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

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 7

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. Final simplification 100.0%

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

Alternative 2: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-59} \lor \neg \left(y \leq 2.3 \cdot 10^{-21}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -2.4e+36)
     t_0
     (if (<= y -3.6e-47)
       (/ y (- y z))
       (if (or (<= y -2.8e-59) (not (<= y 2.3e-21))) t_0 (/ x (- z y)))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -2.4e+36) {
		tmp = t_0;
	} else if (y <= -3.6e-47) {
		tmp = y / (y - z);
	} else if ((y <= -2.8e-59) || !(y <= 2.3e-21)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    if (y <= (-2.4d+36)) then
        tmp = t_0
    else if (y <= (-3.6d-47)) then
        tmp = y / (y - z)
    else if ((y <= (-2.8d-59)) .or. (.not. (y <= 2.3d-21))) then
        tmp = t_0
    else
        tmp = x / (z - y)
    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 <= -2.4e+36) {
		tmp = t_0;
	} else if (y <= -3.6e-47) {
		tmp = y / (y - z);
	} else if ((y <= -2.8e-59) || !(y <= 2.3e-21)) {
		tmp = t_0;
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -2.4e+36:
		tmp = t_0
	elif y <= -3.6e-47:
		tmp = y / (y - z)
	elif (y <= -2.8e-59) or not (y <= 2.3e-21):
		tmp = t_0
	else:
		tmp = x / (z - y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -2.4e+36)
		tmp = t_0;
	elseif (y <= -3.6e-47)
		tmp = Float64(y / Float64(y - z));
	elseif ((y <= -2.8e-59) || !(y <= 2.3e-21))
		tmp = t_0;
	else
		tmp = Float64(x / Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -2.4e+36)
		tmp = t_0;
	elseif (y <= -3.6e-47)
		tmp = y / (y - z);
	elseif ((y <= -2.8e-59) || ~((y <= 2.3e-21)))
		tmp = t_0;
	else
		tmp = x / (z - y);
	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, -2.4e+36], t$95$0, If[LessEqual[y, -3.6e-47], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -2.8e-59], N[Not[LessEqual[y, 2.3e-21]], $MachinePrecision]], t$95$0, N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.39999999999999992e36 or -3.59999999999999991e-47 < y < -2.79999999999999981e-59 or 2.29999999999999999e-21 < y

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. neg-sub0 99.9%

         \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]

      4. associate-+l- 99.9%

         \[\leadsto \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]

      5. sub0-neg 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. distribute-frac-neg2 99.9%

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

      8. sub-neg 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. remove-double-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 85.3%

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

      1. div-sub 85.4%

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

      2. *-inverses 85.4%

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

   7. Simplified 85.4%

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

   if -2.39999999999999992e36 < y < -3.59999999999999991e-47

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]

      5. sub0-neg 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. remove-double-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 80.5%

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

   if -2.79999999999999981e-59 < y < 2.29999999999999999e-21

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]

      5. sub0-neg 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. remove-double-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 86.1%

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

      1. neg-mul-1 86.1%

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

      2. distribute-neg-frac2 86.1%

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

      3. neg-sub0 86.1%

         \[\leadsto \frac{x}{\color{blue}{0 - \left(y - z\right)}} \]

      4. sub-neg 86.1%

         \[\leadsto \frac{x}{0 - \color{blue}{\left(y + \left(-z\right)\right)}} \]

      5. mul-1-neg 86.1%

         \[\leadsto \frac{x}{0 - \left(y + \color{blue}{-1 \cdot z}\right)} \]

      6. +-commutative 86.1%

         \[\leadsto \frac{x}{0 - \color{blue}{\left(-1 \cdot z + y\right)}} \]

      7. associate--r+ 86.1%

         \[\leadsto \frac{x}{\color{blue}{\left(0 - -1 \cdot z\right) - y}} \]

      8. neg-sub0 86.1%

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

      9. mul-1-neg 86.1%

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

      10. remove-double-neg 86.1%

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

   7. Simplified 86.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+36}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-59} \lor \neg \left(y \leq 2.3 \cdot 10^{-21}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.56 \cdot 10^{+41}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.56e+41)
   1.0
   (if (<= y -1.9e-38) (/ y (- z)) (if (<= y 1.35e-21) (/ x z) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.56e+41) {
		tmp = 1.0;
	} else if (y <= -1.9e-38) {
		tmp = y / -z;
	} else if (y <= 1.35e-21) {
		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 <= (-1.56d+41)) then
        tmp = 1.0d0
    else if (y <= (-1.9d-38)) then
        tmp = y / -z
    else if (y <= 1.35d-21) 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 <= -1.56e+41) {
		tmp = 1.0;
	} else if (y <= -1.9e-38) {
		tmp = y / -z;
	} else if (y <= 1.35e-21) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.56e+41:
		tmp = 1.0
	elif y <= -1.9e-38:
		tmp = y / -z
	elif y <= 1.35e-21:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.56e+41)
		tmp = 1.0;
	elseif (y <= -1.9e-38)
		tmp = Float64(y / Float64(-z));
	elseif (y <= 1.35e-21)
		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 <= -1.56e+41)
		tmp = 1.0;
	elseif (y <= -1.9e-38)
		tmp = y / -z;
	elseif (y <= 1.35e-21)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.56e+41], 1.0, If[LessEqual[y, -1.9e-38], N[(y / (-z)), $MachinePrecision], If[LessEqual[y, 1.35e-21], N[(x / z), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.56e41 or 1.3500000000000001e-21 < y

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. neg-sub0 99.9%

         \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]

      4. associate-+l- 99.9%

         \[\leadsto \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]

      5. sub0-neg 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. distribute-frac-neg2 99.9%

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

      8. sub-neg 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. remove-double-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 68.3%

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

   if -1.56e41 < y < -1.9e-38

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]

      5. sub0-neg 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. remove-double-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 73.7%

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

   6. Taylor expanded in y around 0 65.4%

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

      1. associate-*r/ 65.4%

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

      2. neg-mul-1 65.4%

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

   8. Simplified 65.4%

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

   if -1.9e-38 < y < 1.3500000000000001e-21

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]

      5. sub0-neg 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. remove-double-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 70.1%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.56 \cdot 10^{+41}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-38}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-71} \lor \neg \left(y \leq 1.95 \cdot 10^{-25}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.6e-71) (not (<= y 1.95e-25))) (- 1.0 (/ x y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.6e-71) || !(y <= 1.95e-25)) {
		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 <= (-5.6d-71)) .or. (.not. (y <= 1.95d-25))) 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 <= -5.6e-71) || !(y <= 1.95e-25)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.6e-71) or not (y <= 1.95e-25):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.6e-71) || !(y <= 1.95e-25))
		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 <= -5.6e-71) || ~((y <= 1.95e-25)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.6e-71], N[Not[LessEqual[y, 1.95e-25]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.60000000000000001e-71 or 1.95e-25 < y

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. neg-sub0 99.9%

         \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]

      4. associate-+l- 99.9%

         \[\leadsto \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]

      5. sub0-neg 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. distribute-frac-neg2 99.9%

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

      8. sub-neg 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. remove-double-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 81.3%

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

      1. div-sub 81.3%

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

      2. *-inverses 81.3%

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

   7. Simplified 81.3%

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

   if -5.60000000000000001e-71 < y < 1.95e-25

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]

      5. sub0-neg 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. remove-double-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 71.8%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-71} \lor \neg \left(y \leq 1.95 \cdot 10^{-25}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-59} \lor \neg \left(y \leq 1.1 \cdot 10^{-21}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.2e-59) (not (<= y 1.1e-21))) (- 1.0 (/ x y)) (/ x (- z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.2e-59) || !(y <= 1.1e-21)) {
		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 <= (-6.2d-59)) .or. (.not. (y <= 1.1d-21))) 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 <= -6.2e-59) || !(y <= 1.1e-21)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6.2e-59) or not (y <= 1.1e-21):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (z - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.2e-59) || !(y <= 1.1e-21))
		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 <= -6.2e-59) || ~((y <= 1.1e-21)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.2e-59], N[Not[LessEqual[y, 1.1e-21]], $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 < -6.19999999999999998e-59 or 1.1e-21 < y

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. neg-sub0 99.9%

         \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]

      4. associate-+l- 99.9%

         \[\leadsto \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]

      5. sub0-neg 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. distribute-frac-neg2 99.9%

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

      8. sub-neg 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. remove-double-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 81.6%

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

      1. div-sub 81.6%

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

      2. *-inverses 81.6%

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

   7. Simplified 81.6%

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

   if -6.19999999999999998e-59 < y < 1.1e-21

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]

      5. sub0-neg 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. remove-double-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 86.1%

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

      1. neg-mul-1 86.1%

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

      2. distribute-neg-frac2 86.1%

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

      3. neg-sub0 86.1%

         \[\leadsto \frac{x}{\color{blue}{0 - \left(y - z\right)}} \]

      4. sub-neg 86.1%

         \[\leadsto \frac{x}{0 - \color{blue}{\left(y + \left(-z\right)\right)}} \]

      5. mul-1-neg 86.1%

         \[\leadsto \frac{x}{0 - \left(y + \color{blue}{-1 \cdot z}\right)} \]

      6. +-commutative 86.1%

         \[\leadsto \frac{x}{0 - \color{blue}{\left(-1 \cdot z + y\right)}} \]

      7. associate--r+ 86.1%

         \[\leadsto \frac{x}{\color{blue}{\left(0 - -1 \cdot z\right) - y}} \]

      8. neg-sub0 86.1%

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

      9. mul-1-neg 86.1%

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

      10. remove-double-neg 86.1%

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

   7. Simplified 86.1%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-59} \lor \neg \left(y \leq 1.1 \cdot 10^{-21}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-28}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.45e-28) 1.0 (if (<= y 1.6e-21) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.45e-28) {
		tmp = 1.0;
	} else if (y <= 1.6e-21) {
		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 <= (-1.45d-28)) then
        tmp = 1.0d0
    else if (y <= 1.6d-21) 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 <= -1.45e-28) {
		tmp = 1.0;
	} else if (y <= 1.6e-21) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.45e-28:
		tmp = 1.0
	elif y <= 1.6e-21:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.45e-28)
		tmp = 1.0;
	elseif (y <= 1.6e-21)
		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 <= -1.45e-28)
		tmp = 1.0;
	elseif (y <= 1.6e-21)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.45e-28], 1.0, If[LessEqual[y, 1.6e-21], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45000000000000006e-28 or 1.6000000000000001e-21 < y

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. neg-sub0 99.9%

         \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]

      4. associate-+l- 99.9%

         \[\leadsto \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]

      5. sub0-neg 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. distribute-frac-neg2 99.9%

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

      8. sub-neg 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. remove-double-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 64.7%

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

   if -1.45000000000000006e-28 < y < 1.6000000000000001e-21

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]

      5. sub0-neg 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. distribute-frac-neg2 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. remove-double-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 69.2%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-28}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.7% 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. Step-by-step derivation

   1. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. neg-sub0 100.0%

      \[\leadsto \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]

   4. associate-+l- 100.0%

      \[\leadsto \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]

   5. sub0-neg 100.0%

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

   6. distribute-frac-neg 100.0%

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

   7. distribute-frac-neg2 100.0%

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

   8. sub-neg 100.0%

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

   9. distribute-neg-in 100.0%

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

   10. remove-double-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. sub-neg 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in y around inf 38.3%

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

6. Final simplification 38.3%

   \[\leadsto 1 \]
7. 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 2024048 
(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)))