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

Percentage Accurate: 100.0% → 100.0%

Time: 7.9s

Alternatives: 10

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. Add Preprocessing

Alternative 2: 68.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-186}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{-220}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{-25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (<= t_0 -1e+103)
     (/ x (- y))
     (if (<= t_0 -5e-186)
       (/ x z)
       (if (<= t_0 1e-220)
         (/ (- y) z)
         (if (<= t_0 1e-25)
           (/ x z)
           (if (<= t_0 2.0) (- 1.0 (/ x y)) (/ x z))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= -1e+103) {
		tmp = x / -y;
	} else if (t_0 <= -5e-186) {
		tmp = x / z;
	} else if (t_0 <= 1e-220) {
		tmp = -y / z;
	} else if (t_0 <= 1e-25) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (z - y)
    if (t_0 <= (-1d+103)) then
        tmp = x / -y
    else if (t_0 <= (-5d-186)) then
        tmp = x / z
    else if (t_0 <= 1d-220) then
        tmp = -y / z
    else if (t_0 <= 1d-25) then
        tmp = x / z
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= -1e+103) {
		tmp = x / -y;
	} else if (t_0 <= -5e-186) {
		tmp = x / z;
	} else if (t_0 <= 1e-220) {
		tmp = -y / z;
	} else if (t_0 <= 1e-25) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	tmp = 0
	if t_0 <= -1e+103:
		tmp = x / -y
	elif t_0 <= -5e-186:
		tmp = x / z
	elif t_0 <= 1e-220:
		tmp = -y / z
	elif t_0 <= 1e-25:
		tmp = x / z
	elif t_0 <= 2.0:
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -1e+103)
		tmp = Float64(x / Float64(-y));
	elseif (t_0 <= -5e-186)
		tmp = Float64(x / z);
	elseif (t_0 <= 1e-220)
		tmp = Float64(Float64(-y) / z);
	elseif (t_0 <= 1e-25)
		tmp = Float64(x / z);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_0 <= -1e+103)
		tmp = x / -y;
	elseif (t_0 <= -5e-186)
		tmp = x / z;
	elseif (t_0 <= 1e-220)
		tmp = -y / z;
	elseif (t_0 <= 1e-25)
		tmp = x / z;
	elseif (t_0 <= 2.0)
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+103], N[(x / (-y)), $MachinePrecision], If[LessEqual[t$95$0, -5e-186], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 1e-220], N[((-y) / z), $MachinePrecision], If[LessEqual[t$95$0, 1e-25], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 82.6%

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

   if -1e103 < (/.f64 (-.f64 x y) (-.f64 z y)) < -5e-186 or 9.99999999999999992e-221 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000004e-25 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 66.0

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

   5. Applied rewrites 66.0%

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

   if -5e-186 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.99999999999999992e-221

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 82.7%

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

   if 1.00000000000000004e-25 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 96.1

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

   5. Applied rewrites 96.1%

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

   7. Applied rewrites 96.2%

      \[\leadsto 1 - \color{blue}{\frac{x}{y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 68.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-186}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{-220}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{-25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (<= t_0 -1e+103)
     (/ x (- y))
     (if (<= t_0 -5e-186)
       (/ x z)
       (if (<= t_0 1e-220)
         (/ (- y) z)
         (if (<= t_0 1e-25) (/ x z) (if (<= t_0 2.0) 1.0 (/ x z))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= -1e+103) {
		tmp = x / -y;
	} else if (t_0 <= -5e-186) {
		tmp = x / z;
	} else if (t_0 <= 1e-220) {
		tmp = -y / z;
	} else if (t_0 <= 1e-25) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (z - y)
    if (t_0 <= (-1d+103)) then
        tmp = x / -y
    else if (t_0 <= (-5d-186)) then
        tmp = x / z
    else if (t_0 <= 1d-220) then
        tmp = -y / z
    else if (t_0 <= 1d-25) then
        tmp = x / z
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= -1e+103) {
		tmp = x / -y;
	} else if (t_0 <= -5e-186) {
		tmp = x / z;
	} else if (t_0 <= 1e-220) {
		tmp = -y / z;
	} else if (t_0 <= 1e-25) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	tmp = 0
	if t_0 <= -1e+103:
		tmp = x / -y
	elif t_0 <= -5e-186:
		tmp = x / z
	elif t_0 <= 1e-220:
		tmp = -y / z
	elif t_0 <= 1e-25:
		tmp = x / z
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -1e+103)
		tmp = Float64(x / Float64(-y));
	elseif (t_0 <= -5e-186)
		tmp = Float64(x / z);
	elseif (t_0 <= 1e-220)
		tmp = Float64(Float64(-y) / z);
	elseif (t_0 <= 1e-25)
		tmp = Float64(x / z);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_0 <= -1e+103)
		tmp = x / -y;
	elseif (t_0 <= -5e-186)
		tmp = x / z;
	elseif (t_0 <= 1e-220)
		tmp = -y / z;
	elseif (t_0 <= 1e-25)
		tmp = x / z;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+103], N[(x / (-y)), $MachinePrecision], If[LessEqual[t$95$0, -5e-186], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 1e-220], N[((-y) / z), $MachinePrecision], If[LessEqual[t$95$0, 1e-25], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(x / z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 82.6%

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

   if -1e103 < (/.f64 (-.f64 x y) (-.f64 z y)) < -5e-186 or 9.99999999999999992e-221 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000004e-25 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 66.0

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

   5. Applied rewrites 66.0%

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

   if -5e-186 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.99999999999999992e-221

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 82.7%

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

   if 1.00000000000000004e-25 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 92.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y - z}\\ t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z - y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-186}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-220}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-25}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (- y z))) (t_1 (/ (- x y) (- z y))) (t_2 (/ x (- z y))))
   (if (<= t_1 -5e-186)
     t_2
     (if (<= t_1 1e-220)
       t_0
       (if (<= t_1 1e-25) t_2 (if (<= t_1 2.0) t_0 t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double t_1 = (x - y) / (z - y);
	double t_2 = x / (z - y);
	double tmp;
	if (t_1 <= -5e-186) {
		tmp = t_2;
	} else if (t_1 <= 1e-220) {
		tmp = t_0;
	} else if (t_1 <= 1e-25) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = y / (y - z)
    t_1 = (x - y) / (z - y)
    t_2 = x / (z - y)
    if (t_1 <= (-5d-186)) then
        tmp = t_2
    else if (t_1 <= 1d-220) then
        tmp = t_0
    else if (t_1 <= 1d-25) then
        tmp = t_2
    else if (t_1 <= 2.0d0) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double t_1 = (x - y) / (z - y);
	double t_2 = x / (z - y);
	double tmp;
	if (t_1 <= -5e-186) {
		tmp = t_2;
	} else if (t_1 <= 1e-220) {
		tmp = t_0;
	} else if (t_1 <= 1e-25) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (y - z)
	t_1 = (x - y) / (z - y)
	t_2 = x / (z - y)
	tmp = 0
	if t_1 <= -5e-186:
		tmp = t_2
	elif t_1 <= 1e-220:
		tmp = t_0
	elif t_1 <= 1e-25:
		tmp = t_2
	elif t_1 <= 2.0:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(y - z))
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -5e-186)
		tmp = t_2;
	elseif (t_1 <= 1e-220)
		tmp = t_0;
	elseif (t_1 <= 1e-25)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (y - z);
	t_1 = (x - y) / (z - y);
	t_2 = x / (z - y);
	tmp = 0.0;
	if (t_1 <= -5e-186)
		tmp = t_2;
	elseif (t_1 <= 1e-220)
		tmp = t_0;
	elseif (t_1 <= 1e-25)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-186], t$95$2, If[LessEqual[t$95$1, 1e-220], t$95$0, If[LessEqual[t$95$1, 1e-25], t$95$2, If[LessEqual[t$95$1, 2.0], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e-186 or 9.99999999999999992e-221 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000004e-25 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 85.4

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

   5. Applied rewrites 85.4%

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

   if -5e-186 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.99999999999999992e-221 or 1.00000000000000004e-25 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 92.5

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

   5. Applied rewrites 92.5%

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

Alternative 5: 83.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-186}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-220}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{-25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))) (t_1 (/ x (- z y))))
   (if (<= t_0 -5e-186)
     t_1
     (if (<= t_0 1e-220)
       (/ (- y) z)
       (if (<= t_0 1e-25) t_1 (if (<= t_0 2.0) (- 1.0 (/ x y)) t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -5e-186) {
		tmp = t_1;
	} else if (t_0 <= 1e-220) {
		tmp = -y / z;
	} else if (t_0 <= 1e-25) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = t_1;
	}
	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 = (x - y) / (z - y)
    t_1 = x / (z - y)
    if (t_0 <= (-5d-186)) then
        tmp = t_1
    else if (t_0 <= 1d-220) then
        tmp = -y / z
    else if (t_0 <= 1d-25) then
        tmp = t_1
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -5e-186) {
		tmp = t_1;
	} else if (t_0 <= 1e-220) {
		tmp = -y / z;
	} else if (t_0 <= 1e-25) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	t_1 = x / (z - y)
	tmp = 0
	if t_0 <= -5e-186:
		tmp = t_1
	elif t_0 <= 1e-220:
		tmp = -y / z
	elif t_0 <= 1e-25:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = 1.0 - (x / y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -5e-186)
		tmp = t_1;
	elseif (t_0 <= 1e-220)
		tmp = Float64(Float64(-y) / z);
	elseif (t_0 <= 1e-25)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -5e-186)
		tmp = t_1;
	elseif (t_0 <= 1e-220)
		tmp = -y / z;
	elseif (t_0 <= 1e-25)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = 1.0 - (x / y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-186], t$95$1, If[LessEqual[t$95$0, 1e-220], N[((-y) / z), $MachinePrecision], If[LessEqual[t$95$0, 1e-25], t$95$1, If[LessEqual[t$95$0, 2.0], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e-186 or 9.99999999999999992e-221 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000004e-25 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 85.4

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

   5. Applied rewrites 85.4%

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

   if -5e-186 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.99999999999999992e-221

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 82.7%

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

   if 1.00000000000000004e-25 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 96.1

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

   5. Applied rewrites 96.1%

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

   7. Applied rewrites 96.2%

      \[\leadsto 1 - \color{blue}{\frac{x}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{z - x}{y} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))) (t_1 (/ x (- z y))))
   (if (<= t_0 -2e+15)
     t_1
     (if (<= t_0 1e-5)
       (/ (- x y) z)
       (if (<= t_0 2.0) (- (/ (- z x) y) -1.0) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -2e+15) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = ((z - x) / y) - -1.0;
	} else {
		tmp = t_1;
	}
	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 = (x - y) / (z - y)
    t_1 = x / (z - y)
    if (t_0 <= (-2d+15)) then
        tmp = t_1
    else if (t_0 <= 1d-5) then
        tmp = (x - y) / z
    else if (t_0 <= 2.0d0) then
        tmp = ((z - x) / y) - (-1.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -2e+15) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = ((z - x) / y) - -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	t_1 = x / (z - y)
	tmp = 0
	if t_0 <= -2e+15:
		tmp = t_1
	elif t_0 <= 1e-5:
		tmp = (x - y) / z
	elif t_0 <= 2.0:
		tmp = ((z - x) / y) - -1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -2e+15)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = Float64(Float64(x - y) / z);
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(Float64(z - x) / y) - -1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -2e+15)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = (x - y) / z;
	elseif (t_0 <= 2.0)
		tmp = ((z - x) / y) - -1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+15], t$95$1, If[LessEqual[t$95$0, 1e-5], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] - -1.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2e15 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if -2e15 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. distribute-lft-out-- N/A

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

      15. lower-/.f64 N/A

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

      16. cancel-sign-sub-inv N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. +-commutative N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 7: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))) (t_1 (/ x (- z y))))
   (if (<= t_0 -2e+15)
     t_1
     (if (<= t_0 1e-5) (/ (- x y) z) (if (<= t_0 2.0) (- 1.0 (/ x y)) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -2e+15) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = t_1;
	}
	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 = (x - y) / (z - y)
    t_1 = x / (z - y)
    if (t_0 <= (-2d+15)) then
        tmp = t_1
    else if (t_0 <= 1d-5) then
        tmp = (x - y) / z
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -2e+15) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	t_1 = x / (z - y)
	tmp = 0
	if t_0 <= -2e+15:
		tmp = t_1
	elif t_0 <= 1e-5:
		tmp = (x - y) / z
	elif t_0 <= 2.0:
		tmp = 1.0 - (x / y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -2e+15)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = Float64(Float64(x - y) / z);
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -2e+15)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = (x - y) / z;
	elseif (t_0 <= 2.0)
		tmp = 1.0 - (x / y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+15], t$95$1, If[LessEqual[t$95$0, 1e-5], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2e15 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if -2e15 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 98.3

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 98.3%

      \[\leadsto 1 - \color{blue}{\frac{x}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;t\_0 \leq 10^{-25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (<= t_0 -1e+103)
     (/ x (- y))
     (if (<= t_0 1e-25) (/ x z) (if (<= t_0 2.0) 1.0 (/ x z))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= -1e+103) {
		tmp = x / -y;
	} else if (t_0 <= 1e-25) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (z - y)
    if (t_0 <= (-1d+103)) then
        tmp = x / -y
    else if (t_0 <= 1d-25) then
        tmp = x / z
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= -1e+103) {
		tmp = x / -y;
	} else if (t_0 <= 1e-25) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	tmp = 0
	if t_0 <= -1e+103:
		tmp = x / -y
	elif t_0 <= 1e-25:
		tmp = x / z
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -1e+103)
		tmp = Float64(x / Float64(-y));
	elseif (t_0 <= 1e-25)
		tmp = Float64(x / z);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_0 <= -1e+103)
		tmp = x / -y;
	elseif (t_0 <= 1e-25)
		tmp = x / z;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+103], N[(x / (-y)), $MachinePrecision], If[LessEqual[t$95$0, 1e-25], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(x / z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 82.6%

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

   if -1e103 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000004e-25 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.2

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

   5. Applied rewrites 64.2%

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

   if 1.00000000000000004e-25 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 92.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq 10^{-25}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (<= t_0 1e-25) (/ x z) (if (<= t_0 2.0) 1.0 (/ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= 1e-25) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (z - y)
    if (t_0 <= 1d-25) then
        tmp = x / z
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= 1e-25) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	tmp = 0
	if t_0 <= 1e-25:
		tmp = x / z
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_0 <= 1e-25)
		tmp = Float64(x / z);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_0 <= 1e-25)
		tmp = x / z;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-25], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(x / z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000004e-25 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if 1.00000000000000004e-25 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 92.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 34.5% accurate, 18.0× 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

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

5. Applied rewrites 31.9%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer Target 1: 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 2024235 
(FPCore (x y z)
  :name "Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1"
  :precision binary64

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

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