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

Percentage Accurate: 100.0% → 100.0%

Time: 7.5s

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 100.0%

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

   1. div-sub 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 77.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -2.45 \cdot 10^{+48}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z} - \frac{y}{z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -2.45e+48)
     (/ y (- y z))
     (if (<= y -1.85e-7)
       t_0
       (if (<= y -3.7e-60)
         (- (/ x z) (/ y z))
         (if (<= y 2.1e-16) (/ x (- z y)) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -2.45e+48) {
		tmp = y / (y - z);
	} else if (y <= -1.85e-7) {
		tmp = t_0;
	} else if (y <= -3.7e-60) {
		tmp = (x / z) - (y / z);
	} else if (y <= 2.1e-16) {
		tmp = x / (z - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    if (y <= (-2.45d+48)) then
        tmp = y / (y - z)
    else if (y <= (-1.85d-7)) then
        tmp = t_0
    else if (y <= (-3.7d-60)) then
        tmp = (x / z) - (y / z)
    else if (y <= 2.1d-16) then
        tmp = x / (z - y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -2.45e+48) {
		tmp = y / (y - z);
	} else if (y <= -1.85e-7) {
		tmp = t_0;
	} else if (y <= -3.7e-60) {
		tmp = (x / z) - (y / z);
	} else if (y <= 2.1e-16) {
		tmp = x / (z - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -2.45e+48:
		tmp = y / (y - z)
	elif y <= -1.85e-7:
		tmp = t_0
	elif y <= -3.7e-60:
		tmp = (x / z) - (y / z)
	elif y <= 2.1e-16:
		tmp = x / (z - y)
	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 <= -2.45e+48)
		tmp = Float64(y / Float64(y - z));
	elseif (y <= -1.85e-7)
		tmp = t_0;
	elseif (y <= -3.7e-60)
		tmp = Float64(Float64(x / z) - Float64(y / z));
	elseif (y <= 2.1e-16)
		tmp = Float64(x / Float64(z - y));
	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 <= -2.45e+48)
		tmp = y / (y - z);
	elseif (y <= -1.85e-7)
		tmp = t_0;
	elseif (y <= -3.7e-60)
		tmp = (x / z) - (y / z);
	elseif (y <= 2.1e-16)
		tmp = x / (z - y);
	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, -2.45e+48], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.85e-7], t$95$0, If[LessEqual[y, -3.7e-60], N[(N[(x / z), $MachinePrecision] - N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e-16], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.45000000000000015e48

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 87.1%

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

      1. neg-mul-1 87.1%

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

      2. distribute-neg-frac2 87.1%

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

      3. sub-neg 87.1%

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

      4. +-commutative 87.1%

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

      5. distribute-neg-in 87.1%

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

      6. remove-double-neg 87.1%

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

      7. sub-neg 87.1%

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

   5. Simplified 87.1%

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

   if -2.45000000000000015e48 < y < -1.85000000000000002e-7 or 2.1000000000000001e-16 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 83.0%

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

      1. associate-*r/ 83.0%

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

      2. neg-mul-1 83.0%

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

      3. sub-neg 83.0%

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

      4. +-commutative 83.0%

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

      5. distribute-neg-in 83.0%

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

      6. remove-double-neg 83.0%

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

      7. sub-neg 83.0%

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

      8. div-sub 83.0%

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

      9. *-inverses 83.0%

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

   5. Simplified 83.0%

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

   if -1.85000000000000002e-7 < y < -3.70000000000000025e-60

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 93.1%

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

      1. div-sub 93.1%

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

   5. Applied egg-rr 93.1%

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

   if -3.70000000000000025e-60 < y < 2.1000000000000001e-16

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 88.1%

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

Alternative 3: 77.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -7.4 \cdot 10^{+50}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq -1.16 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -7.4e+50)
     (/ y (- y z))
     (if (<= y -1.16e-8)
       t_0
       (if (<= y -8.5e-60)
         (/ (- x y) z)
         (if (<= y 3.4e-13) (/ x (- z y)) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -7.4e+50) {
		tmp = y / (y - z);
	} else if (y <= -1.16e-8) {
		tmp = t_0;
	} else if (y <= -8.5e-60) {
		tmp = (x - y) / z;
	} else if (y <= 3.4e-13) {
		tmp = x / (z - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    if (y <= (-7.4d+50)) then
        tmp = y / (y - z)
    else if (y <= (-1.16d-8)) then
        tmp = t_0
    else if (y <= (-8.5d-60)) then
        tmp = (x - y) / z
    else if (y <= 3.4d-13) then
        tmp = x / (z - y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -7.4e+50) {
		tmp = y / (y - z);
	} else if (y <= -1.16e-8) {
		tmp = t_0;
	} else if (y <= -8.5e-60) {
		tmp = (x - y) / z;
	} else if (y <= 3.4e-13) {
		tmp = x / (z - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -7.4e+50:
		tmp = y / (y - z)
	elif y <= -1.16e-8:
		tmp = t_0
	elif y <= -8.5e-60:
		tmp = (x - y) / z
	elif y <= 3.4e-13:
		tmp = x / (z - y)
	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 <= -7.4e+50)
		tmp = Float64(y / Float64(y - z));
	elseif (y <= -1.16e-8)
		tmp = t_0;
	elseif (y <= -8.5e-60)
		tmp = Float64(Float64(x - y) / z);
	elseif (y <= 3.4e-13)
		tmp = Float64(x / Float64(z - y));
	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 <= -7.4e+50)
		tmp = y / (y - z);
	elseif (y <= -1.16e-8)
		tmp = t_0;
	elseif (y <= -8.5e-60)
		tmp = (x - y) / z;
	elseif (y <= 3.4e-13)
		tmp = x / (z - y);
	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, -7.4e+50], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.16e-8], t$95$0, If[LessEqual[y, -8.5e-60], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 3.4e-13], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.4000000000000001e50

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 87.1%

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

      1. neg-mul-1 87.1%

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

      2. distribute-neg-frac2 87.1%

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

      3. sub-neg 87.1%

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

      4. +-commutative 87.1%

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

      5. distribute-neg-in 87.1%

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

      6. remove-double-neg 87.1%

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

      7. sub-neg 87.1%

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

   5. Simplified 87.1%

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

   if -7.4000000000000001e50 < y < -1.15999999999999996e-8 or 3.40000000000000015e-13 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 83.0%

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

      1. associate-*r/ 83.0%

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

      2. neg-mul-1 83.0%

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

      3. sub-neg 83.0%

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

      4. +-commutative 83.0%

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

      5. distribute-neg-in 83.0%

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

      6. remove-double-neg 83.0%

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

      7. sub-neg 83.0%

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

      8. div-sub 83.0%

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

      9. *-inverses 83.0%

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

   5. Simplified 83.0%

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

   if -1.15999999999999996e-8 < y < -8.50000000000000044e-60

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 93.1%

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

   if -8.50000000000000044e-60 < y < 3.40000000000000015e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 88.1%

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

Alternative 4: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-8} \lor \neg \left(y \leq 3.3 \cdot 10^{-13}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.1e-8) (not (<= y 3.3e-13))) (- 1.0 (/ x y)) (/ x (- z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.1e-8) || !(y <= 3.3e-13)) {
		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 <= (-4.1d-8)) .or. (.not. (y <= 3.3d-13))) 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 <= -4.1e-8) || !(y <= 3.3e-13)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.1e-8) or not (y <= 3.3e-13):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (z - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.1e-8) || !(y <= 3.3e-13))
		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 <= -4.1e-8) || ~((y <= 3.3e-13)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.1e-8], N[Not[LessEqual[y, 3.3e-13]], $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 < -4.10000000000000032e-8 or 3.3000000000000001e-13 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 81.5%

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

      1. associate-*r/ 81.5%

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

      2. neg-mul-1 81.5%

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

      3. sub-neg 81.5%

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

      4. +-commutative 81.5%

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

      5. distribute-neg-in 81.5%

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

      6. remove-double-neg 81.5%

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

      7. sub-neg 81.5%

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

      8. div-sub 81.5%

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

      9. *-inverses 81.5%

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

   5. Simplified 81.5%

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

   if -4.10000000000000032e-8 < y < 3.3000000000000001e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 83.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-8} \lor \neg \left(y \leq 3.3 \cdot 10^{-13}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.1 \cdot 10^{-9} \lor \neg \left(y \leq 3.3 \cdot 10^{-18}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7.1e-9) (not (<= y 3.3e-18))) (- 1.0 (/ x y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.1e-9) || !(y <= 3.3e-18)) {
		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 <= (-7.1d-9)) .or. (.not. (y <= 3.3d-18))) 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 <= -7.1e-9) || !(y <= 3.3e-18)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7.1e-9) or not (y <= 3.3e-18):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7.1e-9) || !(y <= 3.3e-18))
		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 <= -7.1e-9) || ~((y <= 3.3e-18)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7.1e-9], N[Not[LessEqual[y, 3.3e-18]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.09999999999999988e-9 or 3.3000000000000002e-18 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 81.5%

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

      1. associate-*r/ 81.5%

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

      2. neg-mul-1 81.5%

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

      3. sub-neg 81.5%

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

      4. +-commutative 81.5%

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

      5. distribute-neg-in 81.5%

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

      6. remove-double-neg 81.5%

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

      7. sub-neg 81.5%

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

      8. div-sub 81.5%

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

      9. *-inverses 81.5%

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

   5. Simplified 81.5%

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

   if -7.09999999999999988e-9 < y < 3.3000000000000002e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 72.6%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.1 \cdot 10^{-9} \lor \neg \left(y \leq 3.3 \cdot 10^{-18}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.8e+48)
   (/ y (- y z))
   (if (<= y 8.5e-11) (/ x (- z y)) (- 1.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.8e+48) {
		tmp = y / (y - z);
	} else if (y <= 8.5e-11) {
		tmp = x / (z - y);
	} 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) :: tmp
    if (y <= (-2.8d+48)) then
        tmp = y / (y - z)
    else if (y <= 8.5d-11) then
        tmp = x / (z - y)
    else
        tmp = 1.0d0 - (x / 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) {
		tmp = y / (y - z);
	} else if (y <= 8.5e-11) {
		tmp = x / (z - y);
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.8e+48:
		tmp = y / (y - z)
	elif y <= 8.5e-11:
		tmp = x / (z - y)
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.8e+48)
		tmp = Float64(y / Float64(y - z));
	elseif (y <= 8.5e-11)
		tmp = Float64(x / Float64(z - y));
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.8e+48)
		tmp = y / (y - z);
	elseif (y <= 8.5e-11)
		tmp = x / (z - y);
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.8e+48], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e-11], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.80000000000000012e48

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 87.1%

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

      1. neg-mul-1 87.1%

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

      2. distribute-neg-frac2 87.1%

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

      3. sub-neg 87.1%

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

      4. +-commutative 87.1%

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

      5. distribute-neg-in 87.1%

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

      6. remove-double-neg 87.1%

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

      7. sub-neg 87.1%

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

   5. Simplified 87.1%

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

   if -2.80000000000000012e48 < y < 8.50000000000000037e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 82.8%

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

   if 8.50000000000000037e-11 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 82.0%

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

      1. associate-*r/ 82.0%

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

      2. neg-mul-1 82.0%

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

      3. sub-neg 82.0%

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

      4. +-commutative 82.0%

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

      5. distribute-neg-in 82.0%

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

      6. remove-double-neg 82.0%

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

      7. sub-neg 82.0%

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

      8. div-sub 82.0%

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

      9. *-inverses 82.0%

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

   5. Simplified 82.0%

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

Alternative 7: 62.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.72 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.72e-6) 1.0 (if (<= y 3e-15) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.72e-6) {
		tmp = 1.0;
	} else if (y <= 3e-15) {
		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 <= (-1.72d-6)) then
        tmp = 1.0d0
    else if (y <= 3d-15) 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 <= -1.72e-6) {
		tmp = 1.0;
	} else if (y <= 3e-15) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.72e-6:
		tmp = 1.0
	elif y <= 3e-15:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.72e-6)
		tmp = 1.0;
	elseif (y <= 3e-15)
		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 <= -1.72e-6)
		tmp = 1.0;
	elseif (y <= 3e-15)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.72e-6], 1.0, If[LessEqual[y, 3e-15], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.72e-6 or 3e-15 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 64.2%

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

   if -1.72e-6 < y < 3e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 72.6%

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

Alternative 8: 100.0% accurate, 1.0× speedup

Math

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

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 9: 35.1% 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 35.6%

   \[\leadsto \color{blue}{1} \]
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 2024180 
(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)))