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

Percentage Accurate: 100.0% → 100.0%

Time: 8.5s

Alternatives: 8

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: 58.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{y}\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{+62}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.12 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-37}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x) y)))
   (if (<= y -9.2e+62)
     1.0
     (if (<= y -1.12e+18)
       t_0
       (if (<= y -5.5e-37)
         1.0
         (if (<= y 2.1e-153)
           (/ x z)
           (if (<= y 1.05e+42) t_0 (if (<= y 1.65e+49) (/ x z) 1.0))))))))

C

double code(double x, double y, double z) {
	double t_0 = -x / y;
	double tmp;
	if (y <= -9.2e+62) {
		tmp = 1.0;
	} else if (y <= -1.12e+18) {
		tmp = t_0;
	} else if (y <= -5.5e-37) {
		tmp = 1.0;
	} else if (y <= 2.1e-153) {
		tmp = x / z;
	} else if (y <= 1.05e+42) {
		tmp = t_0;
	} else if (y <= 1.65e+49) {
		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) :: t_0
    real(8) :: tmp
    t_0 = -x / y
    if (y <= (-9.2d+62)) then
        tmp = 1.0d0
    else if (y <= (-1.12d+18)) then
        tmp = t_0
    else if (y <= (-5.5d-37)) then
        tmp = 1.0d0
    else if (y <= 2.1d-153) then
        tmp = x / z
    else if (y <= 1.05d+42) then
        tmp = t_0
    else if (y <= 1.65d+49) 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 t_0 = -x / y;
	double tmp;
	if (y <= -9.2e+62) {
		tmp = 1.0;
	} else if (y <= -1.12e+18) {
		tmp = t_0;
	} else if (y <= -5.5e-37) {
		tmp = 1.0;
	} else if (y <= 2.1e-153) {
		tmp = x / z;
	} else if (y <= 1.05e+42) {
		tmp = t_0;
	} else if (y <= 1.65e+49) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -x / y
	tmp = 0
	if y <= -9.2e+62:
		tmp = 1.0
	elif y <= -1.12e+18:
		tmp = t_0
	elif y <= -5.5e-37:
		tmp = 1.0
	elif y <= 2.1e-153:
		tmp = x / z
	elif y <= 1.05e+42:
		tmp = t_0
	elif y <= 1.65e+49:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-x) / y)
	tmp = 0.0
	if (y <= -9.2e+62)
		tmp = 1.0;
	elseif (y <= -1.12e+18)
		tmp = t_0;
	elseif (y <= -5.5e-37)
		tmp = 1.0;
	elseif (y <= 2.1e-153)
		tmp = Float64(x / z);
	elseif (y <= 1.05e+42)
		tmp = t_0;
	elseif (y <= 1.65e+49)
		tmp = Float64(x / z);
	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 <= -9.2e+62)
		tmp = 1.0;
	elseif (y <= -1.12e+18)
		tmp = t_0;
	elseif (y <= -5.5e-37)
		tmp = 1.0;
	elseif (y <= 2.1e-153)
		tmp = x / z;
	elseif (y <= 1.05e+42)
		tmp = t_0;
	elseif (y <= 1.65e+49)
		tmp = x / z;
	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, -9.2e+62], 1.0, If[LessEqual[y, -1.12e+18], t$95$0, If[LessEqual[y, -5.5e-37], 1.0, If[LessEqual[y, 2.1e-153], N[(x / z), $MachinePrecision], If[LessEqual[y, 1.05e+42], t$95$0, If[LessEqual[y, 1.65e+49], N[(x / z), $MachinePrecision], 1.0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.19999999999999936e62 or -1.12e18 < y < -5.4999999999999998e-37 or 1.6499999999999999e49 < 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 64.8%

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

   if -9.19999999999999936e62 < y < -1.12e18 or 2.10000000000000004e-153 < y < 1.04999999999999998e42

   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 z around 0 58.9%

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

   6. Taylor expanded in y around 0 43.9%

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

      1. neg-mul-1 43.9%

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

      2. distribute-neg-frac 43.9%

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

   8. Simplified 43.9%

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

   if -5.4999999999999998e-37 < y < 2.10000000000000004e-153 or 1.04999999999999998e42 < y < 1.6499999999999999e49

   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 80.1%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+62}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.12 \cdot 10^{+18}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-37}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+42}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y - z}\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (- y z))))
   (if (<= y -1.55e-65)
     t_0
     (if (<= y 2.1e-153)
       (/ x z)
       (if (<= y 9.8e-80) (/ (- x) y) (if (<= y 2.85e-52) (/ x z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double tmp;
	if (y <= -1.55e-65) {
		tmp = t_0;
	} else if (y <= 2.1e-153) {
		tmp = x / z;
	} else if (y <= 9.8e-80) {
		tmp = -x / y;
	} else if (y <= 2.85e-52) {
		tmp = x / z;
	} 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 = y / (y - z)
    if (y <= (-1.55d-65)) then
        tmp = t_0
    else if (y <= 2.1d-153) then
        tmp = x / z
    else if (y <= 9.8d-80) then
        tmp = -x / y
    else if (y <= 2.85d-52) then
        tmp = x / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double tmp;
	if (y <= -1.55e-65) {
		tmp = t_0;
	} else if (y <= 2.1e-153) {
		tmp = x / z;
	} else if (y <= 9.8e-80) {
		tmp = -x / y;
	} else if (y <= 2.85e-52) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (y - z)
	tmp = 0
	if y <= -1.55e-65:
		tmp = t_0
	elif y <= 2.1e-153:
		tmp = x / z
	elif y <= 9.8e-80:
		tmp = -x / y
	elif y <= 2.85e-52:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(y - z))
	tmp = 0.0
	if (y <= -1.55e-65)
		tmp = t_0;
	elseif (y <= 2.1e-153)
		tmp = Float64(x / z);
	elseif (y <= 9.8e-80)
		tmp = Float64(Float64(-x) / y);
	elseif (y <= 2.85e-52)
		tmp = Float64(x / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (y - z);
	tmp = 0.0;
	if (y <= -1.55e-65)
		tmp = t_0;
	elseif (y <= 2.1e-153)
		tmp = x / z;
	elseif (y <= 9.8e-80)
		tmp = -x / y;
	elseif (y <= 2.85e-52)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.55e-65], t$95$0, If[LessEqual[y, 2.1e-153], N[(x / z), $MachinePrecision], If[LessEqual[y, 9.8e-80], N[((-x) / y), $MachinePrecision], If[LessEqual[y, 2.85e-52], N[(x / z), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.55000000000000008e-65 or 2.8499999999999999e-52 < 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 x around 0 67.7%

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

   if -1.55000000000000008e-65 < y < 2.10000000000000004e-153 or 9.79999999999999981e-80 < y < 2.8499999999999999e-52

   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 79.0%

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

   if 2.10000000000000004e-153 < y < 9.79999999999999981e-80

   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* 99.9%

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

      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 z around 0 72.1%

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

   6. Taylor expanded in y around 0 59.2%

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

      1. neg-mul-1 59.2%

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

      2. distribute-neg-frac 59.2%

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

   8. Simplified 59.2%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y - z}\\ t_1 := \frac{x - y}{z}\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-82}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (- y z))) (t_1 (/ (- x y) z)))
   (if (<= y -3.8e-35)
     t_0
     (if (<= y 2.1e-153)
       t_1
       (if (<= y 3.3e-82) (/ (- x) y) (if (<= y 4.1e+49) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double t_1 = (x - y) / z;
	double tmp;
	if (y <= -3.8e-35) {
		tmp = t_0;
	} else if (y <= 2.1e-153) {
		tmp = t_1;
	} else if (y <= 3.3e-82) {
		tmp = -x / y;
	} else if (y <= 4.1e+49) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = y / (y - z)
    t_1 = (x - y) / z
    if (y <= (-3.8d-35)) then
        tmp = t_0
    else if (y <= 2.1d-153) then
        tmp = t_1
    else if (y <= 3.3d-82) then
        tmp = -x / y
    else if (y <= 4.1d+49) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double t_1 = (x - y) / z;
	double tmp;
	if (y <= -3.8e-35) {
		tmp = t_0;
	} else if (y <= 2.1e-153) {
		tmp = t_1;
	} else if (y <= 3.3e-82) {
		tmp = -x / y;
	} else if (y <= 4.1e+49) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (y - z)
	t_1 = (x - y) / z
	tmp = 0
	if y <= -3.8e-35:
		tmp = t_0
	elif y <= 2.1e-153:
		tmp = t_1
	elif y <= 3.3e-82:
		tmp = -x / y
	elif y <= 4.1e+49:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(y - z))
	t_1 = Float64(Float64(x - y) / z)
	tmp = 0.0
	if (y <= -3.8e-35)
		tmp = t_0;
	elseif (y <= 2.1e-153)
		tmp = t_1;
	elseif (y <= 3.3e-82)
		tmp = Float64(Float64(-x) / y);
	elseif (y <= 4.1e+49)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (y - z);
	t_1 = (x - y) / z;
	tmp = 0.0;
	if (y <= -3.8e-35)
		tmp = t_0;
	elseif (y <= 2.1e-153)
		tmp = t_1;
	elseif (y <= 3.3e-82)
		tmp = -x / y;
	elseif (y <= 4.1e+49)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -3.8e-35], t$95$0, If[LessEqual[y, 2.1e-153], t$95$1, If[LessEqual[y, 3.3e-82], N[((-x) / y), $MachinePrecision], If[LessEqual[y, 4.1e+49], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8000000000000001e-35 or 4.1e49 < 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 x around 0 71.6%

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

   if -3.8000000000000001e-35 < y < 2.10000000000000004e-153 or 3.30000000000000022e-82 < y < 4.1e49

   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 z around inf 79.9%

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

      1. associate-*r/ 79.9%

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

      2. neg-mul-1 79.9%

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

      3. neg-sub0 79.9%

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

      4. associate--r- 79.9%

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

      5. neg-sub0 79.9%

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

   7. Simplified 79.9%

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

   8. Taylor expanded in y around 0 79.9%

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

      1. +-commutative 79.9%

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

      2. mul-1-neg 79.9%

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

      3. sub-neg 79.9%

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

      4. div-sub 79.9%

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

   10. Simplified 79.9%

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

   if 2.10000000000000004e-153 < y < 3.30000000000000022e-82

   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* 99.9%

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

      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 z around 0 72.1%

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

   6. Taylor expanded in y around 0 59.2%

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

      1. neg-mul-1 59.2%

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

      2. distribute-neg-frac 59.2%

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

   8. Simplified 59.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-82}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+49}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{y - x}{y}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{-x}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.3e-35)
   (/ (- y x) y)
   (if (<= y 2.8e+94) (/ (- x) (- y z)) (/ y (- y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.3e-35) {
		tmp = (y - x) / y;
	} else if (y <= 2.8e+94) {
		tmp = -x / (y - z);
	} 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 (y <= (-4.3d-35)) then
        tmp = (y - x) / y
    else if (y <= 2.8d+94) then
        tmp = -x / (y - z)
    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 (y <= -4.3e-35) {
		tmp = (y - x) / y;
	} else if (y <= 2.8e+94) {
		tmp = -x / (y - z);
	} else {
		tmp = y / (y - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.3e-35:
		tmp = (y - x) / y
	elif y <= 2.8e+94:
		tmp = -x / (y - z)
	else:
		tmp = y / (y - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.3e-35)
		tmp = Float64(Float64(y - x) / y);
	elseif (y <= 2.8e+94)
		tmp = Float64(Float64(-x) / Float64(y - z));
	else
		tmp = Float64(y / Float64(y - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.3e-35)
		tmp = (y - x) / y;
	elseif (y <= 2.8e+94)
		tmp = -x / (y - z);
	else
		tmp = y / (y - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.3e-35], N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 2.8e+94], N[((-x) / N[(y - z), $MachinePrecision]), $MachinePrecision], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.3000000000000002e-35

   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 z around 0 78.7%

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

   if -4.3000000000000002e-35 < y < 2.79999999999999998e94

   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 x around inf 83.0%

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

      1. neg-mul-1 83.0%

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

      2. distribute-neg-frac 83.0%

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

   7. Simplified 83.0%

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

   if 2.79999999999999998e94 < 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 x around 0 83.4%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{y - x}{y}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{-x}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-61} \lor \neg \left(z \leq 7.6 \cdot 10^{+74}\right):\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8.5e-61) (not (<= z 7.6e+74))) (/ (- x y) z) (/ (- y x) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.5e-61) || !(z <= 7.6e+74)) {
		tmp = (x - y) / z;
	} else {
		tmp = (y - x) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-8.5d-61)) .or. (.not. (z <= 7.6d+74))) then
        tmp = (x - y) / z
    else
        tmp = (y - x) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.5e-61) || !(z <= 7.6e+74)) {
		tmp = (x - y) / z;
	} else {
		tmp = (y - x) / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8.5e-61) or not (z <= 7.6e+74):
		tmp = (x - y) / z
	else:
		tmp = (y - x) / y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8.5e-61) || !(z <= 7.6e+74))
		tmp = Float64(Float64(x - y) / z);
	else
		tmp = Float64(Float64(y - x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8.5e-61) || ~((z <= 7.6e+74)))
		tmp = (x - y) / z;
	else
		tmp = (y - x) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8.5e-61], N[Not[LessEqual[z, 7.6e+74]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.50000000000000016e-61 or 7.5999999999999997e74 < z

   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 z around inf 79.1%

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

      1. associate-*r/ 79.1%

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

      2. neg-mul-1 79.1%

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

      3. neg-sub0 79.1%

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

      4. associate--r- 79.1%

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

      5. neg-sub0 79.1%

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

   7. Simplified 79.1%

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

   8. Taylor expanded in y around 0 79.1%

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

      1. +-commutative 79.1%

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

      2. mul-1-neg 79.1%

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

      3. sub-neg 79.1%

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

      4. div-sub 79.1%

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

   10. Simplified 79.1%

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

   if -8.50000000000000016e-61 < z < 7.5999999999999997e74

   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 z around 0 81.4%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-61} \lor \neg \left(z \leq 7.6 \cdot 10^{+74}\right):\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{-38}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.9e-38) 1.0 (if (<= y 1.45e+52) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.9e-38) {
		tmp = 1.0;
	} else if (y <= 1.45e+52) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.9d-38)) then
        tmp = 1.0d0
    else if (y <= 1.45d+52) then
        tmp = x / z
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.9e-38) {
		tmp = 1.0;
	} else if (y <= 1.45e+52) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.9e-38:
		tmp = 1.0
	elif y <= 1.45e+52:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.9e-38)
		tmp = 1.0;
	elseif (y <= 1.45e+52)
		tmp = Float64(x / z);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.9e-38)
		tmp = 1.0;
	elseif (y <= 1.45e+52)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.9e-38], 1.0, If[LessEqual[y, 1.45e+52], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3.8999999999999999e-38 or 1.45e52 < 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 59.6%

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

   if -3.8999999999999999e-38 < y < 1.45e52

   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 65.0%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{-38}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

double code(double x, double y, double z) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0;
}

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{z - y} \]
2. 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 inf 34.6%

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

6. Final simplification 34.6%

   \[\leadsto 1 \]
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 2024019 
(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)))