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

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 9

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: 59.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{z}{y}\\ t_1 := \frac{-y}{z}\\ \mathbf{if}\;y \leq -5 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 0.0002:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ z y))) (t_1 (/ (- y) z)))
   (if (<= y -5e+93)
     t_0
     (if (<= y -6.5e-69)
       t_1
       (if (<= y 3.3e-51)
         (/ x z)
         (if (<= y 0.0002) 1.0 (if (<= y 2.4e+73) t_1 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z / y);
	double t_1 = -y / z;
	double tmp;
	if (y <= -5e+93) {
		tmp = t_0;
	} else if (y <= -6.5e-69) {
		tmp = t_1;
	} else if (y <= 3.3e-51) {
		tmp = x / z;
	} else if (y <= 0.0002) {
		tmp = 1.0;
	} else if (y <= 2.4e+73) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (z / y)
    t_1 = -y / z
    if (y <= (-5d+93)) then
        tmp = t_0
    else if (y <= (-6.5d-69)) then
        tmp = t_1
    else if (y <= 3.3d-51) then
        tmp = x / z
    else if (y <= 0.0002d0) then
        tmp = 1.0d0
    else if (y <= 2.4d+73) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (z / y);
	double t_1 = -y / z;
	double tmp;
	if (y <= -5e+93) {
		tmp = t_0;
	} else if (y <= -6.5e-69) {
		tmp = t_1;
	} else if (y <= 3.3e-51) {
		tmp = x / z;
	} else if (y <= 0.0002) {
		tmp = 1.0;
	} else if (y <= 2.4e+73) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z / y)
	t_1 = -y / z
	tmp = 0
	if y <= -5e+93:
		tmp = t_0
	elif y <= -6.5e-69:
		tmp = t_1
	elif y <= 3.3e-51:
		tmp = x / z
	elif y <= 0.0002:
		tmp = 1.0
	elif y <= 2.4e+73:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z / y))
	t_1 = Float64(Float64(-y) / z)
	tmp = 0.0
	if (y <= -5e+93)
		tmp = t_0;
	elseif (y <= -6.5e-69)
		tmp = t_1;
	elseif (y <= 3.3e-51)
		tmp = Float64(x / z);
	elseif (y <= 0.0002)
		tmp = 1.0;
	elseif (y <= 2.4e+73)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z / y);
	t_1 = -y / z;
	tmp = 0.0;
	if (y <= -5e+93)
		tmp = t_0;
	elseif (y <= -6.5e-69)
		tmp = t_1;
	elseif (y <= 3.3e-51)
		tmp = x / z;
	elseif (y <= 0.0002)
		tmp = 1.0;
	elseif (y <= 2.4e+73)
		tmp = t_1;
	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[((-y) / z), $MachinePrecision]}, If[LessEqual[y, -5e+93], t$95$0, If[LessEqual[y, -6.5e-69], t$95$1, If[LessEqual[y, 3.3e-51], N[(x / z), $MachinePrecision], If[LessEqual[y, 0.0002], 1.0, If[LessEqual[y, 2.4e+73], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.0000000000000001e93 or 2.40000000000000002e73 < 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 y around inf 86.7%

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

      1. associate--l+ 86.7%

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

      2. distribute-lft-out-- 86.7%

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

      3. div-sub 86.7%

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

      4. mul-1-neg 86.7%

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

      5. unsub-neg 86.7%

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

   6. Simplified 86.7%

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

   7. Taylor expanded in x around 0 66.7%

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

      1. neg-mul-1 66.7%

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

      2. distribute-neg-frac 66.7%

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

   9. Simplified 66.7%

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

       1. div-inv 66.7%

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

       2. cancel-sign-sub 66.7%

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

       3. div-inv 66.7%

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

       4. +-commutative 66.7%

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

   11. Applied egg-rr 66.7%

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

   if -5.0000000000000001e93 < y < -6.49999999999999951e-69 or 2.0000000000000001e-4 < y < 2.40000000000000002e73

   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* 99.6%

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

      5. neg-mul-1 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-neg-out 99.6%

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

      9. remove-double-neg 99.6%

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

      10. sub-neg 99.6%

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

      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 inf 62.6%

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

      1. associate-*r/ 62.6%

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

      2. neg-mul-1 62.6%

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

      3. neg-sub0 62.6%

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

      4. associate--r- 62.6%

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

      5. neg-sub0 62.6%

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

   6. Simplified 62.6%

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

   7. Taylor expanded in y around inf 48.8%

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

      1. mul-1-neg 48.8%

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

      2. distribute-neg-frac 48.8%

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

   9. Simplified 48.8%

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

   if -6.49999999999999951e-69 < y < 3.29999999999999973e-51

   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/ 98.6%

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

      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 y around 0 84.4%

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

   if 3.29999999999999973e-51 < y < 2.0000000000000001e-4

   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* 99.7%

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

      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 inf 60.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+93}:\\ \;\;\;\;1 + \frac{z}{y}\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 0.0002:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+73}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{z}{y}\\ \end{array} \]

Alternative 3: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z}\\ t_1 := 1 + \frac{z - x}{y}\\ \mathbf{if}\;y \leq -20000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 12200:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) z)) (t_1 (+ 1.0 (/ (- z x) y))))
   (if (<= y -20000.0)
     t_1
     (if (<= y 1.1e-50)
       t_0
       (if (<= y 12200.0) (- 1.0 (/ x y)) (if (<= y 2.2e+71) t_0 t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / z;
	double t_1 = 1.0 + ((z - x) / y);
	double tmp;
	if (y <= -20000.0) {
		tmp = t_1;
	} else if (y <= 1.1e-50) {
		tmp = t_0;
	} else if (y <= 12200.0) {
		tmp = 1.0 - (x / y);
	} else if (y <= 2.2e+71) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 = (x - y) / z
    t_1 = 1.0d0 + ((z - x) / y)
    if (y <= (-20000.0d0)) then
        tmp = t_1
    else if (y <= 1.1d-50) then
        tmp = t_0
    else if (y <= 12200.0d0) then
        tmp = 1.0d0 - (x / y)
    else if (y <= 2.2d+71) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / z;
	double t_1 = 1.0 + ((z - x) / y);
	double tmp;
	if (y <= -20000.0) {
		tmp = t_1;
	} else if (y <= 1.1e-50) {
		tmp = t_0;
	} else if (y <= 12200.0) {
		tmp = 1.0 - (x / y);
	} else if (y <= 2.2e+71) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / z
	t_1 = 1.0 + ((z - x) / y)
	tmp = 0
	if y <= -20000.0:
		tmp = t_1
	elif y <= 1.1e-50:
		tmp = t_0
	elif y <= 12200.0:
		tmp = 1.0 - (x / y)
	elif y <= 2.2e+71:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / z)
	t_1 = Float64(1.0 + Float64(Float64(z - x) / y))
	tmp = 0.0
	if (y <= -20000.0)
		tmp = t_1;
	elseif (y <= 1.1e-50)
		tmp = t_0;
	elseif (y <= 12200.0)
		tmp = Float64(1.0 - Float64(x / y));
	elseif (y <= 2.2e+71)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / z;
	t_1 = 1.0 + ((z - x) / y);
	tmp = 0.0;
	if (y <= -20000.0)
		tmp = t_1;
	elseif (y <= 1.1e-50)
		tmp = t_0;
	elseif (y <= 12200.0)
		tmp = 1.0 - (x / y);
	elseif (y <= 2.2e+71)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -20000.0], t$95$1, If[LessEqual[y, 1.1e-50], t$95$0, If[LessEqual[y, 12200.0], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+71], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e4 or 2.19999999999999995e71 < 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 82.8%

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

      1. associate--l+ 82.8%

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

      2. distribute-lft-out-- 82.8%

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

      3. div-sub 82.8%

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

      4. mul-1-neg 82.8%

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

      5. unsub-neg 82.8%

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

   6. Simplified 82.8%

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

   if -2e4 < y < 1.0999999999999999e-50 or 12200 < y < 2.19999999999999995e71

   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/ 98.9%

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

      4. associate-/l* 98.8%

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

      5. neg-mul-1 98.8%

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

      6. sub-neg 98.8%

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

      7. +-commutative 98.8%

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

      8. distribute-neg-out 98.8%

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

      9. remove-double-neg 98.8%

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

      10. sub-neg 98.8%

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

      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 inf 86.4%

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

      1. associate-*r/ 86.4%

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

      2. neg-mul-1 86.4%

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

      3. neg-sub0 86.4%

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

      4. associate--r- 86.4%

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

      5. neg-sub0 86.4%

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

   6. Simplified 86.4%

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

   7. Taylor expanded in y around 0 86.4%

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

      1. +-commutative 86.4%

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

      2. mul-1-neg 86.4%

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

      3. sub-neg 86.4%

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

      4. div-sub 86.4%

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

   9. Simplified 86.4%

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

   if 1.0999999999999999e-50 < y < 12200

   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* 99.7%

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

      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 z around 0 87.1%

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

      1. div-sub 87.1%

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

      2. *-inverses 87.1%

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

   6. Simplified 87.1%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -20000:\\ \;\;\;\;1 + \frac{z - x}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;y \leq 12200:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{z - x}{y}\\ \end{array} \]

Alternative 4: 59.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-y}{z}\\ \mathbf{if}\;y \leq -3.35 \cdot 10^{+93}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.25 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y) z)))
   (if (<= y -3.35e+93)
     1.0
     (if (<= y -6.5e-69)
       t_0
       (if (<= y 4.25e-50)
         (/ x z)
         (if (<= y 9e-5) 1.0 (if (<= y 5.2e+54) t_0 1.0)))))))

C

double code(double x, double y, double z) {
	double t_0 = -y / z;
	double tmp;
	if (y <= -3.35e+93) {
		tmp = 1.0;
	} else if (y <= -6.5e-69) {
		tmp = t_0;
	} else if (y <= 4.25e-50) {
		tmp = x / z;
	} else if (y <= 9e-5) {
		tmp = 1.0;
	} else if (y <= 5.2e+54) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -y / z
    if (y <= (-3.35d+93)) then
        tmp = 1.0d0
    else if (y <= (-6.5d-69)) then
        tmp = t_0
    else if (y <= 4.25d-50) then
        tmp = x / z
    else if (y <= 9d-5) then
        tmp = 1.0d0
    else if (y <= 5.2d+54) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -y / z;
	double tmp;
	if (y <= -3.35e+93) {
		tmp = 1.0;
	} else if (y <= -6.5e-69) {
		tmp = t_0;
	} else if (y <= 4.25e-50) {
		tmp = x / z;
	} else if (y <= 9e-5) {
		tmp = 1.0;
	} else if (y <= 5.2e+54) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -y / z
	tmp = 0
	if y <= -3.35e+93:
		tmp = 1.0
	elif y <= -6.5e-69:
		tmp = t_0
	elif y <= 4.25e-50:
		tmp = x / z
	elif y <= 9e-5:
		tmp = 1.0
	elif y <= 5.2e+54:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-y) / z)
	tmp = 0.0
	if (y <= -3.35e+93)
		tmp = 1.0;
	elseif (y <= -6.5e-69)
		tmp = t_0;
	elseif (y <= 4.25e-50)
		tmp = Float64(x / z);
	elseif (y <= 9e-5)
		tmp = 1.0;
	elseif (y <= 5.2e+54)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -y / z;
	tmp = 0.0;
	if (y <= -3.35e+93)
		tmp = 1.0;
	elseif (y <= -6.5e-69)
		tmp = t_0;
	elseif (y <= 4.25e-50)
		tmp = x / z;
	elseif (y <= 9e-5)
		tmp = 1.0;
	elseif (y <= 5.2e+54)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-y) / z), $MachinePrecision]}, If[LessEqual[y, -3.35e+93], 1.0, If[LessEqual[y, -6.5e-69], t$95$0, If[LessEqual[y, 4.25e-50], N[(x / z), $MachinePrecision], If[LessEqual[y, 9e-5], 1.0, If[LessEqual[y, 5.2e+54], t$95$0, 1.0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.34999999999999983e93 or 4.25000000000000006e-50 < y < 9.00000000000000057e-5 or 5.20000000000000013e54 < 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 y around inf 63.6%

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

   if -3.34999999999999983e93 < y < -6.49999999999999951e-69 or 9.00000000000000057e-5 < y < 5.20000000000000013e54

   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* 99.6%

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

      5. neg-mul-1 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-neg-out 99.6%

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

      9. remove-double-neg 99.6%

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

      10. sub-neg 99.6%

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

      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 inf 65.1%

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

      1. associate-*r/ 65.1%

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

      2. neg-mul-1 65.1%

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

      3. neg-sub0 65.1%

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

      4. associate--r- 65.1%

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

      5. neg-sub0 65.1%

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

   6. Simplified 65.1%

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

   7. Taylor expanded in y around inf 51.9%

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

      1. mul-1-neg 51.9%

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

      2. distribute-neg-frac 51.9%

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

   9. Simplified 51.9%

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

   if -6.49999999999999951e-69 < y < 4.25000000000000006e-50

   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/ 98.6%

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

      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 y around 0 84.4%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.35 \cdot 10^{+93}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;y \leq 4.25 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -92000 \lor \neg \left(y \leq 9 \cdot 10^{-52} \lor \neg \left(y \leq 0.000155\right) \land y \leq 1.45 \cdot 10^{+52}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -92000.0)
         (not (or (<= y 9e-52) (and (not (<= y 0.000155)) (<= y 1.45e+52)))))
   (- 1.0 (/ x y))
   (/ (- x y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -92000.0) || !((y <= 9e-52) || (!(y <= 0.000155) && (y <= 1.45e+52)))) {
		tmp = 1.0 - (x / y);
	} 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 <= (-92000.0d0)) .or. (.not. (y <= 9d-52) .or. (.not. (y <= 0.000155d0)) .and. (y <= 1.45d+52))) then
        tmp = 1.0d0 - (x / y)
    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 <= -92000.0) || !((y <= 9e-52) || (!(y <= 0.000155) && (y <= 1.45e+52)))) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x - y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -92000.0) or not ((y <= 9e-52) or (not (y <= 0.000155) and (y <= 1.45e+52))):
		tmp = 1.0 - (x / y)
	else:
		tmp = (x - y) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -92000.0) || !((y <= 9e-52) || (!(y <= 0.000155) && (y <= 1.45e+52))))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(Float64(x - y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -92000.0) || ~(((y <= 9e-52) || (~((y <= 0.000155)) && (y <= 1.45e+52)))))
		tmp = 1.0 - (x / y);
	else
		tmp = (x - y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -92000.0], N[Not[Or[LessEqual[y, 9e-52], And[N[Not[LessEqual[y, 0.000155]], $MachinePrecision], LessEqual[y, 1.45e+52]]]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -92000 or 9.0000000000000001e-52 < y < 1.55e-4 or 1.45e52 < 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 81.7%

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

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

   if -92000 < y < 9.0000000000000001e-52 or 1.55e-4 < y < 1.45e52

   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/ 98.8%

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

      4. associate-/l* 98.7%

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

      5. neg-mul-1 98.7%

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

      6. sub-neg 98.7%

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

      7. +-commutative 98.7%

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

      8. distribute-neg-out 98.7%

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

      9. remove-double-neg 98.7%

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

      10. sub-neg 98.7%

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

      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 z around inf 87.5%

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

      1. associate-*r/ 87.5%

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

      2. neg-mul-1 87.5%

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

      3. neg-sub0 87.5%

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

      4. associate--r- 87.5%

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

      5. neg-sub0 87.5%

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

   6. Simplified 87.5%

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

   7. Taylor expanded in y around 0 87.5%

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

      1. +-commutative 87.5%

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

      2. mul-1-neg 87.5%

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

      3. sub-neg 87.5%

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

      4. div-sub 87.5%

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

   9. Simplified 87.5%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -92000 \lor \neg \left(y \leq 9 \cdot 10^{-52} \lor \neg \left(y \leq 0.000155\right) \land y \leq 1.45 \cdot 10^{+52}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z}\\ \end{array} \]

Alternative 6: 70.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+23} \lor \neg \left(y \leq 1.4 \cdot 10^{+51}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -3.2e-49)
     t_0
     (if (<= y 1.45e-50)
       (/ x z)
       (if (or (<= y 6.5e+23) (not (<= y 1.4e+51))) t_0 (/ (- y) z))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -3.2e-49) {
		tmp = t_0;
	} else if (y <= 1.45e-50) {
		tmp = x / z;
	} else if ((y <= 6.5e+23) || !(y <= 1.4e+51)) {
		tmp = t_0;
	} else {
		tmp = -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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    if (y <= (-3.2d-49)) then
        tmp = t_0
    else if (y <= 1.45d-50) then
        tmp = x / z
    else if ((y <= 6.5d+23) .or. (.not. (y <= 1.4d+51))) then
        tmp = t_0
    else
        tmp = -y / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -3.2e-49) {
		tmp = t_0;
	} else if (y <= 1.45e-50) {
		tmp = x / z;
	} else if ((y <= 6.5e+23) || !(y <= 1.4e+51)) {
		tmp = t_0;
	} else {
		tmp = -y / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -3.2e-49:
		tmp = t_0
	elif y <= 1.45e-50:
		tmp = x / z
	elif (y <= 6.5e+23) or not (y <= 1.4e+51):
		tmp = t_0
	else:
		tmp = -y / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -3.2e-49)
		tmp = t_0;
	elseif (y <= 1.45e-50)
		tmp = Float64(x / z);
	elseif ((y <= 6.5e+23) || !(y <= 1.4e+51))
		tmp = t_0;
	else
		tmp = Float64(Float64(-y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -3.2e-49)
		tmp = t_0;
	elseif (y <= 1.45e-50)
		tmp = x / z;
	elseif ((y <= 6.5e+23) || ~((y <= 1.4e+51)))
		tmp = t_0;
	else
		tmp = -y / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e-49], t$95$0, If[LessEqual[y, 1.45e-50], N[(x / z), $MachinePrecision], If[Or[LessEqual[y, 6.5e+23], N[Not[LessEqual[y, 1.4e+51]], $MachinePrecision]], t$95$0, N[((-y) / z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.20000000000000002e-49 or 1.45000000000000004e-50 < y < 6.4999999999999996e23 or 1.40000000000000002e51 < 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 79.1%

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

      1. div-sub 79.1%

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

      2. *-inverses 79.1%

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

   6. Simplified 79.1%

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

   if -3.20000000000000002e-49 < y < 1.45000000000000004e-50

   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/ 98.7%

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

      4. associate-/l* 98.6%

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

      5. neg-mul-1 98.6%

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

      6. sub-neg 98.6%

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

      7. +-commutative 98.6%

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

      8. distribute-neg-out 98.6%

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

      9. remove-double-neg 98.6%

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

      10. sub-neg 98.6%

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

      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 81.0%

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

   if 6.4999999999999996e23 < y < 1.40000000000000002e51

   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/ 99.6%

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

      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 inf 88.2%

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

      1. associate-*r/ 88.2%

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

      2. neg-mul-1 88.2%

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

      3. neg-sub0 88.2%

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

      4. associate--r- 88.2%

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

      5. neg-sub0 88.2%

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

   6. Simplified 88.2%

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

   7. Taylor expanded in y around inf 61.8%

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

      1. mul-1-neg 61.8%

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

      2. distribute-neg-frac 61.8%

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

   9. Simplified 61.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-49}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+23} \lor \neg \left(y \leq 1.4 \cdot 10^{+51}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{z}\\ \end{array} \]

Alternative 7: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6e-71) (/ y (- y z)) (if (<= y 1.85e-50) (/ x z) (- 1.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6e-71) {
		tmp = y / (y - z);
	} else if (y <= 1.85e-50) {
		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 <= (-6d-71)) then
        tmp = y / (y - z)
    else if (y <= 1.85d-50) 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 <= -6e-71) {
		tmp = y / (y - z);
	} else if (y <= 1.85e-50) {
		tmp = x / z;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6e-71:
		tmp = y / (y - z)
	elif y <= 1.85e-50:
		tmp = x / z
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6e-71)
		tmp = Float64(y / Float64(y - z));
	elseif (y <= 1.85e-50)
		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 <= -6e-71)
		tmp = y / (y - z);
	elseif (y <= 1.85e-50)
		tmp = x / z;
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6e-71], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e-50], N[(x / z), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.0000000000000003e-71

   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 76.8%

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

   if -6.0000000000000003e-71 < y < 1.85e-50

   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/ 98.6%

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

      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 y around 0 84.4%

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

   if 1.85e-50 < 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* 99.7%

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

      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 z around 0 74.7%

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

      1. div-sub 74.7%

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

      2. *-inverses 74.7%

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

   6. Simplified 74.7%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 8: 60.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-48}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.32e-48) 1.0 (if (<= y 1.05e-49) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.32e-48) {
		tmp = 1.0;
	} else if (y <= 1.05e-49) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.32d-48)) then
        tmp = 1.0d0
    else if (y <= 1.05d-49) then
        tmp = x / z
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.32e-48) {
		tmp = 1.0;
	} else if (y <= 1.05e-49) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.32e-48:
		tmp = 1.0
	elif y <= 1.05e-49:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.32e-48)
		tmp = 1.0;
	elseif (y <= 1.05e-49)
		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 <= -1.32e-48)
		tmp = 1.0;
	elseif (y <= 1.05e-49)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.32e-48], 1.0, If[LessEqual[y, 1.05e-49], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.32e-48 or 1.0499999999999999e-49 < 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 y around inf 52.9%

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

   if -1.32e-48 < y < 1.0499999999999999e-49

   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/ 98.7%

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

      4. associate-/l* 98.6%

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

      5. neg-mul-1 98.6%

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

      6. sub-neg 98.6%

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

      7. +-commutative 98.6%

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

      8. distribute-neg-out 98.6%

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

      9. remove-double-neg 98.6%

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

      10. sub-neg 98.6%

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

      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 81.0%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-48}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 35.2% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. *-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/ 99.4%

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

   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 inf 33.6%

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

5. Final simplification 33.6%

   \[\leadsto 1 \]

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 2023271 
(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)))