.75e-104:
		tmp = y / -z
	elif y <= 2.6e-8:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(x / y))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (y <= -1.9e+17)
		tmp = t_0;
	elseif (y <= 1.3e-134)
		tmp = t_1;
	elseif (y <= 1.75e-104)
		tmp = Float64(y / Float64(-z));
	elseif (y <= 2.6e-8)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (x / y);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (y <= -1.9e+17)
		tmp = t_0;
	elseif (y <= 1.3e-134)
		tmp = t_1;
	elseif (y <= 1.75e-104)
		tmp = y / -z;
	elseif (y <= 2.6e-8)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.9e+17], t$95$0, If[LessEqual[y, 1.3e-134], t$95$1, If[LessEqual[y, 1.75e-104], N[(y / (-z)), $MachinePrecision], If[LessEqual[y, 2.6e-8], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9e17 or 2.6000000000000001e-8 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 74.2%

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

      1. div-sub 74.2%

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

      2. sub-neg 74.2%

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

      3. *-inverses 74.2%

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

      4. metadata-eval 74.2%

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

      5. distribute-lft-in 74.2%

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

      6. metadata-eval 74.2%

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

      7. +-commutative 74.2%

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

      8. mul-1-neg 74.2%

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

      9. unsub-neg 74.2%

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

   5. Simplified 74.2%

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

   if -1.9e17 < y < 1.30000000000000011e-134 or 1.75000000000000014e-104 < y < 2.6000000000000001e-8

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 85.8%

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

   if 1.30000000000000011e-134 < y < 1.75000000000000014e-104

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac2 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+17}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-104}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y - z}\\ \mathbf{if}\;y \leq -0.04:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-70}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{+33}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (- y z))))
   (if (<= y -0.04)
     t_0
     (if (<= y 3.7e-70)
       (/ (- x y) z)
       (if (<= y 1.28e+33) (/ x (- z y)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double tmp;
	if (y <= -0.04) {
		tmp = t_0;
	} else if (y <= 3.7e-70) {
		tmp = (x - y) / z;
	} else if (y <= 1.28e+33) {
		tmp = x / (z - y);
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = y / (y - z)
    if (y <= (-0.04d0)) then
        tmp = t_0
    else if (y <= 3.7d-70) then
        tmp = (x - y) / z
    else if (y <= 1.28d+33) then
        tmp = x / (z - y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double tmp;
	if (y <= -0.04) {
		tmp = t_0;
	} else if (y <= 3.7e-70) {
		tmp = (x - y) / z;
	} else if (y <= 1.28e+33) {
		tmp = x / (z - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (y - z)
	tmp = 0
	if y <= -0.04:
		tmp = t_0
	elif y <= 3.7e-70:
		tmp = (x - y) / z
	elif y <= 1.28e+33:
		tmp = x / (z - y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(y - z))
	tmp = 0.0
	if (y <= -0.04)
		tmp = t_0;
	elseif (y <= 3.7e-70)
		tmp = Float64(Float64(x - y) / z);
	elseif (y <= 1.28e+33)
		tmp = Float64(x / Float64(z - y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (y - z);
	tmp = 0.0;
	if (y <= -0.04)
		tmp = t_0;
	elseif (y <= 3.7e-70)
		tmp = (x - y) / z;
	elseif (y <= 1.28e+33)
		tmp = x / (z - y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.04], t$95$0, If[LessEqual[y, 3.7e-70], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 1.28e+33], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0400000000000000008 or 1.28e33 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 85.0%

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

      1. neg-mul-1 85.0%

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

      2. distribute-neg-frac 85.0%

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

   5. Simplified 85.0%

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

      1. frac-2neg 85.0%

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

      2. div-inv 84.8%

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

      3. remove-double-neg 84.8%

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

      4. sub-neg 84.8%

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

      5. distribute-neg-in 84.8%

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

      6. remove-double-neg 84.8%

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

   7. Applied egg-rr 84.8%

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

      1. associate-*r/ 85.0%

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

      2. *-rgt-identity 85.0%

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

      3. +-commutative 85.0%

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

      4. unsub-neg 85.0%

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

   9. Simplified 85.0%

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

   if -0.0400000000000000008 < y < 3.7e-70

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 88.6%

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

   if 3.7e-70 < y < 1.28e33

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 70.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.04:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-70}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{+33}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -160000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-67}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -160000000.0)
   1.0
   (if (<= y 3.1e-67) (/ x z) (if (<= y 4.9e+35) (/ x (- y)) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -160000000.0) {
		tmp = 1.0;
	} else if (y <= 3.1e-67) {
		tmp = x / z;
	} else if (y <= 4.9e+35) {
		tmp = x / -y;
	} else {
		tmp = 1.0;
	}
	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 <= (-160000000.0d0)) then
        tmp = 1.0d0
    else if (y <= 3.1d-67) then
        tmp = x / z
    else if (y <= 4.9d+35) then
        tmp = x / -y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -160000000.0) {
		tmp = 1.0;
	} else if (y <= 3.1e-67) {
		tmp = x / z;
	} else if (y <= 4.9e+35) {
		tmp = x / -y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -160000000.0:
		tmp = 1.0
	elif y <= 3.1e-67:
		tmp = x / z
	elif y <= 4.9e+35:
		tmp = x / -y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -160000000.0)
		tmp = 1.0;
	elseif (y <= 3.1e-67)
		tmp = Float64(x / z);
	elseif (y <= 4.9e+35)
		tmp = Float64(x / Float64(-y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -160000000.0)
		tmp = 1.0;
	elseif (y <= 3.1e-67)
		tmp = x / z;
	elseif (y <= 4.9e+35)
		tmp = x / -y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -160000000.0], 1.0, If[LessEqual[y, 3.1e-67], N[(x / z), $MachinePrecision], If[LessEqual[y, 4.9e+35], N[(x / (-y)), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6e8 or 4.90000000000000025e35 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 66.7%

      \[\leadsto \color{blue}{1} \]

   if -1.6e8 < y < 3.1000000000000003e-67

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 73.9%

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

   if 3.1000000000000003e-67 < y < 4.90000000000000025e35

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 62.7%

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

      1. div-sub 62.7%

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

      2. sub-neg 62.7%

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

      3. *-inverses 62.7%

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

      4. metadata-eval 62.7%

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

      5. distribute-lft-in 62.7%

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

      6. metadata-eval 62.7%

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

      7. +-commutative 62.7%

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

      8. mul-1-neg 62.7%

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

      9. unsub-neg 62.7%

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

   5. Simplified 62.7%

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

   6. Taylor expanded in x around inf 51.0%

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

      1. mul-1-neg 51.0%

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

      2. distribute-frac-neg2 51.0%

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

   8. Simplified 51.0%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -160000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-67}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -225000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -225000.0) 1.0 (if (<= y 1.9e-8) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -225000.0) {
		tmp = 1.0;
	} else if (y <= 1.9e-8) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	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 <= (-225000.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.9d-8) then
        tmp = x / z
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -225000.0) {
		tmp = 1.0;
	} else if (y <= 1.9e-8) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -225000.0:
		tmp = 1.0
	elif y <= 1.9e-8:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -225000.0)
		tmp = 1.0;
	elseif (y <= 1.9e-8)
		tmp = Float64(x / z);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -225000.0)
		tmp = 1.0;
	elseif (y <= 1.9e-8)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -225000.0], 1.0, If[LessEqual[y, 1.9e-8], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -225000 or 1.90000000000000014e-8 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 62.4%

      \[\leadsto \color{blue}{1} \]

   if -225000 < y < 1.90000000000000014e-8

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 69.5%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -225000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0
end function

Java

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

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{z - y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf 34.5%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 34.5%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x / (z - y)) - (y / (z - y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024057 
(FPCore (x y z)
  :name "Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1"
  :precision binary64

  :alt
  (- (/ x (- z y)) (/ y (- z y)))

  (/ (- x y) (- z y)))