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

Percentage Accurate: 100.0% → 100.0%

Time: 8.6s

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: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{z - y}\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 490000:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{+71} \lor \neg \left(x \leq 9.4 \cdot 10^{+104}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- z y))))
   (if (<= x -2.4e+53)
     t_0
     (if (<= x 490000.0)
       (/ y (- y z))
       (if (or (<= x 6.1e+71) (not (<= x 9.4e+104))) t_0 (- 1.0 (/ x y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x / (z - y);
	double tmp;
	if (x <= -2.4e+53) {
		tmp = t_0;
	} else if (x <= 490000.0) {
		tmp = y / (y - z);
	} else if ((x <= 6.1e+71) || !(x <= 9.4e+104)) {
		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 / (z - y)
    if (x <= (-2.4d+53)) then
        tmp = t_0
    else if (x <= 490000.0d0) then
        tmp = y / (y - z)
    else if ((x <= 6.1d+71) .or. (.not. (x <= 9.4d+104))) 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 / (z - y);
	double tmp;
	if (x <= -2.4e+53) {
		tmp = t_0;
	} else if (x <= 490000.0) {
		tmp = y / (y - z);
	} else if ((x <= 6.1e+71) || !(x <= 9.4e+104)) {
		tmp = t_0;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (z - y)
	tmp = 0
	if x <= -2.4e+53:
		tmp = t_0
	elif x <= 490000.0:
		tmp = y / (y - z)
	elif (x <= 6.1e+71) or not (x <= 9.4e+104):
		tmp = t_0
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (x <= -2.4e+53)
		tmp = t_0;
	elseif (x <= 490000.0)
		tmp = Float64(y / Float64(y - z));
	elseif ((x <= 6.1e+71) || !(x <= 9.4e+104))
		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 / (z - y);
	tmp = 0.0;
	if (x <= -2.4e+53)
		tmp = t_0;
	elseif (x <= 490000.0)
		tmp = y / (y - z);
	elseif ((x <= 6.1e+71) || ~((x <= 9.4e+104)))
		tmp = t_0;
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.4e+53], t$95$0, If[LessEqual[x, 490000.0], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 6.1e+71], N[Not[LessEqual[x, 9.4e+104]], $MachinePrecision]], t$95$0, N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4e53 or 4.9e5 < x < 6.1000000000000003e71 or 9.40000000000000034e104 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 85.0%

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

   if -2.4e53 < x < 4.9e5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 80.4%

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

      1. neg-mul-1 80.4%

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

      2. distribute-neg-frac2 80.4%

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

      3. sub-neg 80.4%

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

      4. +-commutative 80.4%

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

      5. distribute-neg-in 80.4%

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

      6. remove-double-neg 80.4%

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

      7. sub-neg 80.4%

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

   5. Simplified 80.4%

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

   if 6.1000000000000003e71 < x < 9.40000000000000034e104

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 86.2%

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

      1. associate-*r/ 86.2%

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

      2. neg-mul-1 86.2%

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

      3. sub-neg 86.2%

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

      4. +-commutative 86.2%

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

      5. distribute-neg-in 86.2%

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

      6. remove-double-neg 86.2%

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

      7. sub-neg 86.2%

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

      8. div-sub 86.4%

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

      9. *-inverses 86.4%

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

   5. Simplified 86.4%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;x \leq 490000:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{+71} \lor \neg \left(x \leq 9.4 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z}\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-100}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-126}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+29}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) z)))
   (if (<= z -9.5e-8)
     t_0
     (if (<= z -4.1e-100)
       (/ y (- y z))
       (if (<= z -7.6e-126)
         (/ x (- z y))
         (if (<= z 2.1e+29) (- 1.0 (/ x y)) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / z;
	double tmp;
	if (z <= -9.5e-8) {
		tmp = t_0;
	} else if (z <= -4.1e-100) {
		tmp = y / (y - z);
	} else if (z <= -7.6e-126) {
		tmp = x / (z - y);
	} else if (z <= 2.1e+29) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / z
    if (z <= (-9.5d-8)) then
        tmp = t_0
    else if (z <= (-4.1d-100)) then
        tmp = y / (y - z)
    else if (z <= (-7.6d-126)) then
        tmp = x / (z - y)
    else if (z <= 2.1d+29) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / z;
	double tmp;
	if (z <= -9.5e-8) {
		tmp = t_0;
	} else if (z <= -4.1e-100) {
		tmp = y / (y - z);
	} else if (z <= -7.6e-126) {
		tmp = x / (z - y);
	} else if (z <= 2.1e+29) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / z
	tmp = 0
	if z <= -9.5e-8:
		tmp = t_0
	elif z <= -4.1e-100:
		tmp = y / (y - z)
	elif z <= -7.6e-126:
		tmp = x / (z - y)
	elif z <= 2.1e+29:
		tmp = 1.0 - (x / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / z)
	tmp = 0.0
	if (z <= -9.5e-8)
		tmp = t_0;
	elseif (z <= -4.1e-100)
		tmp = Float64(y / Float64(y - z));
	elseif (z <= -7.6e-126)
		tmp = Float64(x / Float64(z - y));
	elseif (z <= 2.1e+29)
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / z;
	tmp = 0.0;
	if (z <= -9.5e-8)
		tmp = t_0;
	elseif (z <= -4.1e-100)
		tmp = y / (y - z);
	elseif (z <= -7.6e-126)
		tmp = x / (z - y);
	elseif (z <= 2.1e+29)
		tmp = 1.0 - (x / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -9.5e-8], t$95$0, If[LessEqual[z, -4.1e-100], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -7.6e-126], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.1e+29], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9.50000000000000036e-8 or 2.1000000000000002e29 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 79.1%

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

   if -9.50000000000000036e-8 < z < -4.0999999999999999e-100

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 70.0%

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

      1. neg-mul-1 70.0%

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

      2. distribute-neg-frac2 70.0%

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

      3. sub-neg 70.0%

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

      4. +-commutative 70.0%

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

      5. distribute-neg-in 70.0%

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

      6. remove-double-neg 70.0%

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

      7. sub-neg 70.0%

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

   5. Simplified 70.0%

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

   if -4.0999999999999999e-100 < z < -7.5999999999999997e-126

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 86.2%

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

   if -7.5999999999999997e-126 < z < 2.1000000000000002e29

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 85.5%

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

      1. associate-*r/ 85.5%

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

      2. neg-mul-1 85.5%

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

      3. sub-neg 85.5%

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

      4. +-commutative 85.5%

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

      5. distribute-neg-in 85.5%

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

      6. remove-double-neg 85.5%

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

      7. sub-neg 85.5%

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

      8. div-sub 85.5%

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

      9. *-inverses 85.5%

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

   5. Simplified 85.5%

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

Alternative 4: 61.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+83}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.46 \cdot 10^{-42}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;y \leq 0.00024:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.7e+83)
   1.0
   (if (<= y -2.46e-42) (/ y (- z)) (if (<= y 0.00024) (/ x z) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.7e+83) {
		tmp = 1.0;
	} else if (y <= -2.46e-42) {
		tmp = y / -z;
	} else if (y <= 0.00024) {
		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 <= (-2.7d+83)) then
        tmp = 1.0d0
    else if (y <= (-2.46d-42)) then
        tmp = y / -z
    else if (y <= 0.00024d0) 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 <= -2.7e+83) {
		tmp = 1.0;
	} else if (y <= -2.46e-42) {
		tmp = y / -z;
	} else if (y <= 0.00024) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.7e+83:
		tmp = 1.0
	elif y <= -2.46e-42:
		tmp = y / -z
	elif y <= 0.00024:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.7e+83)
		tmp = 1.0;
	elseif (y <= -2.46e-42)
		tmp = Float64(y / Float64(-z));
	elseif (y <= 0.00024)
		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 <= -2.7e+83)
		tmp = 1.0;
	elseif (y <= -2.46e-42)
		tmp = y / -z;
	elseif (y <= 0.00024)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.7e+83], 1.0, If[LessEqual[y, -2.46e-42], N[(y / (-z)), $MachinePrecision], If[LessEqual[y, 0.00024], N[(x / z), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.70000000000000007e83 or 2.40000000000000006e-4 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 63.0%

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

   if -2.70000000000000007e83 < y < -2.45999999999999989e-42

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 51.2%

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

      1. neg-mul-1 51.2%

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

      2. distribute-neg-frac2 51.2%

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

      3. sub-neg 51.2%

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

      4. +-commutative 51.2%

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

      5. distribute-neg-in 51.2%

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

      6. remove-double-neg 51.2%

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

      7. sub-neg 51.2%

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

   5. Simplified 51.2%

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

   6. Taylor expanded in y around 0 36.2%

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

      1. associate-*r/ 36.2%

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

      2. neg-mul-1 36.2%

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

   8. Simplified 36.2%

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

   if -2.45999999999999989e-42 < y < 2.40000000000000006e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 66.2%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+83}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.46 \cdot 10^{-42}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;y \leq 0.00024:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+82}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.4e+82)
   1.0
   (if (<= y -7.6e-35) (/ x (- y)) (if (<= y 1.1e-5) (/ x z) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.4e+82) {
		tmp = 1.0;
	} else if (y <= -7.6e-35) {
		tmp = x / -y;
	} else if (y <= 1.1e-5) {
		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 <= (-3.4d+82)) then
        tmp = 1.0d0
    else if (y <= (-7.6d-35)) then
        tmp = x / -y
    else if (y <= 1.1d-5) 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 <= -3.4e+82) {
		tmp = 1.0;
	} else if (y <= -7.6e-35) {
		tmp = x / -y;
	} else if (y <= 1.1e-5) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.4e+82:
		tmp = 1.0
	elif y <= -7.6e-35:
		tmp = x / -y
	elif y <= 1.1e-5:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.4e+82)
		tmp = 1.0;
	elseif (y <= -7.6e-35)
		tmp = Float64(x / Float64(-y));
	elseif (y <= 1.1e-5)
		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 <= -3.4e+82)
		tmp = 1.0;
	elseif (y <= -7.6e-35)
		tmp = x / -y;
	elseif (y <= 1.1e-5)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.4e+82], 1.0, If[LessEqual[y, -7.6e-35], N[(x / (-y)), $MachinePrecision], If[LessEqual[y, 1.1e-5], N[(x / z), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.39999999999999994e82 or 1.1e-5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 62.5%

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

   if -3.39999999999999994e82 < y < -7.6000000000000002e-35

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 52.4%

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

      1. associate-*r/ 52.4%

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

      2. neg-mul-1 52.4%

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

      3. sub-neg 52.4%

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

      4. +-commutative 52.4%

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

      5. distribute-neg-in 52.4%

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

      6. remove-double-neg 52.4%

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

      7. sub-neg 52.4%

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

      8. div-sub 52.4%

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

      9. *-inverses 52.4%

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

   5. Simplified 52.4%

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

   6. Taylor expanded in x around inf 40.6%

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

      1. mul-1-neg 40.6%

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

      2. distribute-frac-neg2 40.6%

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

   8. Simplified 40.6%

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

   if -7.6000000000000002e-35 < y < 1.1e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 64.6%

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

Alternative 6: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-82} \lor \neg \left(y \leq 0.000116\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5e-82) (not (<= y 0.000116))) (- 1.0 (/ x y)) (/ x (- z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e-82) || !(y <= 0.000116)) {
		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 <= (-5d-82)) .or. (.not. (y <= 0.000116d0))) 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 <= -5e-82) || !(y <= 0.000116)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5e-82) or not (y <= 0.000116):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (z - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5e-82) || !(y <= 0.000116))
		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 <= -5e-82) || ~((y <= 0.000116)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5e-82], N[Not[LessEqual[y, 0.000116]], $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 < -4.9999999999999998e-82 or 1.16e-4 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 72.8%

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

      1. associate-*r/ 72.8%

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

      2. neg-mul-1 72.8%

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

      3. sub-neg 72.8%

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

      4. +-commutative 72.8%

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

      5. distribute-neg-in 72.8%

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

      6. remove-double-neg 72.8%

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

      7. sub-neg 72.8%

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

      8. div-sub 72.9%

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

      9. *-inverses 72.9%

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

   5. Simplified 72.9%

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

   if -4.9999999999999998e-82 < y < 1.16e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 84.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-82} \lor \neg \left(y \leq 0.000116\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-124} \lor \neg \left(y \leq 2.6 \cdot 10^{-92}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.2e-124) (not (<= y 2.6e-92))) (- 1.0 (/ x y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.2e-124) || !(y <= 2.6e-92)) {
		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 <= (-1.2d-124)) .or. (.not. (y <= 2.6d-92))) 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 <= -1.2e-124) || !(y <= 2.6e-92)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.2e-124) or not (y <= 2.6e-92):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.2e-124) || !(y <= 2.6e-92))
		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 <= -1.2e-124) || ~((y <= 2.6e-92)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.2e-124], N[Not[LessEqual[y, 2.6e-92]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.19999999999999996e-124 or 2.6e-92 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 69.8%

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

      1. associate-*r/ 69.8%

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

      2. neg-mul-1 69.8%

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

      3. sub-neg 69.8%

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

      4. +-commutative 69.8%

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

      5. distribute-neg-in 69.8%

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

      6. remove-double-neg 69.8%

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

      7. sub-neg 69.8%

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

      8. div-sub 69.8%

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

      9. *-inverses 69.8%

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

   5. Simplified 69.8%

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

   if -1.19999999999999996e-124 < y < 2.6e-92

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 79.0%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-124} \lor \neg \left(y \leq 2.6 \cdot 10^{-92}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.55 \cdot 10^{-74}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.55e-74) 1.0 (if (<= y 3.1e-7) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.55e-74) {
		tmp = 1.0;
	} else if (y <= 3.1e-7) {
		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 <= (-2.55d-74)) then
        tmp = 1.0d0
    else if (y <= 3.1d-7) 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 <= -2.55e-74) {
		tmp = 1.0;
	} else if (y <= 3.1e-7) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.55e-74:
		tmp = 1.0
	elif y <= 3.1e-7:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.55e-74)
		tmp = 1.0;
	elseif (y <= 3.1e-7)
		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 <= -2.55e-74)
		tmp = 1.0;
	elseif (y <= 3.1e-7)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.55e-74], 1.0, If[LessEqual[y, 3.1e-7], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5499999999999998e-74 or 3.1e-7 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 53.7%

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

   if -2.5499999999999998e-74 < y < 3.1e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 68.0%

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

Alternative 9: 35.2% 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 35.3%

   \[\leadsto \color{blue}{1} \]
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 2024110 
(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)))