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

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

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. Add Preprocessing

Alternative 2: 75.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z}\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-82} \lor \neg \left(z \leq 3.8 \cdot 10^{+25}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) z)))
   (if (<= z -2.05e-12)
     t_0
     (if (<= z -1.9e-38)
       (/ y (- y z))
       (if (or (<= z -3.3e-82) (not (<= z 3.8e+25))) t_0 (/ (- y x) y))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / z;
	double tmp;
	if (z <= -2.05e-12) {
		tmp = t_0;
	} else if (z <= -1.9e-38) {
		tmp = y / (y - z);
	} else if ((z <= -3.3e-82) || !(z <= 3.8e+25)) {
		tmp = t_0;
	} else {
		tmp = (y - x) / 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) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / z
    if (z <= (-2.05d-12)) then
        tmp = t_0
    else if (z <= (-1.9d-38)) then
        tmp = y / (y - z)
    else if ((z <= (-3.3d-82)) .or. (.not. (z <= 3.8d+25))) then
        tmp = t_0
    else
        tmp = (y - x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / z;
	double tmp;
	if (z <= -2.05e-12) {
		tmp = t_0;
	} else if (z <= -1.9e-38) {
		tmp = y / (y - z);
	} else if ((z <= -3.3e-82) || !(z <= 3.8e+25)) {
		tmp = t_0;
	} else {
		tmp = (y - x) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / z
	tmp = 0
	if z <= -2.05e-12:
		tmp = t_0
	elif z <= -1.9e-38:
		tmp = y / (y - z)
	elif (z <= -3.3e-82) or not (z <= 3.8e+25):
		tmp = t_0
	else:
		tmp = (y - x) / y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / z)
	tmp = 0.0
	if (z <= -2.05e-12)
		tmp = t_0;
	elseif (z <= -1.9e-38)
		tmp = Float64(y / Float64(y - z));
	elseif ((z <= -3.3e-82) || !(z <= 3.8e+25))
		tmp = t_0;
	else
		tmp = Float64(Float64(y - x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / z;
	tmp = 0.0;
	if (z <= -2.05e-12)
		tmp = t_0;
	elseif (z <= -1.9e-38)
		tmp = y / (y - z);
	elseif ((z <= -3.3e-82) || ~((z <= 3.8e+25)))
		tmp = t_0;
	else
		tmp = (y - x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -2.05e-12], t$95$0, If[LessEqual[z, -1.9e-38], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -3.3e-82], N[Not[LessEqual[z, 3.8e+25]], $MachinePrecision]], t$95$0, N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.04999999999999995e-12 or -1.9e-38 < z < -3.30000000000000022e-82 or 3.8e25 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 77.9%

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

   if -2.04999999999999995e-12 < z < -1.9e-38

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. neg-mul-1 99.7%

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

      2. distribute-neg-frac2 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

   5. Simplified 99.7%

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

   if -3.30000000000000022e-82 < z < 3.8e25

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 86.1%

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

      1. associate-*r/ 86.1%

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

      2. neg-mul-1 86.1%

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

      3. sub-neg 86.1%

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

      4. +-commutative 86.1%

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

      5. distribute-neg-in 86.1%

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

      6. remove-double-neg 86.1%

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

      7. sub-neg 86.1%

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

   5. Simplified 86.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{-12}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-82} \lor \neg \left(z \leq 3.8 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z}\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-84} \lor \neg \left(z \leq 5.2 \cdot 10^{+25}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) z)))
   (if (<= z -1.55e-12)
     t_0
     (if (<= z -5.4e-38)
       (/ y (- y z))
       (if (or (<= z -1.85e-84) (not (<= z 5.2e+25))) t_0 (- 1.0 (/ x y)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / z;
	double tmp;
	if (z <= -1.55e-12) {
		tmp = t_0;
	} else if (z <= -5.4e-38) {
		tmp = y / (y - z);
	} else if ((z <= -1.85e-84) || !(z <= 5.2e+25)) {
		tmp = t_0;
	} else {
		tmp = 1.0 - (x / 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) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / z
    if (z <= (-1.55d-12)) then
        tmp = t_0
    else if (z <= (-5.4d-38)) then
        tmp = y / (y - z)
    else if ((z <= (-1.85d-84)) .or. (.not. (z <= 5.2d+25))) then
        tmp = t_0
    else
        tmp = 1.0d0 - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / z;
	double tmp;
	if (z <= -1.55e-12) {
		tmp = t_0;
	} else if (z <= -5.4e-38) {
		tmp = y / (y - z);
	} else if ((z <= -1.85e-84) || !(z <= 5.2e+25)) {
		tmp = t_0;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / z
	tmp = 0
	if z <= -1.55e-12:
		tmp = t_0
	elif z <= -5.4e-38:
		tmp = y / (y - z)
	elif (z <= -1.85e-84) or not (z <= 5.2e+25):
		tmp = t_0
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / z)
	tmp = 0.0
	if (z <= -1.55e-12)
		tmp = t_0;
	elseif (z <= -5.4e-38)
		tmp = Float64(y / Float64(y - z));
	elseif ((z <= -1.85e-84) || !(z <= 5.2e+25))
		tmp = t_0;
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / z;
	tmp = 0.0;
	if (z <= -1.55e-12)
		tmp = t_0;
	elseif (z <= -5.4e-38)
		tmp = y / (y - z);
	elseif ((z <= -1.85e-84) || ~((z <= 5.2e+25)))
		tmp = t_0;
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -1.55e-12], t$95$0, If[LessEqual[z, -5.4e-38], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -1.85e-84], N[Not[LessEqual[z, 5.2e+25]], $MachinePrecision]], t$95$0, N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5500000000000001e-12 or -5.40000000000000011e-38 < z < -1.85e-84 or 5.1999999999999997e25 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 77.9%

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

   if -1.5500000000000001e-12 < z < -5.40000000000000011e-38

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. neg-mul-1 99.7%

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

      2. distribute-neg-frac2 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

   5. Simplified 99.7%

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

   if -1.85e-84 < z < 5.1999999999999997e25

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 86.1%

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

      1. associate-*r/ 86.1%

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

      2. neg-mul-1 86.1%

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

      3. sub-neg 86.1%

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

      4. +-commutative 86.1%

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

      5. distribute-neg-in 86.1%

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

      6. remove-double-neg 86.1%

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

      7. sub-neg 86.1%

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

      8. div-sub 86.1%

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

      9. *-inverses 86.1%

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

   5. Simplified 86.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-12}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-84} \lor \neg \left(z \leq 5.2 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-23} \lor \neg \left(x \leq 3.5 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.6e-23) (not (<= x 3.5e-31))) (/ x (- z y)) (/ y (- y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.6e-23) || !(x <= 3.5e-31)) {
		tmp = x / (z - y);
	} else {
		tmp = y / (y - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-8.6d-23)) .or. (.not. (x <= 3.5d-31))) then
        tmp = x / (z - y)
    else
        tmp = y / (y - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.6e-23) || !(x <= 3.5e-31)) {
		tmp = x / (z - y);
	} else {
		tmp = y / (y - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.6e-23) or not (x <= 3.5e-31):
		tmp = x / (z - y)
	else:
		tmp = y / (y - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.6e-23) || !(x <= 3.5e-31))
		tmp = Float64(x / Float64(z - y));
	else
		tmp = Float64(y / Float64(y - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.6e-23) || ~((x <= 3.5e-31)))
		tmp = x / (z - y);
	else
		tmp = y / (y - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.6e-23], N[Not[LessEqual[x, 3.5e-31]], $MachinePrecision]], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.60000000000000004e-23 or 3.49999999999999985e-31 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 74.6%

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

   if -8.60000000000000004e-23 < x < 3.49999999999999985e-31

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 84.0%

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

      1. neg-mul-1 84.0%

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

      2. distribute-neg-frac2 84.0%

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

      3. sub-neg 84.0%

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

      4. +-commutative 84.0%

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

      5. distribute-neg-in 84.0%

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

      6. remove-double-neg 84.0%

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

      7. sub-neg 84.0%

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

   5. Simplified 84.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-23} \lor \neg \left(x \leq 3.5 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+48} \lor \neg \left(y \leq 6.5 \cdot 10^{-82}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.8e+48) (not (<= y 6.5e-82))) (- 1.0 (/ x y)) (/ x (- z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.8e+48) || !(y <= 6.5e-82)) {
		tmp = 1.0 - (x / y);
	} 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 <= (-2.8d+48)) .or. (.not. (y <= 6.5d-82))) then
        tmp = 1.0d0 - (x / y)
    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 <= -2.8e+48) || !(y <= 6.5e-82)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.8e+48) or not (y <= 6.5e-82):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (z - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.8e+48) || !(y <= 6.5e-82))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.8e+48) || ~((y <= 6.5e-82)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.8e+48], N[Not[LessEqual[y, 6.5e-82]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.80000000000000012e48 or 6.4999999999999997e-82 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 76.9%

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

      1. associate-*r/ 76.9%

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

      2. neg-mul-1 76.9%

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

      3. sub-neg 76.9%

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

      4. +-commutative 76.9%

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

      5. distribute-neg-in 76.9%

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

      6. remove-double-neg 76.9%

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

      7. sub-neg 76.9%

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

      8. div-sub 76.9%

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

      9. *-inverses 76.9%

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

   5. Simplified 76.9%

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

   if -2.80000000000000012e48 < y < 6.4999999999999997e-82

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+48} \lor \neg \left(y \leq 6.5 \cdot 10^{-82}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-23} \lor \neg \left(y \leq 4.4 \cdot 10^{-82}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -9.5e-23) (not (<= y 4.4e-82))) (- 1.0 (/ x y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -9.5e-23) || !(y <= 4.4e-82)) {
		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) :: tmp
    if ((y <= (-9.5d-23)) .or. (.not. (y <= 4.4d-82))) 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 tmp;
	if ((y <= -9.5e-23) || !(y <= 4.4e-82)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -9.5e-23) or not (y <= 4.4e-82):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -9.5e-23) || !(y <= 4.4e-82))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -9.5e-23) || ~((y <= 4.4e-82)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -9.5e-23], N[Not[LessEqual[y, 4.4e-82]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.50000000000000058e-23 or 4.39999999999999971e-82 < 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. associate-*r/ 74.2%

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

      2. neg-mul-1 74.2%

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

      3. sub-neg 74.2%

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

      4. +-commutative 74.2%

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

      5. distribute-neg-in 74.2%

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

      6. remove-double-neg 74.2%

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

      7. sub-neg 74.2%

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

      8. div-sub 74.2%

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

      9. *-inverses 74.2%

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

   5. Simplified 74.2%

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

   if -9.50000000000000058e-23 < y < 4.39999999999999971e-82

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 69.4%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-23} \lor \neg \left(y \leq 4.4 \cdot 10^{-82}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+70}:\\ \;\;\;\;1 + \frac{z}{y}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.1e+70) (+ 1.0 (/ z y)) (if (<= y 5.8e-81) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.1e+70) {
		tmp = 1.0 + (z / y);
	} else if (y <= 5.8e-81) {
		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 <= (-1.1d+70)) then
        tmp = 1.0d0 + (z / y)
    else if (y <= 5.8d-81) 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 <= -1.1e+70) {
		tmp = 1.0 + (z / y);
	} else if (y <= 5.8e-81) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.1e+70:
		tmp = 1.0 + (z / y)
	elif y <= 5.8e-81:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.1e+70)
		tmp = Float64(1.0 + Float64(z / y));
	elseif (y <= 5.8e-81)
		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 <= -1.1e+70)
		tmp = 1.0 + (z / y);
	elseif (y <= 5.8e-81)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.1e+70], N[(1.0 + N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e-81], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1e70

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 86.2%

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

      1. neg-mul-1 86.2%

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

      2. distribute-neg-frac2 86.2%

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

      3. sub-neg 86.2%

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

      4. +-commutative 86.2%

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

      5. distribute-neg-in 86.2%

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

      6. remove-double-neg 86.2%

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

      7. sub-neg 86.2%

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

   5. Simplified 86.2%

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

   6. Taylor expanded in y around inf 71.9%

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

   if -1.1e70 < y < 5.79999999999999978e-81

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 62.3%

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

   if 5.79999999999999978e-81 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 55.0%

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

Alternative 8: 60.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{+70}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.6e+70) 1.0 (if (<= y 7.8e-81) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.6e+70) {
		tmp = 1.0;
	} else if (y <= 7.8e-81) {
		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 <= (-7.6d+70)) then
        tmp = 1.0d0
    else if (y <= 7.8d-81) 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 <= -7.6e+70) {
		tmp = 1.0;
	} else if (y <= 7.8e-81) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.6e+70:
		tmp = 1.0
	elif y <= 7.8e-81:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.6e+70)
		tmp = 1.0;
	elseif (y <= 7.8e-81)
		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 <= -7.6e+70)
		tmp = 1.0;
	elseif (y <= 7.8e-81)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.6e+70], 1.0, If[LessEqual[y, 7.8e-81], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5999999999999996e70 or 7.7999999999999997e-81 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 60.6%

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

   if -7.5999999999999996e70 < y < 7.7999999999999997e-81

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 62.3%

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

Alternative 9: 35.4% 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 36.6%

   \[\leadsto \color{blue}{1} \]
4. 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 2024111 
(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)))