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

Percentage Accurate: 100.0% → 100.0%

Time: 12.1s

Alternatives: 11

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, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-lft-identity 100.0%

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

   2. metadata-eval 100.0%

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

   3. associate-/r/ 99.7%

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

   4. associate-/l* 99.5%

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

   5. neg-mul-1 99.5%

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

   6. sub-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. distribute-neg-out 99.5%

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

   9. remove-double-neg 99.5%

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

   10. sub-neg 99.5%

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

   11. associate-/l* 100.0%

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

   12. neg-mul-1 100.0%

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

   13. sub-neg 100.0%

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

   14. distribute-neg-out 100.0%

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

   15. remove-double-neg 100.0%

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

   16. +-commutative 100.0%

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

   17. 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. div-sub 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 60.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{z}{y}\\ t_1 := \frac{-x}{y}\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ z y))) (t_1 (/ (- x) y)))
   (if (<= y -9.2e+51)
     t_0
     (if (<= y -3.2e-49)
       t_1
       (if (<= y 8e-107)
         (/ x z)
         (if (<= y 2.15e-17) t_1 (if (<= y 2.3e+27) (/ x z) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z / y);
	double t_1 = -x / y;
	double tmp;
	if (y <= -9.2e+51) {
		tmp = t_0;
	} else if (y <= -3.2e-49) {
		tmp = t_1;
	} else if (y <= 8e-107) {
		tmp = x / z;
	} else if (y <= 2.15e-17) {
		tmp = t_1;
	} else if (y <= 2.3e+27) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (z / y)
    t_1 = -x / y
    if (y <= (-9.2d+51)) then
        tmp = t_0
    else if (y <= (-3.2d-49)) then
        tmp = t_1
    else if (y <= 8d-107) then
        tmp = x / z
    else if (y <= 2.15d-17) then
        tmp = t_1
    else if (y <= 2.3d+27) then
        tmp = x / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (z / y);
	double t_1 = -x / y;
	double tmp;
	if (y <= -9.2e+51) {
		tmp = t_0;
	} else if (y <= -3.2e-49) {
		tmp = t_1;
	} else if (y <= 8e-107) {
		tmp = x / z;
	} else if (y <= 2.15e-17) {
		tmp = t_1;
	} else if (y <= 2.3e+27) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z / y)
	t_1 = -x / y
	tmp = 0
	if y <= -9.2e+51:
		tmp = t_0
	elif y <= -3.2e-49:
		tmp = t_1
	elif y <= 8e-107:
		tmp = x / z
	elif y <= 2.15e-17:
		tmp = t_1
	elif y <= 2.3e+27:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z / y))
	t_1 = Float64(Float64(-x) / y)
	tmp = 0.0
	if (y <= -9.2e+51)
		tmp = t_0;
	elseif (y <= -3.2e-49)
		tmp = t_1;
	elseif (y <= 8e-107)
		tmp = Float64(x / z);
	elseif (y <= 2.15e-17)
		tmp = t_1;
	elseif (y <= 2.3e+27)
		tmp = Float64(x / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z / y);
	t_1 = -x / y;
	tmp = 0.0;
	if (y <= -9.2e+51)
		tmp = t_0;
	elseif (y <= -3.2e-49)
		tmp = t_1;
	elseif (y <= 8e-107)
		tmp = x / z;
	elseif (y <= 2.15e-17)
		tmp = t_1;
	elseif (y <= 2.3e+27)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / y), $MachinePrecision]}, If[LessEqual[y, -9.2e+51], t$95$0, If[LessEqual[y, -3.2e-49], t$95$1, If[LessEqual[y, 8e-107], N[(x / z), $MachinePrecision], If[LessEqual[y, 2.15e-17], t$95$1, If[LessEqual[y, 2.3e+27], N[(x / z), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.2000000000000002e51 or 2.3000000000000001e27 < y

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.8%

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

      5. neg-mul-1 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-out 99.8%

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

      9. remove-double-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 0 83.3%

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

   5. Taylor expanded in y around inf 71.3%

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

   if -9.2000000000000002e51 < y < -3.20000000000000002e-49 or 8e-107 < y < 2.15000000000000012e-17

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 98.5%

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

      4. associate-/l* 98.5%

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

      5. neg-mul-1 98.5%

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

      6. sub-neg 98.5%

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

      7. +-commutative 98.5%

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

      8. distribute-neg-out 98.5%

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

      9. remove-double-neg 98.5%

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

      10. sub-neg 98.5%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 50.3%

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

      1. neg-mul-1 50.3%

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

      2. distribute-neg-frac 50.3%

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

   6. Simplified 50.3%

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

   7. Taylor expanded in y around inf 41.4%

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

      1. associate-*r/ 41.4%

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

      2. mul-1-neg 41.4%

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

   9. Simplified 41.4%

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

   if -3.20000000000000002e-49 < y < 8e-107 or 2.15000000000000012e-17 < y < 2.3000000000000001e27

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.7%

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

      5. neg-mul-1 99.7%

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

      6. sub-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-neg-out 99.7%

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

      9. remove-double-neg 99.7%

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

      10. sub-neg 99.7%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 80.5%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+51}:\\ \;\;\;\;1 + \frac{z}{y}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-17}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{z}{y}\\ \end{array} \]

Alternative 3: 60.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-49}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.55e-49)
   1.0
   (if (<= y 9.8e-107)
     (/ x z)
     (if (<= y 3.2e-17) (/ (- x) y) (if (<= y 5.4e+51) (/ x z) 1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.55e-49) {
		tmp = 1.0;
	} else if (y <= 9.8e-107) {
		tmp = x / z;
	} else if (y <= 3.2e-17) {
		tmp = -x / y;
	} else if (y <= 5.4e+51) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.55d-49)) then
        tmp = 1.0d0
    else if (y <= 9.8d-107) then
        tmp = x / z
    else if (y <= 3.2d-17) then
        tmp = -x / y
    else if (y <= 5.4d+51) then
        tmp = x / z
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.55e-49) {
		tmp = 1.0;
	} else if (y <= 9.8e-107) {
		tmp = x / z;
	} else if (y <= 3.2e-17) {
		tmp = -x / y;
	} else if (y <= 5.4e+51) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.55e-49:
		tmp = 1.0
	elif y <= 9.8e-107:
		tmp = x / z
	elif y <= 3.2e-17:
		tmp = -x / y
	elif y <= 5.4e+51:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.55e-49)
		tmp = 1.0;
	elseif (y <= 9.8e-107)
		tmp = Float64(x / z);
	elseif (y <= 3.2e-17)
		tmp = Float64(Float64(-x) / y);
	elseif (y <= 5.4e+51)
		tmp = Float64(x / z);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.55e-49)
		tmp = 1.0;
	elseif (y <= 9.8e-107)
		tmp = x / z;
	elseif (y <= 3.2e-17)
		tmp = -x / y;
	elseif (y <= 5.4e+51)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.55e-49], 1.0, If[LessEqual[y, 9.8e-107], N[(x / z), $MachinePrecision], If[LessEqual[y, 3.2e-17], N[((-x) / y), $MachinePrecision], If[LessEqual[y, 5.4e+51], N[(x / z), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.55e-49 or 5.39999999999999983e51 < y

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.8%

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

      5. neg-mul-1 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-out 99.8%

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

      9. remove-double-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 64.3%

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

   if -1.55e-49 < y < 9.79999999999999959e-107 or 3.2000000000000002e-17 < y < 5.39999999999999983e51

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.8%

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

      4. associate-/l* 99.7%

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

      5. neg-mul-1 99.7%

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

      6. sub-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-neg-out 99.7%

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

      9. remove-double-neg 99.7%

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

      10. sub-neg 99.7%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 78.2%

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

   if 9.79999999999999959e-107 < y < 3.2000000000000002e-17

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 97.2%

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

      4. associate-/l* 97.1%

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

      5. neg-mul-1 97.1%

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

      6. sub-neg 97.1%

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

      7. +-commutative 97.1%

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

      8. distribute-neg-out 97.1%

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

      9. remove-double-neg 97.1%

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

      10. sub-neg 97.1%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 50.6%

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

      1. neg-mul-1 50.6%

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

      2. distribute-neg-frac 50.6%

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

   6. Simplified 50.6%

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

   7. Taylor expanded in y around inf 37.7%

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

      1. associate-*r/ 37.7%

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

      2. mul-1-neg 37.7%

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

   9. Simplified 37.7%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-49}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 76.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6000000000000:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+56}:\\ \;\;\;\;\frac{-x}{y - z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{z - x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6000000000000.0)
   (/ y (- y z))
   (if (<= y 7.4e+56) (/ (- x) (- y z)) (+ 1.0 (/ (- z x) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6000000000000.0) {
		tmp = y / (y - z);
	} else if (y <= 7.4e+56) {
		tmp = -x / (y - z);
	} else {
		tmp = 1.0 + ((z - 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 (y <= (-6000000000000.0d0)) then
        tmp = y / (y - z)
    else if (y <= 7.4d+56) then
        tmp = -x / (y - z)
    else
        tmp = 1.0d0 + ((z - x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -6000000000000.0) {
		tmp = y / (y - z);
	} else if (y <= 7.4e+56) {
		tmp = -x / (y - z);
	} else {
		tmp = 1.0 + ((z - x) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6000000000000.0:
		tmp = y / (y - z)
	elif y <= 7.4e+56:
		tmp = -x / (y - z)
	else:
		tmp = 1.0 + ((z - x) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6000000000000.0)
		tmp = Float64(y / Float64(y - z));
	elseif (y <= 7.4e+56)
		tmp = Float64(Float64(-x) / Float64(y - z));
	else
		tmp = Float64(1.0 + Float64(Float64(z - x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -6000000000000.0)
		tmp = y / (y - z);
	elseif (y <= 7.4e+56)
		tmp = -x / (y - z);
	else
		tmp = 1.0 + ((z - x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6000000000000.0], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.4e+56], N[((-x) / N[(y - z), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6e12

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.8%

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

      5. neg-mul-1 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-out 99.8%

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

      9. remove-double-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. associate-/l* 99.9%

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

      12. neg-mul-1 99.9%

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

      13. sub-neg 99.9%

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

      14. distribute-neg-out 99.9%

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

      15. remove-double-neg 99.9%

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

      16. +-commutative 99.9%

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

      17. 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 86.4%

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

   if -6e12 < y < 7.39999999999999994e56

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.5%

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

      4. associate-/l* 99.4%

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

      5. neg-mul-1 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-neg-out 99.4%

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

      9. remove-double-neg 99.4%

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

      10. sub-neg 99.4%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 80.8%

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

      1. neg-mul-1 80.8%

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

      2. distribute-neg-frac 80.8%

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

   6. Simplified 80.8%

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

   if 7.39999999999999994e56 < y

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.8%

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

      5. neg-mul-1 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-out 99.8%

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

      9. remove-double-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 86.2%

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

      1. associate--l+ 86.2%

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

      2. distribute-lft-out-- 86.2%

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

      3. div-sub 86.2%

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

      4. mul-1-neg 86.2%

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

      5. unsub-neg 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6000000000000:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+56}:\\ \;\;\;\;\frac{-x}{y - z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{z - x}{y}\\ \end{array} \]

Alternative 5: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -520000000:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+56}:\\ \;\;\;\;\frac{-x}{y - z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -520000000.0)
   (/ y (- y z))
   (if (<= y 7e+56) (/ (- x) (- y z)) (- 1.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -520000000.0) {
		tmp = y / (y - z);
	} else if (y <= 7e+56) {
		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 (y <= (-520000000.0d0)) then
        tmp = y / (y - z)
    else if (y <= 7d+56) 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 (y <= -520000000.0) {
		tmp = y / (y - z);
	} else if (y <= 7e+56) {
		tmp = -x / (y - z);
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -520000000.0:
		tmp = y / (y - z)
	elif y <= 7e+56:
		tmp = -x / (y - z)
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -520000000.0)
		tmp = Float64(y / Float64(y - z));
	elseif (y <= 7e+56)
		tmp = Float64(Float64(-x) / Float64(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 (y <= -520000000.0)
		tmp = y / (y - z);
	elseif (y <= 7e+56)
		tmp = -x / (y - z);
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -520000000.0], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+56], N[((-x) / N[(y - z), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.2e8

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.8%

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

      5. neg-mul-1 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-out 99.8%

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

      9. remove-double-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. associate-/l* 99.9%

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

      12. neg-mul-1 99.9%

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

      13. sub-neg 99.9%

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

      14. distribute-neg-out 99.9%

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

      15. remove-double-neg 99.9%

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

      16. +-commutative 99.9%

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

      17. 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 86.4%

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

   if -5.2e8 < y < 6.99999999999999999e56

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.5%

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

      4. associate-/l* 99.4%

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

      5. neg-mul-1 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-neg-out 99.4%

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

      9. remove-double-neg 99.4%

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

      10. sub-neg 99.4%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 80.8%

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

      1. neg-mul-1 80.8%

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

      2. distribute-neg-frac 80.8%

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

   6. Simplified 80.8%

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

   if 6.99999999999999999e56 < y

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.8%

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

      5. neg-mul-1 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-out 99.8%

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

      9. remove-double-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 z around 0 84.8%

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

      1. div-sub 84.8%

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

      2. *-inverses 84.8%

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

   6. Simplified 84.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -520000000:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+56}:\\ \;\;\;\;\frac{-x}{y - z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 6: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-49} \lor \neg \left(y \leq 1.16 \cdot 10^{-107}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.6e-49) (not (<= y 1.16e-107))) (- 1.0 (/ x y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.6e-49) || !(y <= 1.16e-107)) {
		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 <= (-1.6d-49)) .or. (.not. (y <= 1.16d-107))) 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 <= -1.6e-49) || !(y <= 1.16e-107)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.6e-49) or not (y <= 1.16e-107):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.6e-49) || !(y <= 1.16e-107))
		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 <= -1.6e-49) || ~((y <= 1.16e-107)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.6e-49], N[Not[LessEqual[y, 1.16e-107]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.60000000000000001e-49 or 1.16e-107 < y

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.5%

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

      4. associate-/l* 99.4%

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

      5. neg-mul-1 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-neg-out 99.4%

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

      9. remove-double-neg 99.4%

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

      10. sub-neg 99.4%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 z around 0 71.9%

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

      1. div-sub 71.9%

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

      2. *-inverses 71.9%

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

   6. Simplified 71.9%

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

   if -1.60000000000000001e-49 < y < 1.16e-107

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.7%

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

      5. neg-mul-1 99.7%

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

      6. sub-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-neg-out 99.7%

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

      9. remove-double-neg 99.7%

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

      10. sub-neg 99.7%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 81.2%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-49} \lor \neg \left(y \leq 1.16 \cdot 10^{-107}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 7: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-109}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -9.6e-51)
   (/ y (- y z))
   (if (<= y 3.9e-109) (/ x z) (- 1.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -9.6e-51) {
		tmp = y / (y - z);
	} else if (y <= 3.9e-109) {
		tmp = x / 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 (y <= (-9.6d-51)) then
        tmp = y / (y - z)
    else if (y <= 3.9d-109) then
        tmp = x / 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 (y <= -9.6e-51) {
		tmp = y / (y - z);
	} else if (y <= 3.9e-109) {
		tmp = x / z;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -9.6e-51:
		tmp = y / (y - z)
	elif y <= 3.9e-109:
		tmp = x / z
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -9.6e-51)
		tmp = Float64(y / Float64(y - z));
	elseif (y <= 3.9e-109)
		tmp = Float64(x / 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 (y <= -9.6e-51)
		tmp = y / (y - z);
	elseif (y <= 3.9e-109)
		tmp = x / z;
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -9.6e-51], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e-109], N[(x / z), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.6000000000000001e-51

   1. Initial program 99.9%

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

      1. *-lft-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. associate-/r/ 99.8%

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

      4. associate-/l* 99.8%

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

      5. neg-mul-1 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-out 99.8%

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

      9. remove-double-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. associate-/l* 99.9%

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

      12. neg-mul-1 99.9%

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

      13. sub-neg 99.9%

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

      14. distribute-neg-out 99.9%

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

      15. remove-double-neg 99.9%

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

      16. +-commutative 99.9%

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

      17. 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 80.3%

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

   if -9.6000000000000001e-51 < y < 3.90000000000000023e-109

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.7%

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

      5. neg-mul-1 99.7%

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

      6. sub-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-neg-out 99.7%

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

      9. remove-double-neg 99.7%

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

      10. sub-neg 99.7%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 81.2%

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

   if 3.90000000000000023e-109 < y

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.2%

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

      4. associate-/l* 99.2%

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

      5. neg-mul-1 99.2%

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

      6. sub-neg 99.2%

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

      7. +-commutative 99.2%

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

      8. distribute-neg-out 99.2%

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

      9. remove-double-neg 99.2%

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

      10. sub-neg 99.2%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 z around 0 70.1%

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

      1. div-sub 70.1%

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

      2. *-inverses 70.1%

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

   6. Simplified 70.1%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-109}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 8: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.1e-56)
   (/ y (- y z))
   (if (<= y 1.2e+38) (/ (- x y) z) (- 1.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e-56) {
		tmp = y / (y - z);
	} else if (y <= 1.2e+38) {
		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 (y <= (-3.1d-56)) then
        tmp = y / (y - z)
    else if (y <= 1.2d+38) 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 (y <= -3.1e-56) {
		tmp = y / (y - z);
	} else if (y <= 1.2e+38) {
		tmp = (x - y) / z;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.1e-56:
		tmp = y / (y - z)
	elif y <= 1.2e+38:
		tmp = (x - y) / z
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.1e-56)
		tmp = Float64(y / Float64(y - z));
	elseif (y <= 1.2e+38)
		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 (y <= -3.1e-56)
		tmp = y / (y - z);
	elseif (y <= 1.2e+38)
		tmp = (x - y) / z;
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.1e-56], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+38], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.09999999999999987e-56

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.8%

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

      4. associate-/l* 99.8%

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

      5. neg-mul-1 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-out 99.8%

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

      9. remove-double-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 0 79.4%

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

   if -3.09999999999999987e-56 < y < 1.20000000000000009e38

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.5%

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

      4. associate-/l* 99.3%

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

      5. neg-mul-1 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-neg-out 99.3%

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

      9. remove-double-neg 99.3%

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

      10. sub-neg 99.3%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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. div-sub 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 90.7%

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

      1. associate-*r/ 90.7%

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

      2. neg-mul-1 90.7%

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

   8. Simplified 90.7%

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

   9. Taylor expanded in z around inf 83.2%

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

       1. cancel-sign-sub-inv 83.2%

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

       2. metadata-eval 83.2%

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

       3. *-lft-identity 83.2%

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

       4. +-commutative 83.2%

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

       5. mul-1-neg 83.2%

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

       6. sub-neg 83.2%

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

   11. Simplified 83.2%

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

   if 1.20000000000000009e38 < y

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.8%

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

      4. associate-/l* 99.8%

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

      5. neg-mul-1 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-out 99.8%

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

      9. remove-double-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 z around 0 81.6%

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

      1. div-sub 81.7%

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

      2. *-inverses 81.7%

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

   6. Simplified 81.7%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 9: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-49}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.3e-49) 1.0 (if (<= y 1.02e+52) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.3e-49) {
		tmp = 1.0;
	} else if (y <= 1.02e+52) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.3d-49)) then
        tmp = 1.0d0
    else if (y <= 1.02d+52) then
        tmp = x / z
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.3e-49) {
		tmp = 1.0;
	} else if (y <= 1.02e+52) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.3e-49:
		tmp = 1.0
	elif y <= 1.02e+52:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.3e-49)
		tmp = 1.0;
	elseif (y <= 1.02e+52)
		tmp = Float64(x / z);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.3e-49)
		tmp = 1.0;
	elseif (y <= 1.02e+52)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.3e-49], 1.0, If[LessEqual[y, 1.02e+52], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3e-49 or 1.02000000000000002e52 < y

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.9%

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

      4. associate-/l* 99.8%

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

      5. neg-mul-1 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-out 99.8%

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

      9. remove-double-neg 99.8%

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

      10. sub-neg 99.8%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 64.3%

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

   if -3.3e-49 < y < 1.02000000000000002e52

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-/r/ 99.5%

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

      4. associate-/l* 99.3%

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

      5. neg-mul-1 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-neg-out 99.3%

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

      9. remove-double-neg 99.3%

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

      10. sub-neg 99.3%

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

      11. associate-/l* 100.0%

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

      12. neg-mul-1 100.0%

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

      13. sub-neg 100.0%

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

      14. distribute-neg-out 100.0%

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

      15. remove-double-neg 100.0%

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

      16. +-commutative 100.0%

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

      17. 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 69.4%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-49}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 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 11: 35.0% 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. *-lft-identity 100.0%

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

   2. metadata-eval 100.0%

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

   3. associate-/r/ 99.7%

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

   4. associate-/l* 99.5%

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

   5. neg-mul-1 99.5%

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

   6. sub-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. distribute-neg-out 99.5%

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

   9. remove-double-neg 99.5%

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

   10. sub-neg 99.5%

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

   11. associate-/l* 100.0%

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

   12. neg-mul-1 100.0%

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

   13. sub-neg 100.0%

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

   14. distribute-neg-out 100.0%

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

   15. remove-double-neg 100.0%

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

   16. +-commutative 100.0%

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

   17. 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 33.7%

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

5. Final simplification 33.7%

   \[\leadsto 1 \]

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 2023275 
(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)))