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

Percentage Accurate: 100.0% → 100.0%

Time: 11.2s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (y / (y - z)) + (x / (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 = (y / (y - z)) + (x / (z - y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. neg-sub0 N/A

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

   4. associate-+l- N/A

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

   5. sub0-neg N/A

      \[\leadsto \frac{x - y}{\mathsf{neg}\left(\left(y - z\right)\right)} \]

   6. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{neg}\left(\frac{x - y}{y - z}\right) \]

   7. distribute-neg-frac N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. distribute-neg-out N/A

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

   11. remove-double-neg N/A

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

   12. sub-neg N/A

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

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - z\right)}\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - z\right)\right) \]

   15. --lowering--.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

3. Simplified 100.0%

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

   1. div-sub N/A

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

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{y}{y - z}\right), \color{blue}{\left(\frac{x}{y - z}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(y, \left(y - z\right)\right), \left(\frac{\color{blue}{x}}{y - z}\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(y, z\right)\right), \left(\frac{x}{y - z}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{/.f64}\left(x, \color{blue}{\left(y - z\right)}\right)\right) \]

   6. --lowering--.f64 100.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{z} - \frac{y}{z}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-83}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.1e-52)
   (- (/ x z) (/ y z))
   (if (<= z 5.2e-83) (- 1.0 (/ x y)) (/ (- x y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.1e-52) {
		tmp = (x / z) - (y / z);
	} else if (z <= 5.2e-83) {
		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 (z <= (-5.1d-52)) then
        tmp = (x / z) - (y / z)
    else if (z <= 5.2d-83) 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 (z <= -5.1e-52) {
		tmp = (x / z) - (y / z);
	} else if (z <= 5.2e-83) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = (x - y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.1e-52:
		tmp = (x / z) - (y / z)
	elif z <= 5.2e-83:
		tmp = 1.0 - (x / y)
	else:
		tmp = (x - y) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.1e-52)
		tmp = Float64(Float64(x / z) - Float64(y / z));
	elseif (z <= 5.2e-83)
		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 (z <= -5.1e-52)
		tmp = (x / z) - (y / z);
	elseif (z <= 5.2e-83)
		tmp = 1.0 - (x / y);
	else
		tmp = (x - y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.1e-52], N[(N[(x / z), $MachinePrecision] - N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.2e-83], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.09999999999999989e-52

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

         \[\leadsto \frac{x - y}{\mathsf{neg}\left(\left(y - z\right)\right)} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x - y}{y - z}\right) \]

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - z\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - z\right)\right) \]

      15. --lowering--.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{z}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), z\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), z\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), z\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), z\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y\right), z\right) \]

      9. --lowering--.f64 86.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), z\right) \]

   7. Simplified 86.3%

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

      1. div-sub N/A

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

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{y}{z}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\color{blue}{y}}{z}\right)\right) \]

      4. /-lowering-/.f64 86.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   9. Applied egg-rr 86.3%

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

   if -5.09999999999999989e-52 < z < 5.20000000000000018e-83

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

         \[\leadsto \frac{x - y}{\mathsf{neg}\left(\left(y - z\right)\right)} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x - y}{y - z}\right) \]

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - z\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - z\right)\right) \]

      15. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right) \]

      4. /-lowering-/.f64 84.7%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   7. Simplified 84.7%

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

   if 5.20000000000000018e-83 < z

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

         \[\leadsto \frac{x - y}{\mathsf{neg}\left(\left(y - z\right)\right)} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x - y}{y - z}\right) \]

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - z\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - z\right)\right) \]

      15. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{z}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), z\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), z\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), z\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), z\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y\right), z\right) \]

      9. --lowering--.f64 73.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), z\right) \]

   7. Simplified 73.6%

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

Alternative 3: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z}\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-83}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) z)))
   (if (<= z -1.9e-49) t_0 (if (<= z 2.7e-83) (- 1.0 (/ x y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / z;
	double tmp;
	if (z <= -1.9e-49) {
		tmp = t_0;
	} else if (z <= 2.7e-83) {
		tmp = 1.0 - (x / y);
	} 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) :: tmp
    t_0 = (x - y) / z
    if (z <= (-1.9d-49)) then
        tmp = t_0
    else if (z <= 2.7d-83) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / z;
	double tmp;
	if (z <= -1.9e-49) {
		tmp = t_0;
	} else if (z <= 2.7e-83) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / z
	tmp = 0
	if z <= -1.9e-49:
		tmp = t_0
	elif z <= 2.7e-83:
		tmp = 1.0 - (x / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / z)
	tmp = 0.0
	if (z <= -1.9e-49)
		tmp = t_0;
	elseif (z <= 2.7e-83)
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / z;
	tmp = 0.0;
	if (z <= -1.9e-49)
		tmp = t_0;
	elseif (z <= 2.7e-83)
		tmp = 1.0 - (x / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -1.9e-49], t$95$0, If[LessEqual[z, 2.7e-83], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8999999999999999e-49 or 2.69999999999999991e-83 < z

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

         \[\leadsto \frac{x - y}{\mathsf{neg}\left(\left(y - z\right)\right)} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x - y}{y - z}\right) \]

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - z\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - z\right)\right) \]

      15. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(y - x\right)\right), \color{blue}{z}\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(y - x\right)\right)\right), z\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), z\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)\right)\right), z\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right), z\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y\right), z\right) \]

      9. --lowering--.f64 79.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), z\right) \]

   7. Simplified 79.8%

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

   if -1.8999999999999999e-49 < z < 2.69999999999999991e-83

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

         \[\leadsto \frac{x - y}{\mathsf{neg}\left(\left(y - z\right)\right)} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x - y}{y - z}\right) \]

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - z\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - z\right)\right) \]

      15. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right) \]

      4. /-lowering-/.f64 84.7%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   7. Simplified 84.7%

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

Alternative 4: 69.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y - z}\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{-71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (- y z))))
   (if (<= y -1.45e-71) t_0 (if (<= y 2.8e+86) (/ x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double tmp;
	if (y <= -1.45e-71) {
		tmp = t_0;
	} else if (y <= 2.8e+86) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y - z)
    if (y <= (-1.45d-71)) then
        tmp = t_0
    else if (y <= 2.8d+86) then
        tmp = x / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double tmp;
	if (y <= -1.45e-71) {
		tmp = t_0;
	} else if (y <= 2.8e+86) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (y - z)
	tmp = 0
	if y <= -1.45e-71:
		tmp = t_0
	elif y <= 2.8e+86:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(y - z))
	tmp = 0.0
	if (y <= -1.45e-71)
		tmp = t_0;
	elseif (y <= 2.8e+86)
		tmp = Float64(x / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (y - z);
	tmp = 0.0;
	if (y <= -1.45e-71)
		tmp = t_0;
	elseif (y <= 2.8e+86)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.45e-71], t$95$0, If[LessEqual[y, 2.8e+86], N[(x / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4499999999999999e-71 or 2.80000000000000004e86 < y

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

         \[\leadsto \frac{x - y}{\mathsf{neg}\left(\left(y - z\right)\right)} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x - y}{y - z}\right) \]

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - z\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - z\right)\right) \]

      15. --lowering--.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(y - z\right)}\right) \]

      2. --lowering--.f64 72.7%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Simplified 72.7%

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

   if -1.4499999999999999e-71 < y < 2.80000000000000004e86

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

         \[\leadsto \frac{x - y}{\mathsf{neg}\left(\left(y - z\right)\right)} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x - y}{y - z}\right) \]

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - z\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - z\right)\right) \]

      15. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 70.8%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   7. Simplified 70.8%

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

Alternative 5: 62.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.65 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-83}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.65e+18) (/ x z) (if (<= z 9.2e-83) (- 1.0 (/ x y)) (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.65e+18) {
		tmp = x / z;
	} else if (z <= 9.2e-83) {
		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 (z <= (-3.65d+18)) then
        tmp = x / z
    else if (z <= 9.2d-83) 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 (z <= -3.65e+18) {
		tmp = x / z;
	} else if (z <= 9.2e-83) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.65e+18:
		tmp = x / z
	elif z <= 9.2e-83:
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.65e+18)
		tmp = Float64(x / z);
	elseif (z <= 9.2e-83)
		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 (z <= -3.65e+18)
		tmp = x / z;
	elseif (z <= 9.2e-83)
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.65e+18], N[(x / z), $MachinePrecision], If[LessEqual[z, 9.2e-83], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.65e18 or 9.19999999999999959e-83 < z

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

         \[\leadsto \frac{x - y}{\mathsf{neg}\left(\left(y - z\right)\right)} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x - y}{y - z}\right) \]

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - z\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - z\right)\right) \]

      15. --lowering--.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 62.8%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   7. Simplified 62.8%

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

   if -3.65e18 < z < 9.19999999999999959e-83

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

         \[\leadsto \frac{x - y}{\mathsf{neg}\left(\left(y - z\right)\right)} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x - y}{y - z}\right) \]

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - z\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - z\right)\right) \]

      15. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{x}{y}\right)}\right) \]

      4. /-lowering-/.f64 81.3%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   7. Simplified 81.3%

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

Alternative 6: 61.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+56}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.5e+56) 1.0 (if (<= y 3.5e+63) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.5e+56) {
		tmp = 1.0;
	} else if (y <= 3.5e+63) {
		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 <= (-4.5d+56)) then
        tmp = 1.0d0
    else if (y <= 3.5d+63) 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 <= -4.5e+56) {
		tmp = 1.0;
	} else if (y <= 3.5e+63) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.5e+56:
		tmp = 1.0
	elif y <= 3.5e+63:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.5e+56)
		tmp = 1.0;
	elseif (y <= 3.5e+63)
		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 <= -4.5e+56)
		tmp = 1.0;
	elseif (y <= 3.5e+63)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.5e+56], 1.0, If[LessEqual[y, 3.5e+63], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5000000000000003e56 or 3.50000000000000029e63 < y

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

         \[\leadsto \frac{x - y}{\mathsf{neg}\left(\left(y - z\right)\right)} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x - y}{y - z}\right) \]

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - z\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - z\right)\right) \]

      15. --lowering--.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 65.4%

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

   if -4.5000000000000003e56 < y < 3.50000000000000029e63

   1. Initial program 100.0%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

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

      4. associate-+l- N/A

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

      5. sub0-neg N/A

         \[\leadsto \frac{x - y}{\mathsf{neg}\left(\left(y - z\right)\right)} \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x - y}{y - z}\right) \]

      7. distribute-neg-frac N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-out N/A

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

      11. remove-double-neg N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - z\right)}\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - z\right)\right) \]

      15. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 64.9%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   7. Simplified 64.9%

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

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 8: 34.0% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. neg-sub0 N/A

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

   4. associate-+l- N/A

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

   5. sub0-neg N/A

      \[\leadsto \frac{x - y}{\mathsf{neg}\left(\left(y - z\right)\right)} \]

   6. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{neg}\left(\frac{x - y}{y - z}\right) \]

   7. distribute-neg-frac N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. distribute-neg-out N/A

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

   11. remove-double-neg N/A

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

   12. sub-neg N/A

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

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(y - x\right), \color{blue}{\left(y - z\right)}\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\color{blue}{y} - z\right)\right) \]

   15. --lowering--.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{\_.f64}\left(y, \color{blue}{z}\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in y around inf

   \[\leadsto \color{blue}{1} \]
6. Step-by-step derivation

7. Simplified 31.6%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Developer Target 1: 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 2024161 
(FPCore (x y z)
  :name "Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ x (- z y)) (/ y (- z y))))

  (/ (- x y) (- z y)))