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

Percentage Accurate: 100.0% → 100.0%

Time: 8.6s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ (- x y) (- z y)))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x - y) / (z - y)
end function

Java

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

Python

def code(x, y, z):
	return (x - y) / (z - y)

Julia

function code(x, y, z)
	return Float64(Float64(x - y) / Float64(z - y))
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ (- x y) (- z y)))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x - y) / (z - y)
end function

Java

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

Python

def code(x, y, z):
	return (x - y) / (z - y)

Julia

function code(x, y, z)
	return Float64(Float64(x - y) / Float64(z - y))
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (/ (- x y) (- z y)))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x - y) / (z - y)
end function

Java

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

Python

def code(x, y, z):
	return (x - y) / (z - y)

Julia

function code(x, y, z)
	return Float64(Float64(x - y) / Float64(z - y))
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 71.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-79} \lor \neg \left(y \leq 10^{-78}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.3e-79) (not (<= y 1e-78))) (- 1.0 (/ x y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.3e-79) || !(y <= 1e-78)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-4.3d-79)) .or. (.not. (y <= 1d-78))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.3e-79) || !(y <= 1e-78)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.3e-79) or not (y <= 1e-78):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.3e-79) || !(y <= 1e-78))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.3e-79) || ~((y <= 1e-78)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.3e-79], N[Not[LessEqual[y, 1e-78]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.29999999999999982e-79 or 9.99999999999999999e-79 < 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 z around 0 68.5%

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

      1. div-sub 68.5%

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

      2. *-inverses 68.5%

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

   7. Simplified 68.5%

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

   if -4.29999999999999982e-79 < y < 9.99999999999999999e-79

   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* 99.7%

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

      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 y around 0 72.9%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-79} \lor \neg \left(y \leq 10^{-78}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-52} \lor \neg \left(y \leq 3.1 \cdot 10^{+52}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.5e-52) (not (<= y 3.1e+52))) (- 1.0 (/ x y)) (/ x (- z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.5e-52) || !(y <= 3.1e+52)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-6.5d-52)) .or. (.not. (y <= 3.1d+52))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / (z - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.5e-52) || !(y <= 3.1e+52)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6.5e-52) or not (y <= 3.1e+52):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (z - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.5e-52) || !(y <= 3.1e+52))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -6.5e-52) || ~((y <= 3.1e+52)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.5e-52], N[Not[LessEqual[y, 3.1e+52]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5e-52 or 3.1e52 < 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 z around 0 76.9%

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

      1. div-sub 76.9%

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

      2. *-inverses 76.9%

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

   7. Simplified 76.9%

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

   if -6.5e-52 < y < 3.1e52

   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* 99.7%

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

      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 x around inf 77.5%

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

      1. neg-mul-1 77.5%

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

      2. distribute-neg-frac 77.5%

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

   7. Simplified 77.5%

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

      1. frac-2neg 77.5%

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

      2. div-inv 77.1%

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

      3. remove-double-neg 77.1%

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

   9. Applied egg-rr 77.1%

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

   10. Taylor expanded in x around 0 77.5%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-52} \lor \neg \left(y \leq 3.1 \cdot 10^{+52}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-61} \lor \neg \left(y \leq 6.5 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7.6e-61) (not (<= y 6.5e+22))) (/ y (- y z)) (/ x (- z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.6e-61) || !(y <= 6.5e+22)) {
		tmp = y / (y - z);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-7.6d-61)) .or. (.not. (y <= 6.5d+22))) then
        tmp = y / (y - z)
    else
        tmp = x / (z - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.6e-61) || !(y <= 6.5e+22)) {
		tmp = y / (y - z);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7.6e-61) or not (y <= 6.5e+22):
		tmp = y / (y - z)
	else:
		tmp = x / (z - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7.6e-61) || !(y <= 6.5e+22))
		tmp = Float64(y / Float64(y - z));
	else
		tmp = Float64(x / Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -7.6e-61) || ~((y <= 6.5e+22)))
		tmp = y / (y - z);
	else
		tmp = x / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7.6e-61], N[Not[LessEqual[y, 6.5e+22]], $MachinePrecision]], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.59999999999999961e-61 or 6.49999999999999979e22 < 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 81.4%

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

   if -7.59999999999999961e-61 < y < 6.49999999999999979e22

   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* 99.7%

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

      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 x around inf 81.7%

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

      1. neg-mul-1 81.7%

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

      2. distribute-neg-frac 81.7%

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

   7. Simplified 81.7%

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

      1. frac-2neg 81.7%

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

      2. div-inv 81.3%

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

      3. remove-double-neg 81.3%

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

   9. Applied egg-rr 81.3%

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

   10. Taylor expanded in x around 0 81.7%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-61} \lor \neg \left(y \leq 6.5 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+64} \lor \neg \left(z \leq 5 \cdot 10^{-69}\right):\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.2e+64) (not (<= z 5e-69))) (/ (- x y) z) (- 1.0 (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2e+64) || !(z <= 5e-69)) {
		tmp = (x - y) / z;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-5.2d+64)) .or. (.not. (z <= 5d-69))) then
        tmp = (x - y) / z
    else
        tmp = 1.0d0 - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2e+64) || !(z <= 5e-69)) {
		tmp = (x - y) / z;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.2e+64) or not (z <= 5e-69):
		tmp = (x - y) / z
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.2e+64) || !(z <= 5e-69))
		tmp = Float64(Float64(x - y) / z);
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -5.2e+64) || ~((z <= 5e-69)))
		tmp = (x - y) / z;
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.2e+64], N[Not[LessEqual[z, 5e-69]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.19999999999999994e64 or 5.00000000000000033e-69 < 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* 99.8%

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

      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. clear-num 99.4%

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

      2. associate-/r/ 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in z around inf 80.6%

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

      1. associate-*r/ 80.6%

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

      2. sub-neg 80.6%

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

      3. mul-1-neg 80.6%

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

      4. distribute-lft-in 80.6%

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

      5. neg-mul-1 80.6%

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

      6. neg-mul-1 80.6%

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

      7. mul-1-neg 80.6%

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

      8. remove-double-neg 80.6%

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

      9. +-commutative 80.6%

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

      10. unsub-neg 80.6%

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

   9. Simplified 80.6%

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

   if -5.19999999999999994e64 < z < 5.00000000000000033e-69

   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* 99.8%

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

      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* 99.8%

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

      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 83.1%

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

      1. div-sub 83.2%

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

      2. *-inverses 83.2%

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

   7. Simplified 83.2%

      \[\leadsto \color{blue}{1 - \frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+64} \lor \neg \left(z \leq 5 \cdot 10^{-69}\right):\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-45}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.5e-45) 1.0 (if (<= y 2.3e+23) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.5e-45) {
		tmp = 1.0;
	} else if (y <= 2.3e+23) {
		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 <= (-7.5d-45)) then
        tmp = 1.0d0
    else if (y <= 2.3d+23) 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 <= -7.5e-45) {
		tmp = 1.0;
	} else if (y <= 2.3e+23) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.5e-45:
		tmp = 1.0
	elif y <= 2.3e+23:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.5e-45)
		tmp = 1.0;
	elseif (y <= 2.3e+23)
		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 <= -7.5e-45)
		tmp = 1.0;
	elseif (y <= 2.3e+23)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.5e-45], 1.0, If[LessEqual[y, 2.3e+23], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5000000000000006e-45 or 2.3e23 < 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 62.1%

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

   if -7.5000000000000006e-45 < y < 2.3e23

   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* 99.7%

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

      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 y around 0 64.6%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-45}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.2% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{z - y} \]
2. 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* 99.8%

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

   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 35.3%

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

6. Final simplification 35.3%

   \[\leadsto 1 \]
7. Add Preprocessing

Developer target: 100.0% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (/ x (- z y)) (/ y (- z y))))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x / (z - y)) - (y / (z - y))
end function

Java

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

Python

def code(x, y, z):
	return (x / (z - y)) - (y / (z - y))

Julia

function code(x, y, z)
	return Float64(Float64(x / Float64(z - y)) - Float64(y / Float64(z - y)))
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] - N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024031 
(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)))