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

Percentage Accurate: 100.0% → 100.0%

Time: 6.1s

Alternatives: 6

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: 55.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{y}\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+202}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+124}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+129}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x) y)))
   (if (<= y -9.5e+202)
     1.0
     (if (<= y -1.3e-70)
       t_0
       (if (<= y 2e-10)
         (/ x z)
         (if (<= y 1.3e+64)
           t_0
           (if (<= y 4e+124) (/ x z) (if (<= y 3.2e+129) t_0 1.0))))))))

C

double code(double x, double y, double z) {
	double t_0 = -x / y;
	double tmp;
	if (y <= -9.5e+202) {
		tmp = 1.0;
	} else if (y <= -1.3e-70) {
		tmp = t_0;
	} else if (y <= 2e-10) {
		tmp = x / z;
	} else if (y <= 1.3e+64) {
		tmp = t_0;
	} else if (y <= 4e+124) {
		tmp = x / z;
	} else if (y <= 3.2e+129) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = -x / y
    if (y <= (-9.5d+202)) then
        tmp = 1.0d0
    else if (y <= (-1.3d-70)) then
        tmp = t_0
    else if (y <= 2d-10) then
        tmp = x / z
    else if (y <= 1.3d+64) then
        tmp = t_0
    else if (y <= 4d+124) then
        tmp = x / z
    else if (y <= 3.2d+129) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -x / y;
	double tmp;
	if (y <= -9.5e+202) {
		tmp = 1.0;
	} else if (y <= -1.3e-70) {
		tmp = t_0;
	} else if (y <= 2e-10) {
		tmp = x / z;
	} else if (y <= 1.3e+64) {
		tmp = t_0;
	} else if (y <= 4e+124) {
		tmp = x / z;
	} else if (y <= 3.2e+129) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -x / y
	tmp = 0
	if y <= -9.5e+202:
		tmp = 1.0
	elif y <= -1.3e-70:
		tmp = t_0
	elif y <= 2e-10:
		tmp = x / z
	elif y <= 1.3e+64:
		tmp = t_0
	elif y <= 4e+124:
		tmp = x / z
	elif y <= 3.2e+129:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-x) / y)
	tmp = 0.0
	if (y <= -9.5e+202)
		tmp = 1.0;
	elseif (y <= -1.3e-70)
		tmp = t_0;
	elseif (y <= 2e-10)
		tmp = Float64(x / z);
	elseif (y <= 1.3e+64)
		tmp = t_0;
	elseif (y <= 4e+124)
		tmp = Float64(x / z);
	elseif (y <= 3.2e+129)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -x / y;
	tmp = 0.0;
	if (y <= -9.5e+202)
		tmp = 1.0;
	elseif (y <= -1.3e-70)
		tmp = t_0;
	elseif (y <= 2e-10)
		tmp = x / z;
	elseif (y <= 1.3e+64)
		tmp = t_0;
	elseif (y <= 4e+124)
		tmp = x / z;
	elseif (y <= 3.2e+129)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-x) / y), $MachinePrecision]}, If[LessEqual[y, -9.5e+202], 1.0, If[LessEqual[y, -1.3e-70], t$95$0, If[LessEqual[y, 2e-10], N[(x / z), $MachinePrecision], If[LessEqual[y, 1.3e+64], t$95$0, If[LessEqual[y, 4e+124], N[(x / z), $MachinePrecision], If[LessEqual[y, 3.2e+129], t$95$0, 1.0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.50000000000000059e202 or 3.2000000000000002e129 < y

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. +-commutative 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. associate-/r* 100.0%

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

      8. div-sub 100.0%

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

      9. remove-double-neg 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate-/l* 100.0%

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

      12. associate-/r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. *-lft-identity 100.0%

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

      15. remove-double-neg 100.0%

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

      16. neg-mul-1 100.0%

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

      17. associate-/l* 99.7%

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

      18. associate-/r/ 100.0%

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

      19. metadata-eval 100.0%

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

      20. *-lft-identity 100.0%

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

      21. unsub-neg 100.0%

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

      22. remove-double-neg 100.0%

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

      23. +-commutative 100.0%

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

      24. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 76.3%

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

   if -9.50000000000000059e202 < y < -1.30000000000000001e-70 or 2.00000000000000007e-10 < y < 1.29999999999999998e64 or 3.99999999999999979e124 < y < 3.2000000000000002e129

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. remove-double-neg 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. +-commutative 99.9%

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

      5. sub-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. associate-/r* 99.9%

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

      8. div-sub 99.9%

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

      9. remove-double-neg 99.9%

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

      10. neg-mul-1 99.9%

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

      11. associate-/l* 99.9%

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

      12. associate-/r/ 99.9%

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

      13. metadata-eval 99.9%

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

      14. *-lft-identity 99.9%

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

      15. remove-double-neg 99.9%

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

      16. neg-mul-1 99.9%

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

      17. associate-/l* 99.9%

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

      18. associate-/r/ 99.9%

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

      19. metadata-eval 99.9%

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

      20. *-lft-identity 99.9%

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

      21. unsub-neg 99.9%

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

      22. remove-double-neg 99.9%

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

      23. +-commutative 99.9%

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

      24. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 68.9%

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

   6. Taylor expanded in y around 0 46.9%

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

      1. associate-*r/ 46.9%

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

      2. neg-mul-1 46.9%

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

   8. Simplified 46.9%

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

   if -1.30000000000000001e-70 < y < 2.00000000000000007e-10 or 1.29999999999999998e64 < y < 3.99999999999999979e124

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. +-commutative 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. associate-/r* 100.0%

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

      8. div-sub 100.0%

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

      9. remove-double-neg 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate-/l* 99.8%

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

      12. associate-/r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. *-lft-identity 100.0%

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

      15. remove-double-neg 100.0%

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

      16. neg-mul-1 100.0%

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

      17. associate-/l* 99.9%

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

      18. associate-/r/ 100.0%

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

      19. metadata-eval 100.0%

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

      20. *-lft-identity 100.0%

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

      21. unsub-neg 100.0%

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

      22. remove-double-neg 100.0%

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

      23. +-commutative 100.0%

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

      24. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 69.4%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+202}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-70}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+64}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+124}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+129}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9500000 \lor \neg \left(x \leq 8 \cdot 10^{-88}\right):\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9500000.0) (not (<= x 8e-88))) (/ x (- z y)) (/ y (- y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9500000.0) || !(x <= 8e-88)) {
		tmp = x / (z - y);
	} else {
		tmp = y / (y - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-9500000.0d0)) .or. (.not. (x <= 8d-88))) then
        tmp = x / (z - y)
    else
        tmp = y / (y - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9500000.0) || !(x <= 8e-88)) {
		tmp = x / (z - y);
	} else {
		tmp = y / (y - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9500000.0) or not (x <= 8e-88):
		tmp = x / (z - y)
	else:
		tmp = y / (y - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9500000.0) || !(x <= 8e-88))
		tmp = Float64(x / Float64(z - y));
	else
		tmp = Float64(y / Float64(y - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -9500000.0) || ~((x <= 8e-88)))
		tmp = x / (z - y);
	else
		tmp = y / (y - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9500000.0], N[Not[LessEqual[x, 8e-88]], $MachinePrecision]], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5e6 or 7.99999999999999947e-88 < x

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. +-commutative 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. associate-/r* 100.0%

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

      8. div-sub 100.0%

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

      9. remove-double-neg 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate-/l* 99.8%

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

      12. associate-/r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. *-lft-identity 100.0%

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

      15. remove-double-neg 100.0%

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

      16. neg-mul-1 100.0%

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

      17. associate-/l* 99.9%

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

      18. associate-/r/ 100.0%

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

      19. metadata-eval 100.0%

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

      20. *-lft-identity 100.0%

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

      21. unsub-neg 100.0%

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

      22. remove-double-neg 100.0%

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

      23. +-commutative 100.0%

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

      24. sub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. div-sub 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. sub-div 100.0%

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

      2. clear-num 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. frac-2neg 99.7%

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

      2. div-inv 99.6%

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

      3. sub-neg 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. remove-double-neg 99.6%

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

      6. +-commutative 99.6%

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

      7. sub-neg 99.6%

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

   10. Applied egg-rr 99.6%

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

   11. Taylor expanded in x around inf 80.2%

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

   if -9.5e6 < x < 7.99999999999999947e-88

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. +-commutative 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. associate-/r* 100.0%

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

      8. div-sub 100.0%

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

      9. remove-double-neg 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate-/l* 99.9%

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

      12. associate-/r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. *-lft-identity 100.0%

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

      15. remove-double-neg 100.0%

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

      16. neg-mul-1 100.0%

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

      17. associate-/l* 99.7%

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

      18. associate-/r/ 100.0%

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

      19. metadata-eval 100.0%

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

      20. *-lft-identity 100.0%

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

      21. unsub-neg 100.0%

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

      22. remove-double-neg 100.0%

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

      23. +-commutative 100.0%

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

      24. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 84.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9500000 \lor \neg \left(x \leq 8 \cdot 10^{-88}\right):\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+203}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.5e+203) 1.0 (if (<= y 2.8e+129) (/ x (- z y)) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5.5e+203) {
		tmp = 1.0;
	} else if (y <= 2.8e+129) {
		tmp = x / (z - y);
	} 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 <= (-5.5d+203)) then
        tmp = 1.0d0
    else if (y <= 2.8d+129) then
        tmp = x / (z - y)
    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 <= -5.5e+203) {
		tmp = 1.0;
	} else if (y <= 2.8e+129) {
		tmp = x / (z - y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5.5e+203:
		tmp = 1.0
	elif y <= 2.8e+129:
		tmp = x / (z - y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5.5e+203)
		tmp = 1.0;
	elseif (y <= 2.8e+129)
		tmp = Float64(x / Float64(z - y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5.5e+203)
		tmp = 1.0;
	elseif (y <= 2.8e+129)
		tmp = x / (z - y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5.5e+203], 1.0, If[LessEqual[y, 2.8e+129], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -5.50000000000000029e203 or 2.79999999999999975e129 < y

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. +-commutative 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. associate-/r* 100.0%

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

      8. div-sub 100.0%

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

      9. remove-double-neg 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate-/l* 100.0%

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

      12. associate-/r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. *-lft-identity 100.0%

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

      15. remove-double-neg 100.0%

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

      16. neg-mul-1 100.0%

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

      17. associate-/l* 99.7%

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

      18. associate-/r/ 100.0%

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

      19. metadata-eval 100.0%

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

      20. *-lft-identity 100.0%

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

      21. unsub-neg 100.0%

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

      22. remove-double-neg 100.0%

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

      23. +-commutative 100.0%

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

      24. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 76.3%

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

   if -5.50000000000000029e203 < y < 2.79999999999999975e129

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. +-commutative 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. associate-/r* 100.0%

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

      8. div-sub 100.0%

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

      9. remove-double-neg 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate-/l* 99.8%

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

      12. associate-/r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. *-lft-identity 100.0%

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

      15. remove-double-neg 100.0%

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

      16. neg-mul-1 100.0%

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

      17. associate-/l* 99.9%

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

      18. associate-/r/ 100.0%

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

      19. metadata-eval 100.0%

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

      20. *-lft-identity 100.0%

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

      21. unsub-neg 100.0%

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

      22. remove-double-neg 100.0%

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

      23. +-commutative 100.0%

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

      24. sub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. div-sub 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. sub-div 100.0%

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

      2. clear-num 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. frac-2neg 99.7%

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

      2. div-inv 99.5%

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

      3. sub-neg 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. remove-double-neg 99.5%

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

      6. +-commutative 99.5%

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

      7. sub-neg 99.5%

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

   10. Applied egg-rr 99.5%

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

   11. Taylor expanded in x around inf 71.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+203}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1450000000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -9.8e+35) 1.0 (if (<= y 1450000000000.0) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -9.8e+35) {
		tmp = 1.0;
	} else if (y <= 1450000000000.0) {
		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 <= (-9.8d+35)) then
        tmp = 1.0d0
    else if (y <= 1450000000000.0d0) 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 <= -9.8e+35) {
		tmp = 1.0;
	} else if (y <= 1450000000000.0) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -9.8e+35:
		tmp = 1.0
	elif y <= 1450000000000.0:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -9.8e+35)
		tmp = 1.0;
	elseif (y <= 1450000000000.0)
		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 <= -9.8e+35)
		tmp = 1.0;
	elseif (y <= 1450000000000.0)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -9.8e+35], 1.0, If[LessEqual[y, 1450000000000.0], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -9.8000000000000005e35 or 1.45e12 < y

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. +-commutative 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. associate-/r* 100.0%

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

      8. div-sub 100.0%

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

      9. remove-double-neg 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate-/l* 99.9%

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

      12. associate-/r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. *-lft-identity 100.0%

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

      15. remove-double-neg 100.0%

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

      16. neg-mul-1 100.0%

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

      17. associate-/l* 99.8%

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

      18. associate-/r/ 100.0%

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

      19. metadata-eval 100.0%

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

      20. *-lft-identity 100.0%

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

      21. unsub-neg 100.0%

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

      22. remove-double-neg 100.0%

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

      23. +-commutative 100.0%

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

      24. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 49.7%

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

   if -9.8000000000000005e35 < y < 1.45e12

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. +-commutative 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. associate-/r* 100.0%

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

      8. div-sub 100.0%

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

      9. remove-double-neg 100.0%

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

      10. neg-mul-1 100.0%

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

      11. associate-/l* 99.9%

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

      12. associate-/r/ 100.0%

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

      13. metadata-eval 100.0%

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

      14. *-lft-identity 100.0%

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

      15. remove-double-neg 100.0%

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

      16. neg-mul-1 100.0%

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

      17. associate-/l* 99.9%

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

      18. associate-/r/ 100.0%

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

      19. metadata-eval 100.0%

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

      20. *-lft-identity 100.0%

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

      21. unsub-neg 100.0%

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

      22. remove-double-neg 100.0%

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

      23. +-commutative 100.0%

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

      24. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 64.8%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1450000000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.8% 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%

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

   2. remove-double-neg 100.0%

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

   3. distribute-neg-in 100.0%

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

   4. +-commutative 100.0%

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

   5. sub-neg 100.0%

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

   6. neg-mul-1 100.0%

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

   7. associate-/r* 100.0%

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

   8. div-sub 100.0%

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

   9. remove-double-neg 100.0%

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

   10. neg-mul-1 100.0%

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

   11. associate-/l* 99.9%

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

   12. associate-/r/ 100.0%

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

   13. metadata-eval 100.0%

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

   14. *-lft-identity 100.0%

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

   15. remove-double-neg 100.0%

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

   16. neg-mul-1 100.0%

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

   17. associate-/l* 99.8%

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

   18. associate-/r/ 100.0%

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

   19. metadata-eval 100.0%

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

   20. *-lft-identity 100.0%

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

   21. unsub-neg 100.0%

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

   22. remove-double-neg 100.0%

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

   23. +-commutative 100.0%

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

   24. sub-neg 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in y around inf 27.9%

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

6. Final simplification 27.9%

   \[\leadsto 1 \]
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 2024018 
(FPCore (x y z)
  :name "Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (- (/ x (- z y)) (/ y (- z y)))

  (/ (- x y) (- z y)))