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

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 70.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y - z}\\ t_1 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{+210}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (- y z))) (t_1 (- 1.0 (/ x y))))
   (if (<= y -4.7e+210)
     t_0
     (if (<= y -3e+68)
       t_1
       (if (<= y -4.2e-95)
         t_0
         (if (<= y 1.8e-31) (/ x z) (if (<= y 1.1e+42) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double t_1 = 1.0 - (x / y);
	double tmp;
	if (y <= -4.7e+210) {
		tmp = t_0;
	} else if (y <= -3e+68) {
		tmp = t_1;
	} else if (y <= -4.2e-95) {
		tmp = t_0;
	} else if (y <= 1.8e-31) {
		tmp = x / z;
	} else if (y <= 1.1e+42) {
		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 = y / (y - z)
    t_1 = 1.0d0 - (x / y)
    if (y <= (-4.7d+210)) then
        tmp = t_0
    else if (y <= (-3d+68)) then
        tmp = t_1
    else if (y <= (-4.2d-95)) then
        tmp = t_0
    else if (y <= 1.8d-31) then
        tmp = x / z
    else if (y <= 1.1d+42) 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 = y / (y - z);
	double t_1 = 1.0 - (x / y);
	double tmp;
	if (y <= -4.7e+210) {
		tmp = t_0;
	} else if (y <= -3e+68) {
		tmp = t_1;
	} else if (y <= -4.2e-95) {
		tmp = t_0;
	} else if (y <= 1.8e-31) {
		tmp = x / z;
	} else if (y <= 1.1e+42) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (y - z)
	t_1 = 1.0 - (x / y)
	tmp = 0
	if y <= -4.7e+210:
		tmp = t_0
	elif y <= -3e+68:
		tmp = t_1
	elif y <= -4.2e-95:
		tmp = t_0
	elif y <= 1.8e-31:
		tmp = x / z
	elif y <= 1.1e+42:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(y - z))
	t_1 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -4.7e+210)
		tmp = t_0;
	elseif (y <= -3e+68)
		tmp = t_1;
	elseif (y <= -4.2e-95)
		tmp = t_0;
	elseif (y <= 1.8e-31)
		tmp = Float64(x / z);
	elseif (y <= 1.1e+42)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (y - z);
	t_1 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -4.7e+210)
		tmp = t_0;
	elseif (y <= -3e+68)
		tmp = t_1;
	elseif (y <= -4.2e-95)
		tmp = t_0;
	elseif (y <= 1.8e-31)
		tmp = x / z;
	elseif (y <= 1.1e+42)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.7e+210], t$95$0, If[LessEqual[y, -3e+68], t$95$1, If[LessEqual[y, -4.2e-95], t$95$0, If[LessEqual[y, 1.8e-31], N[(x / z), $MachinePrecision], If[LessEqual[y, 1.1e+42], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.7000000000000001e210 or -3.0000000000000002e68 < y < -4.2e-95 or 1.80000000000000002e-31 < y < 1.1000000000000001e42

   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. +-commutative 100.0%

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

      15. distribute-neg-out 100.0%

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

      16. remove-double-neg 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 74.4%

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

   if -4.7000000000000001e210 < y < -3.0000000000000002e68 or 1.1000000000000001e42 < 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. +-commutative 100.0%

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

      15. distribute-neg-out 100.0%

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

      16. remove-double-neg 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 83.4%

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

      1. div-sub 83.4%

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

      2. *-inverses 83.4%

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

   6. Simplified 83.4%

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

   if -4.2e-95 < y < 1.80000000000000002e-31

   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. +-commutative 100.0%

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

      15. distribute-neg-out 100.0%

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

      16. remove-double-neg 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 77.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+210}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+68}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+42}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 3: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y - z}\\ t_1 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -6 \cdot 10^{+212}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-33}:\\ \;\;\;\;\frac{-x}{y - z}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (- y z))) (t_1 (- 1.0 (/ x y))))
   (if (<= y -6e+212)
     t_0
     (if (<= y -2e+73)
       t_1
       (if (<= y -3.4e-14)
         t_0
         (if (<= y 1.95e-33)
           (/ (- x) (- y z))
           (if (<= y 1.9e+43) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double t_1 = 1.0 - (x / y);
	double tmp;
	if (y <= -6e+212) {
		tmp = t_0;
	} else if (y <= -2e+73) {
		tmp = t_1;
	} else if (y <= -3.4e-14) {
		tmp = t_0;
	} else if (y <= 1.95e-33) {
		tmp = -x / (y - z);
	} else if (y <= 1.9e+43) {
		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 = y / (y - z)
    t_1 = 1.0d0 - (x / y)
    if (y <= (-6d+212)) then
        tmp = t_0
    else if (y <= (-2d+73)) then
        tmp = t_1
    else if (y <= (-3.4d-14)) then
        tmp = t_0
    else if (y <= 1.95d-33) then
        tmp = -x / (y - z)
    else if (y <= 1.9d+43) 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 = y / (y - z);
	double t_1 = 1.0 - (x / y);
	double tmp;
	if (y <= -6e+212) {
		tmp = t_0;
	} else if (y <= -2e+73) {
		tmp = t_1;
	} else if (y <= -3.4e-14) {
		tmp = t_0;
	} else if (y <= 1.95e-33) {
		tmp = -x / (y - z);
	} else if (y <= 1.9e+43) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (y - z)
	t_1 = 1.0 - (x / y)
	tmp = 0
	if y <= -6e+212:
		tmp = t_0
	elif y <= -2e+73:
		tmp = t_1
	elif y <= -3.4e-14:
		tmp = t_0
	elif y <= 1.95e-33:
		tmp = -x / (y - z)
	elif y <= 1.9e+43:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(y - z))
	t_1 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -6e+212)
		tmp = t_0;
	elseif (y <= -2e+73)
		tmp = t_1;
	elseif (y <= -3.4e-14)
		tmp = t_0;
	elseif (y <= 1.95e-33)
		tmp = Float64(Float64(-x) / Float64(y - z));
	elseif (y <= 1.9e+43)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (y - z);
	t_1 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -6e+212)
		tmp = t_0;
	elseif (y <= -2e+73)
		tmp = t_1;
	elseif (y <= -3.4e-14)
		tmp = t_0;
	elseif (y <= 1.95e-33)
		tmp = -x / (y - z);
	elseif (y <= 1.9e+43)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6e+212], t$95$0, If[LessEqual[y, -2e+73], t$95$1, If[LessEqual[y, -3.4e-14], t$95$0, If[LessEqual[y, 1.95e-33], N[((-x) / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+43], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6e212 or -1.99999999999999997e73 < y < -3.40000000000000003e-14 or 1.94999999999999987e-33 < y < 1.90000000000000004e43

   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. +-commutative 100.0%

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

      15. distribute-neg-out 100.0%

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

      16. remove-double-neg 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 85.5%

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

   if -6e212 < y < -1.99999999999999997e73 or 1.90000000000000004e43 < 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. +-commutative 100.0%

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

      15. distribute-neg-out 100.0%

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

      16. remove-double-neg 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 83.4%

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

      1. div-sub 83.4%

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

      2. *-inverses 83.4%

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

   6. Simplified 83.4%

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

   if -3.40000000000000003e-14 < y < 1.94999999999999987e-33

   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. +-commutative 100.0%

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

      15. distribute-neg-out 100.0%

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

      16. remove-double-neg 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 85.3%

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

      1. neg-mul-1 85.3%

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

      2. distribute-neg-frac 85.3%

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

   6. Simplified 85.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+212}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq -2 \cdot 10^{+73}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-33}:\\ \;\;\;\;\frac{-x}{y - z}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+43}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 4: 70.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{-27} \lor \neg \left(y \leq 1.9 \cdot 10^{-28}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.35e-27) (not (<= y 1.9e-28))) (- 1.0 (/ x y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.35e-27) || !(y <= 1.9e-28)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.35d-27)) .or. (.not. (y <= 1.9d-28))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.35e-27) || !(y <= 1.9e-28)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.35e-27) or not (y <= 1.9e-28):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.35e-27) || !(y <= 1.9e-28))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.35e-27) || ~((y <= 1.9e-28)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.35e-27], N[Not[LessEqual[y, 1.9e-28]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.35000000000000016e-27 or 1.90000000000000005e-28 < 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. +-commutative 100.0%

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

      15. distribute-neg-out 100.0%

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

      16. remove-double-neg 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 73.0%

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

      1. div-sub 73.1%

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

      2. *-inverses 73.1%

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

   6. Simplified 73.1%

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

   if -2.35000000000000016e-27 < y < 1.90000000000000005e-28

   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. +-commutative 100.0%

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

      15. distribute-neg-out 100.0%

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

      16. remove-double-neg 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 73.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{-27} \lor \neg \left(y \leq 1.9 \cdot 10^{-28}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 5: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+14} \lor \neg \left(z \leq 5.8 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.7e+14) (not (<= z 5.8e+31))) (/ (- x y) z) (- 1.0 (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.7e+14) || !(z <= 5.8e+31)) {
		tmp = (x - y) / z;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-6.7d+14)) .or. (.not. (z <= 5.8d+31))) then
        tmp = (x - y) / z
    else
        tmp = 1.0d0 - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.7e+14) || !(z <= 5.8e+31)) {
		tmp = (x - y) / z;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.7e+14) or not (z <= 5.8e+31):
		tmp = (x - y) / z
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.7e+14) || !(z <= 5.8e+31))
		tmp = Float64(Float64(x - y) / z);
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.7e+14) || ~((z <= 5.8e+31)))
		tmp = (x - y) / z;
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.7e+14], N[Not[LessEqual[z, 5.8e+31]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.7e14 or 5.8000000000000001e31 < z

   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* 99.6%

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

      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. +-commutative 100.0%

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

      15. distribute-neg-out 100.0%

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

      16. remove-double-neg 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. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in y around 0 81.7%

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

   7. Taylor expanded in z around 0 82.0%

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

      1. associate-*r/ 82.0%

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

      2. neg-mul-1 82.0%

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

      3. neg-sub0 82.0%

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

      4. associate--r- 82.0%

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

      5. neg-sub0 82.0%

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

      6. +-commutative 82.0%

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

      7. sub-neg 82.0%

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

   9. Simplified 82.0%

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

   if -6.7e14 < z < 5.8000000000000001e31

   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. +-commutative 100.0%

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

      15. distribute-neg-out 100.0%

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

      16. remove-double-neg 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 79.5%

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

      1. div-sub 79.5%

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

      2. *-inverses 79.5%

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

   6. Simplified 79.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.7 \cdot 10^{+14} \lor \neg \left(z \leq 5.8 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]

Alternative 6: 61.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-25}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.4e-25) 1.0 (if (<= y 1.95e-27) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.4e-25) {
		tmp = 1.0;
	} else if (y <= 1.95e-27) {
		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 <= (-3.4d-25)) then
        tmp = 1.0d0
    else if (y <= 1.95d-27) 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 <= -3.4e-25) {
		tmp = 1.0;
	} else if (y <= 1.95e-27) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.4e-25:
		tmp = 1.0
	elif y <= 1.95e-27:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.4e-25)
		tmp = 1.0;
	elseif (y <= 1.95e-27)
		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 <= -3.4e-25)
		tmp = 1.0;
	elseif (y <= 1.95e-27)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.4e-25], 1.0, If[LessEqual[y, 1.95e-27], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3.40000000000000002e-25 or 1.94999999999999986e-27 < 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. +-commutative 100.0%

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

      15. distribute-neg-out 100.0%

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

      16. remove-double-neg 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 54.4%

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

   if -3.40000000000000002e-25 < y < 1.94999999999999986e-27

   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. +-commutative 100.0%

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

      15. distribute-neg-out 100.0%

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

      16. remove-double-neg 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 73.4%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-25}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 33.8% 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/ 99.8%

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

   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. +-commutative 100.0%

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

   15. distribute-neg-out 100.0%

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

   16. remove-double-neg 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 34.0%

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

5. Final simplification 34.0%

   \[\leadsto 1 \]

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 2023299 
(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)))