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, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\frac{x - y}{z - y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. div-sub 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -480:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-28}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -480.0)
     t_0
     (if (<= y 2.7e-74) (/ x (- z y)) (if (<= y 1.85e-28) (/ y (- z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -480.0) {
		tmp = t_0;
	} else if (y <= 2.7e-74) {
		tmp = x / (z - y);
	} else if (y <= 1.85e-28) {
		tmp = y / -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 <= (-480.0d0)) then
        tmp = t_0
    else if (y <= 2.7d-74) then
        tmp = x / (z - y)
    else if (y <= 1.85d-28) then
        tmp = y / -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 <= -480.0) {
		tmp = t_0;
	} else if (y <= 2.7e-74) {
		tmp = x / (z - y);
	} else if (y <= 1.85e-28) {
		tmp = y / -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -480.0:
		tmp = t_0
	elif y <= 2.7e-74:
		tmp = x / (z - y)
	elif y <= 1.85e-28:
		tmp = y / -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 <= -480.0)
		tmp = t_0;
	elseif (y <= 2.7e-74)
		tmp = Float64(x / Float64(z - y));
	elseif (y <= 1.85e-28)
		tmp = Float64(y / Float64(-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 <= -480.0)
		tmp = t_0;
	elseif (y <= 2.7e-74)
		tmp = x / (z - y);
	elseif (y <= 1.85e-28)
		tmp = y / -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, -480.0], t$95$0, If[LessEqual[y, 2.7e-74], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e-28], N[(y / (-z)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -480 or 1.8500000000000001e-28 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 75.4%

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

      1. associate-*r/ 75.4%

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

      2. neg-mul-1 75.4%

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

      3. sub-neg 75.4%

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

      4. +-commutative 75.4%

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

      5. distribute-neg-in 75.4%

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

      6. remove-double-neg 75.4%

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

      7. sub-neg 75.4%

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

      8. div-sub 75.4%

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

      9. *-inverses 75.4%

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

   5. Simplified 75.4%

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

   if -480 < y < 2.70000000000000018e-74

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 80.6%

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

   if 2.70000000000000018e-74 < y < 1.8500000000000001e-28

   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-frac2 85.1%

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

      3. sub-neg 85.1%

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

      4. +-commutative 85.1%

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

      5. distribute-neg-in 85.1%

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

      6. remove-double-neg 85.1%

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

      7. sub-neg 85.1%

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

   5. Simplified 85.1%

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

   6. Taylor expanded in y around 0 77.5%

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

      1. associate-*r/ 77.5%

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

      2. neg-mul-1 77.5%

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

   8. Simplified 77.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -480:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-28}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{-104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-76}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -9.5e-104)
     t_0
     (if (<= y 9e-76) (/ x z) (if (<= y 2.2e-28) (/ y (- z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -9.5e-104) {
		tmp = t_0;
	} else if (y <= 9e-76) {
		tmp = x / z;
	} else if (y <= 2.2e-28) {
		tmp = y / -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 <= (-9.5d-104)) then
        tmp = t_0
    else if (y <= 9d-76) then
        tmp = x / z
    else if (y <= 2.2d-28) then
        tmp = y / -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 <= -9.5e-104) {
		tmp = t_0;
	} else if (y <= 9e-76) {
		tmp = x / z;
	} else if (y <= 2.2e-28) {
		tmp = y / -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -9.5e-104:
		tmp = t_0
	elif y <= 9e-76:
		tmp = x / z
	elif y <= 2.2e-28:
		tmp = y / -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 <= -9.5e-104)
		tmp = t_0;
	elseif (y <= 9e-76)
		tmp = Float64(x / z);
	elseif (y <= 2.2e-28)
		tmp = Float64(y / Float64(-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 <= -9.5e-104)
		tmp = t_0;
	elseif (y <= 9e-76)
		tmp = x / z;
	elseif (y <= 2.2e-28)
		tmp = y / -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, -9.5e-104], t$95$0, If[LessEqual[y, 9e-76], N[(x / z), $MachinePrecision], If[LessEqual[y, 2.2e-28], N[(y / (-z)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5000000000000002e-104 or 2.19999999999999996e-28 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 72.1%

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

      1. associate-*r/ 72.1%

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

      2. neg-mul-1 72.1%

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

      3. sub-neg 72.1%

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

      4. +-commutative 72.1%

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

      5. distribute-neg-in 72.1%

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

      6. remove-double-neg 72.1%

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

      7. sub-neg 72.1%

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

      8. div-sub 72.1%

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

      9. *-inverses 72.1%

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

   5. Simplified 72.1%

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

   if -9.5000000000000002e-104 < y < 9.0000000000000001e-76

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 75.5%

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

   if 9.0000000000000001e-76 < y < 2.19999999999999996e-28

   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-frac2 85.1%

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

      3. sub-neg 85.1%

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

      4. +-commutative 85.1%

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

      5. distribute-neg-in 85.1%

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

      6. remove-double-neg 85.1%

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

      7. sub-neg 85.1%

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

   5. Simplified 85.1%

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

   6. Taylor expanded in y around 0 77.5%

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

      1. associate-*r/ 77.5%

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

      2. neg-mul-1 77.5%

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

   8. Simplified 77.5%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-104}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-76}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{z}{y}\\ \mathbf{if}\;y \leq -0.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{+63}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ z y))))
   (if (<= y -0.8)
     t_0
     (if (<= y 3e-77) (/ x z) (if (<= y 1.38e+63) (/ y (- z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z / y);
	double tmp;
	if (y <= -0.8) {
		tmp = t_0;
	} else if (y <= 3e-77) {
		tmp = x / z;
	} else if (y <= 1.38e+63) {
		tmp = y / -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 + (z / y)
    if (y <= (-0.8d0)) then
        tmp = t_0
    else if (y <= 3d-77) then
        tmp = x / z
    else if (y <= 1.38d+63) then
        tmp = y / -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 + (z / y);
	double tmp;
	if (y <= -0.8) {
		tmp = t_0;
	} else if (y <= 3e-77) {
		tmp = x / z;
	} else if (y <= 1.38e+63) {
		tmp = y / -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z / y)
	tmp = 0
	if y <= -0.8:
		tmp = t_0
	elif y <= 3e-77:
		tmp = x / z
	elif y <= 1.38e+63:
		tmp = y / -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z / y))
	tmp = 0.0
	if (y <= -0.8)
		tmp = t_0;
	elseif (y <= 3e-77)
		tmp = Float64(x / z);
	elseif (y <= 1.38e+63)
		tmp = Float64(y / Float64(-z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z / y);
	tmp = 0.0;
	if (y <= -0.8)
		tmp = t_0;
	elseif (y <= 3e-77)
		tmp = x / z;
	elseif (y <= 1.38e+63)
		tmp = y / -z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.8], t$95$0, If[LessEqual[y, 3e-77], N[(x / z), $MachinePrecision], If[LessEqual[y, 1.38e+63], N[(y / (-z)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.80000000000000004 or 1.38e63 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 82.0%

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

      1. neg-mul-1 82.0%

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

      2. distribute-neg-frac2 82.0%

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

      3. sub-neg 82.0%

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

      4. +-commutative 82.0%

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

      5. distribute-neg-in 82.0%

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

      6. remove-double-neg 82.0%

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

      7. sub-neg 82.0%

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

   5. Simplified 82.0%

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

   6. Taylor expanded in y around inf 69.9%

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

   if -0.80000000000000004 < y < 3.00000000000000016e-77

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 66.5%

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

   if 3.00000000000000016e-77 < y < 1.38e63

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 75.0%

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

      1. neg-mul-1 75.0%

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

      2. distribute-neg-frac2 75.0%

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

      3. sub-neg 75.0%

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

      4. +-commutative 75.0%

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

      5. distribute-neg-in 75.0%

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

      6. remove-double-neg 75.0%

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

      7. sub-neg 75.0%

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

   5. Simplified 75.0%

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

   6. Taylor expanded in y around 0 54.3%

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

      1. associate-*r/ 54.3%

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

      2. neg-mul-1 54.3%

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

   8. Simplified 54.3%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.8:\\ \;\;\;\;1 + \frac{z}{y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{+63}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -215:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{+63}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -215.0)
   1.0
   (if (<= y 1.2e-74) (/ x z) (if (<= y 1.38e+63) (/ y (- z)) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -215.0) {
		tmp = 1.0;
	} else if (y <= 1.2e-74) {
		tmp = x / z;
	} else if (y <= 1.38e+63) {
		tmp = y / -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 <= (-215.0d0)) then
        tmp = 1.0d0
    else if (y <= 1.2d-74) then
        tmp = x / z
    else if (y <= 1.38d+63) then
        tmp = y / -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 <= -215.0) {
		tmp = 1.0;
	} else if (y <= 1.2e-74) {
		tmp = x / z;
	} else if (y <= 1.38e+63) {
		tmp = y / -z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -215.0:
		tmp = 1.0
	elif y <= 1.2e-74:
		tmp = x / z
	elif y <= 1.38e+63:
		tmp = y / -z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -215.0)
		tmp = 1.0;
	elseif (y <= 1.2e-74)
		tmp = Float64(x / z);
	elseif (y <= 1.38e+63)
		tmp = Float64(y / Float64(-z));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -215.0)
		tmp = 1.0;
	elseif (y <= 1.2e-74)
		tmp = x / z;
	elseif (y <= 1.38e+63)
		tmp = y / -z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -215.0], 1.0, If[LessEqual[y, 1.2e-74], N[(x / z), $MachinePrecision], If[LessEqual[y, 1.38e+63], N[(y / (-z)), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -215 or 1.38e63 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 69.5%

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

   if -215 < y < 1.1999999999999999e-74

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 66.5%

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

   if 1.1999999999999999e-74 < y < 1.38e63

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 75.0%

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

      1. neg-mul-1 75.0%

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

      2. distribute-neg-frac2 75.0%

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

      3. sub-neg 75.0%

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

      4. +-commutative 75.0%

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

      5. distribute-neg-in 75.0%

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

      6. remove-double-neg 75.0%

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

      7. sub-neg 75.0%

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

   5. Simplified 75.0%

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

   6. Taylor expanded in y around 0 54.3%

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

      1. associate-*r/ 54.3%

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

      2. neg-mul-1 54.3%

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

   8. Simplified 54.3%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -215:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{+63}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.045 \lor \neg \left(y \leq 1.05 \cdot 10^{-108}\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.045) (not (<= y 1.05e-108))) (/ y (- y z)) (/ x (- z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.045) || !(y <= 1.05e-108)) {
		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 <= (-0.045d0)) .or. (.not. (y <= 1.05d-108))) 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 <= -0.045) || !(y <= 1.05e-108)) {
		tmp = y / (y - z);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.045) or not (y <= 1.05e-108):
		tmp = y / (y - z)
	else:
		tmp = x / (z - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.045) || !(y <= 1.05e-108))
		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 <= -0.045) || ~((y <= 1.05e-108)))
		tmp = y / (y - z);
	else
		tmp = x / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.045], N[Not[LessEqual[y, 1.05e-108]], $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 < -0.044999999999999998 or 1.05e-108 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 79.3%

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

      1. neg-mul-1 79.3%

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

      2. distribute-neg-frac2 79.3%

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

      3. sub-neg 79.3%

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

      4. +-commutative 79.3%

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

      5. distribute-neg-in 79.3%

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

      6. remove-double-neg 79.3%

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

      7. sub-neg 79.3%

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

   5. Simplified 79.3%

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

   if -0.044999999999999998 < y < 1.05e-108

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 82.4%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.045 \lor \neg \left(y \leq 1.05 \cdot 10^{-108}\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.3) 1.0 (if (<= y 4.5e-39) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.3) {
		tmp = 1.0;
	} else if (y <= 4.5e-39) {
		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 <= (-3.3d0)) then
        tmp = 1.0d0
    else if (y <= 4.5d-39) 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 <= -3.3) {
		tmp = 1.0;
	} else if (y <= 4.5e-39) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.3:
		tmp = 1.0
	elif y <= 4.5e-39:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.3)
		tmp = 1.0;
	elseif (y <= 4.5e-39)
		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 <= -3.3)
		tmp = 1.0;
	elseif (y <= 4.5e-39)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.3], 1.0, If[LessEqual[y, 4.5e-39], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2999999999999998 or 4.5000000000000001e-39 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 63.1%

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

   if -3.2999999999999998 < y < 4.5000000000000001e-39

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.2%

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

Alternative 8: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 9: 35.3% 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 99.9%

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

3. Taylor expanded in y around inf 38.0%

   \[\leadsto \color{blue}{1} \]
4. 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 2024145 
(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)))