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

Percentage Accurate: 100.0% → 100.0%

Time: 7.7s

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. Final simplification 100.0%

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

Alternative 2: 75.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+70} \lor \neg \left(y \leq 6 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.2e+70) (not (<= y 6e-87))) (/ y (- y z)) (/ (- x) (- y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.2e+70) || !(y <= 6e-87)) {
		tmp = y / (y - z);
	} else {
		tmp = -x / (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 <= (-1.2d+70)) .or. (.not. (y <= 6d-87))) then
        tmp = y / (y - z)
    else
        tmp = -x / (y - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.2e+70) || !(y <= 6e-87)) {
		tmp = y / (y - z);
	} else {
		tmp = -x / (y - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.2e+70) or not (y <= 6e-87):
		tmp = y / (y - z)
	else:
		tmp = -x / (y - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.2e+70) || !(y <= 6e-87))
		tmp = Float64(y / Float64(y - z));
	else
		tmp = Float64(Float64(-x) / Float64(y - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.2e+70) || ~((y <= 6e-87)))
		tmp = y / (y - z);
	else
		tmp = -x / (y - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.2e+70], N[Not[LessEqual[y, 6e-87]], $MachinePrecision]], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], N[((-x) / N[(y - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.19999999999999993e70 or 6.00000000000000033e-87 < y

   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. +-commutative 99.9%

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

      3. neg-sub0 99.9%

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

      4. associate-+l- 99.9%

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

      5. sub0-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. associate-/r* 99.9%

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

      8. div-sub 99.9%

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

      9. remove-double-neg 99.9%

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

      10. neg-mul-1 99.9%

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

      11. associate-/l* 99.9%

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

      12. associate-/r/ 99.9%

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

      13. metadata-eval 99.9%

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

      14. *-lft-identity 99.9%

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

      15. remove-double-neg 99.9%

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

      16. neg-mul-1 99.9%

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

      17. associate-/l* 99.7%

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

      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. Taylor expanded in x around 0 81.6%

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

   if -1.19999999999999993e70 < y < 6.00000000000000033e-87

   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. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-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. Taylor expanded in x around inf 85.6%

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

      1. neg-mul-1 85.6%

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

      2. distribute-neg-frac 85.6%

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

   6. Simplified 85.6%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+70} \lor \neg \left(y \leq 6 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{y - z}\\ \end{array} \]

Alternative 3: 71.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-12} \lor \neg \left(y \leq 1.02 \cdot 10^{-85}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.6e-12) (not (<= y 1.02e-85))) (- 1.0 (/ x y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.6e-12) || !(y <= 1.02e-85)) {
		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 <= (-2.6d-12)) .or. (.not. (y <= 1.02d-85))) 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 <= -2.6e-12) || !(y <= 1.02e-85)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.6e-12) or not (y <= 1.02e-85):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.6e-12) || !(y <= 1.02e-85))
		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 <= -2.6e-12) || ~((y <= 1.02e-85)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.6e-12], N[Not[LessEqual[y, 1.02e-85]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.59999999999999983e-12 or 1.02000000000000001e-85 < y

   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. +-commutative 99.9%

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

      3. neg-sub0 99.9%

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

      4. associate-+l- 99.9%

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

      5. sub0-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. associate-/r* 99.9%

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

      8. div-sub 99.9%

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

      9. remove-double-neg 99.9%

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

      10. neg-mul-1 99.9%

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

      11. associate-/l* 99.9%

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

      12. associate-/r/ 99.9%

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

      13. metadata-eval 99.9%

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

      14. *-lft-identity 99.9%

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

      15. remove-double-neg 99.9%

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

      16. neg-mul-1 99.9%

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

      17. associate-/l* 99.7%

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

      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. Taylor expanded in z around 0 72.1%

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

      1. div-sub 72.1%

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

      2. *-inverses 72.1%

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

   6. Simplified 72.1%

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

   if -2.59999999999999983e-12 < y < 1.02000000000000001e-85

   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. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-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. Taylor expanded in y around 0 79.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-12} \lor \neg \left(y \leq 1.02 \cdot 10^{-85}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 4: 68.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+69} \lor \neg \left(y \leq 3.5 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -7.5e+69) (not (<= y 3.5e-87))) (/ y (- y z)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.5e+69) || !(y <= 3.5e-87)) {
		tmp = y / (y - z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-7.5d+69)) .or. (.not. (y <= 3.5d-87))) then
        tmp = y / (y - z)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -7.5e+69) || !(y <= 3.5e-87)) {
		tmp = y / (y - z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -7.5e+69) or not (y <= 3.5e-87):
		tmp = y / (y - z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -7.5e+69) || !(y <= 3.5e-87))
		tmp = Float64(y / Float64(y - z));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -7.5e+69) || ~((y <= 3.5e-87)))
		tmp = y / (y - z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -7.5e+69], N[Not[LessEqual[y, 3.5e-87]], $MachinePrecision]], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.49999999999999939e69 or 3.50000000000000012e-87 < y

   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. +-commutative 99.9%

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

      3. neg-sub0 99.9%

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

      4. associate-+l- 99.9%

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

      5. sub0-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. associate-/r* 99.9%

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

      8. div-sub 99.9%

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

      9. remove-double-neg 99.9%

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

      10. neg-mul-1 99.9%

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

      11. associate-/l* 99.9%

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

      12. associate-/r/ 99.9%

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

      13. metadata-eval 99.9%

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

      14. *-lft-identity 99.9%

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

      15. remove-double-neg 99.9%

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

      16. neg-mul-1 99.9%

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

      17. associate-/l* 99.7%

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

      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. Taylor expanded in x around 0 81.6%

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

   if -7.49999999999999939e69 < y < 3.50000000000000012e-87

   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. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-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. Taylor expanded in y around 0 73.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+69} \lor \neg \left(y \leq 3.5 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 5: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+70} \lor \neg \left(y \leq 6 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.6e+70) (not (<= y 6e-87))) (/ y (- y z)) (/ (- x y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.6e+70) || !(y <= 6e-87)) {
		tmp = y / (y - z);
	} else {
		tmp = (x - 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 <= (-2.6d+70)) .or. (.not. (y <= 6d-87))) then
        tmp = y / (y - z)
    else
        tmp = (x - y) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.6e+70) || !(y <= 6e-87)) {
		tmp = y / (y - z);
	} else {
		tmp = (x - y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.6e+70) or not (y <= 6e-87):
		tmp = y / (y - z)
	else:
		tmp = (x - y) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.6e+70) || !(y <= 6e-87))
		tmp = Float64(y / Float64(y - z));
	else
		tmp = Float64(Float64(x - y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.6e+70) || ~((y <= 6e-87)))
		tmp = y / (y - z);
	else
		tmp = (x - y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.6e+70], N[Not[LessEqual[y, 6e-87]], $MachinePrecision]], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6e70 or 6.00000000000000033e-87 < y

   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. +-commutative 99.9%

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

      3. neg-sub0 99.9%

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

      4. associate-+l- 99.9%

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

      5. sub0-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. associate-/r* 99.9%

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

      8. div-sub 99.9%

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

      9. remove-double-neg 99.9%

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

      10. neg-mul-1 99.9%

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

      11. associate-/l* 99.9%

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

      12. associate-/r/ 99.9%

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

      13. metadata-eval 99.9%

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

      14. *-lft-identity 99.9%

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

      15. remove-double-neg 99.9%

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

      16. neg-mul-1 99.9%

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

      17. associate-/l* 99.7%

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

      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. Taylor expanded in x around 0 81.6%

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

   if -2.6e70 < y < 6.00000000000000033e-87

   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. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in z around inf 81.6%

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

      1. associate-*r/ 81.6%

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

      2. neg-mul-1 81.6%

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

      3. sub-neg 81.6%

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

      4. mul-1-neg 81.6%

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

      5. +-commutative 81.6%

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

      6. distribute-neg-in 81.6%

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

      7. mul-1-neg 81.6%

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

      8. remove-double-neg 81.6%

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

      9. sub-neg 81.6%

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

   8. Simplified 81.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+70} \lor \neg \left(y \leq 6 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z}\\ \end{array} \]

Alternative 6: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+70}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.6e+70) 1.0 (if (<= y 3.15e-84) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.6e+70) {
		tmp = 1.0;
	} else if (y <= 3.15e-84) {
		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 <= (-2.6d+70)) then
        tmp = 1.0d0
    else if (y <= 3.15d-84) 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 <= -2.6e+70) {
		tmp = 1.0;
	} else if (y <= 3.15e-84) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.6e+70:
		tmp = 1.0
	elif y <= 3.15e-84:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.6e+70)
		tmp = 1.0;
	elseif (y <= 3.15e-84)
		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 <= -2.6e+70)
		tmp = 1.0;
	elseif (y <= 3.15e-84)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.6e+70], 1.0, If[LessEqual[y, 3.15e-84], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6e70 or 3.1500000000000002e-84 < y

   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. +-commutative 99.9%

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

      3. neg-sub0 99.9%

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

      4. associate-+l- 99.9%

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

      5. sub0-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. associate-/r* 99.9%

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

      8. div-sub 99.9%

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

      9. remove-double-neg 99.9%

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

      10. neg-mul-1 99.9%

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

      11. associate-/l* 99.9%

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

      12. associate-/r/ 99.9%

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

      13. metadata-eval 99.9%

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

      14. *-lft-identity 99.9%

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

      15. remove-double-neg 99.9%

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

      16. neg-mul-1 99.9%

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

      17. associate-/l* 99.7%

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

      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. Taylor expanded in y around inf 65.5%

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

   if -2.6e70 < y < 3.1500000000000002e-84

   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. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-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. Taylor expanded in y around 0 73.0%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+70}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+70}:\\ \;\;\;\;1 + \frac{z}{y}\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.7e+70) (+ 1.0 (/ z y)) (if (<= y 3.15e-84) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.7e+70) {
		tmp = 1.0 + (z / y);
	} else if (y <= 3.15e-84) {
		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 <= (-2.7d+70)) then
        tmp = 1.0d0 + (z / y)
    else if (y <= 3.15d-84) 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 <= -2.7e+70) {
		tmp = 1.0 + (z / y);
	} else if (y <= 3.15e-84) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.7e+70:
		tmp = 1.0 + (z / y)
	elif y <= 3.15e-84:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.7e+70)
		tmp = Float64(1.0 + Float64(z / y));
	elseif (y <= 3.15e-84)
		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 <= -2.7e+70)
		tmp = 1.0 + (z / y);
	elseif (y <= 3.15e-84)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.7e+70], N[(1.0 + N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.15e-84], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7e70

   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. +-commutative 99.9%

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

      3. neg-sub0 99.9%

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

      4. associate-+l- 99.9%

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

      5. sub0-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. associate-/r* 99.9%

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

      8. div-sub 99.9%

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

      9. remove-double-neg 99.9%

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

      10. neg-mul-1 99.9%

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

      11. associate-/l* 99.9%

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

      12. associate-/r/ 99.9%

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

      13. metadata-eval 99.9%

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

      14. *-lft-identity 99.9%

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

      15. remove-double-neg 99.9%

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

      16. neg-mul-1 99.9%

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

      17. associate-/l* 99.5%

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

      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. Taylor expanded in x around 0 89.9%

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

   5. Taylor expanded in y around inf 74.7%

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

   if -2.7e70 < y < 3.1500000000000002e-84

   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. +-commutative 100.0%

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

      3. neg-sub0 100.0%

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

      4. associate-+l- 100.0%

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

      5. sub0-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. Taylor expanded in y around 0 73.0%

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

   if 3.1500000000000002e-84 < y

   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. +-commutative 99.9%

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

      3. neg-sub0 99.9%

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

      4. associate-+l- 99.9%

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

      5. sub0-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. associate-/r* 99.9%

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

      8. div-sub 99.9%

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

      9. remove-double-neg 99.9%

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

      10. neg-mul-1 99.9%

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

      11. associate-/l* 99.9%

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

      12. associate-/r/ 99.9%

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

      13. metadata-eval 99.9%

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

      14. *-lft-identity 99.9%

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

      15. remove-double-neg 99.9%

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

      16. neg-mul-1 99.9%

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

      17. associate-/l* 99.7%

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

      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. Taylor expanded in y around inf 61.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+70}:\\ \;\;\;\;1 + \frac{z}{y}\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 34.5% 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. +-commutative 100.0%

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

   3. neg-sub0 100.0%

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

   4. associate-+l- 100.0%

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

   5. sub0-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. Taylor expanded in y around inf 39.1%

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

5. Final simplification 39.1%

   \[\leadsto 1 \]

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 2023322 
(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)))