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

Percentage Accurate: 100.0% → 100.0%

Time: 10.9s

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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{y - x}{y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-45}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.8e+15)
   (/ (- y x) y)
   (if (<= y 3e-45) (/ (- x y) z) (/ y (- y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e+15) {
		tmp = (y - x) / y;
	} else if (y <= 3e-45) {
		tmp = (x - y) / z;
	} else {
		tmp = y / (y - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4.8d+15)) then
        tmp = (y - x) / y
    else if (y <= 3d-45) then
        tmp = (x - y) / z
    else
        tmp = y / (y - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e+15) {
		tmp = (y - x) / y;
	} else if (y <= 3e-45) {
		tmp = (x - y) / z;
	} else {
		tmp = y / (y - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.8e+15:
		tmp = (y - x) / y
	elif y <= 3e-45:
		tmp = (x - y) / z
	else:
		tmp = y / (y - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.8e+15)
		tmp = Float64(Float64(y - x) / y);
	elseif (y <= 3e-45)
		tmp = Float64(Float64(x - y) / z);
	else
		tmp = Float64(y / Float64(y - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.8e+15)
		tmp = (y - x) / y;
	elseif (y <= 3e-45)
		tmp = (x - y) / z;
	else
		tmp = y / (y - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.8e+15], N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 3e-45], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8e15

   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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \color{blue}{y}\right) \]
   6. Step-by-step derivation

   7. Simplified 85.3%

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

   if -4.8e15 < y < 3.00000000000000011e-45

   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 83.6%

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

   7. Simplified 83.6%

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

   if 3.00000000000000011e-45 < 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 72.5%

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

   7. Simplified 72.5%

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

Alternative 3: 75.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y - z}\\ \mathbf{if}\;y \leq -440000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (- y z))))
   (if (<= y -440000.0) t_0 (if (<= y 5e-43) (/ (- x y) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double tmp;
	if (y <= -440000.0) {
		tmp = t_0;
	} else if (y <= 5e-43) {
		tmp = (x - y) / 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 <= (-440000.0d0)) then
        tmp = t_0
    else if (y <= 5d-43) then
        tmp = (x - y) / 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 <= -440000.0) {
		tmp = t_0;
	} else if (y <= 5e-43) {
		tmp = (x - y) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (y - z)
	tmp = 0
	if y <= -440000.0:
		tmp = t_0
	elif y <= 5e-43:
		tmp = (x - y) / 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 <= -440000.0)
		tmp = t_0;
	elseif (y <= 5e-43)
		tmp = Float64(Float64(x - y) / 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 <= -440000.0)
		tmp = t_0;
	elseif (y <= 5e-43)
		tmp = (x - y) / 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, -440000.0], t$95$0, If[LessEqual[y, 5e-43], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4e5 or 5.00000000000000019e-43 < y

   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 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 77.1%

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

   7. Simplified 77.1%

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

   if -4.4e5 < y < 5.00000000000000019e-43

   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 83.6%

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

   7. Simplified 83.6%

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

Alternative 4: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{z - y}\\ \mathbf{if}\;x \leq -1.12 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+43}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- z y))))
   (if (<= x -1.12e+39) t_0 (if (<= x 3e+43) (/ y (- y z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x / (z - y);
	double tmp;
	if (x <= -1.12e+39) {
		tmp = t_0;
	} else if (x <= 3e+43) {
		tmp = y / (y - 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 = x / (z - y)
    if (x <= (-1.12d+39)) then
        tmp = t_0
    else if (x <= 3d+43) then
        tmp = y / (y - 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 = x / (z - y);
	double tmp;
	if (x <= -1.12e+39) {
		tmp = t_0;
	} else if (x <= 3e+43) {
		tmp = y / (y - z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (z - y)
	tmp = 0
	if x <= -1.12e+39:
		tmp = t_0
	elif x <= 3e+43:
		tmp = y / (y - z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (x <= -1.12e+39)
		tmp = t_0;
	elseif (x <= 3e+43)
		tmp = Float64(y / Float64(y - z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / (z - y);
	tmp = 0.0;
	if (x <= -1.12e+39)
		tmp = t_0;
	elseif (x <= 3e+43)
		tmp = y / (y - z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.12e+39], t$95$0, If[LessEqual[x, 3e+43], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.12e39 or 3.00000000000000017e43 < x

   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. clear-num N/A

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

      2. frac-2neg N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(0 - \left(y - z\right)\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{x}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(0 - \left(y + \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\left(y - x\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(0 - \left(\left(\mathsf{neg}\left(z\right)\right) + y\right)\right)\right), \left(\mathsf{neg}\left(\left(y - x\right)\right)\right)\right) \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - y\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{x}\right)\right)\right)\right) \]

      10. neg-sub0 N/A

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

      11. remove-double-neg N/A

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

      12. --lowering--.f64 N/A

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

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(z, y\right)\right), \left(0 - \color{blue}{\left(y - x\right)}\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(z, y\right)\right), \left(0 - \left(y + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(z, y\right)\right), \left(0 - \left(\left(\mathsf{neg}\left(x\right)\right) + \color{blue}{y}\right)\right)\right) \]

      16. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(0 - \left(\mathsf{neg}\left(x\right)\right)\right) - \color{blue}{y}\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right)\right) \]

      18. remove-double-neg N/A

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

      19. --lowering--.f64 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

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

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

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

      2. --lowering--.f64 81.0%

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

   9. Simplified 81.0%

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

   if -1.12e39 < x < 3.00000000000000017e43

   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 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 78.5%

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

   7. Simplified 78.5%

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

Alternative 5: 69.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+96}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.5e+19) 1.0 (if (<= y 5.5e+96) (/ x (- z y)) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.5e+19) {
		tmp = 1.0;
	} else if (y <= 5.5e+96) {
		tmp = x / (z - y);
	} 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+19)) then
        tmp = 1.0d0
    else if (y <= 5.5d+96) then
        tmp = x / (z - y)
    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+19) {
		tmp = 1.0;
	} else if (y <= 5.5e+96) {
		tmp = x / (z - y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.5e+19:
		tmp = 1.0
	elif y <= 5.5e+96:
		tmp = x / (z - y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.5e+19)
		tmp = 1.0;
	elseif (y <= 5.5e+96)
		tmp = Float64(x / Float64(z - y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.5e+19)
		tmp = 1.0;
	elseif (y <= 5.5e+96)
		tmp = x / (z - y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.5e+19], 1.0, If[LessEqual[y, 5.5e+96], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5e19 or 5.5000000000000002e96 < y

   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 72.7%

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

   if -4.5e19 < y < 5.5000000000000002e96

   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. clear-num N/A

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

      2. frac-2neg N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

      6. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(0 - \left(y - z\right)\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{x}\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(0 - \left(y + \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\left(y - x\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(0 - \left(\left(\mathsf{neg}\left(z\right)\right) + y\right)\right)\right), \left(\mathsf{neg}\left(\left(y - x\right)\right)\right)\right) \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - y\right)\right), \left(\mathsf{neg}\left(\left(y - \color{blue}{x}\right)\right)\right)\right) \]

      10. neg-sub0 N/A

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

      11. remove-double-neg N/A

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

      12. --lowering--.f64 N/A

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

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(z, y\right)\right), \left(0 - \color{blue}{\left(y - x\right)}\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(z, y\right)\right), \left(0 - \left(y + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(z, y\right)\right), \left(0 - \left(\left(\mathsf{neg}\left(x\right)\right) + \color{blue}{y}\right)\right)\right) \]

      16. associate--r+ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(0 - \left(\mathsf{neg}\left(x\right)\right)\right) - \color{blue}{y}\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y\right)\right) \]

      18. remove-double-neg N/A

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

      19. --lowering--.f64 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf

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

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

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

      2. --lowering--.f64 70.8%

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

   9. Simplified 70.8%

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

Alternative 6: 60.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -480000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -480000000.0) 1.0 (if (<= y 2.1e-7) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -480000000.0) {
		tmp = 1.0;
	} else if (y <= 2.1e-7) {
		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 <= (-480000000.0d0)) then
        tmp = 1.0d0
    else if (y <= 2.1d-7) 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 <= -480000000.0) {
		tmp = 1.0;
	} else if (y <= 2.1e-7) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -480000000.0:
		tmp = 1.0
	elif y <= 2.1e-7:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -480000000.0)
		tmp = 1.0;
	elseif (y <= 2.1e-7)
		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 <= -480000000.0)
		tmp = 1.0;
	elseif (y <= 2.1e-7)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -480000000.0], 1.0, If[LessEqual[y, 2.1e-7], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8e8 or 2.1e-7 < y

   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 63.5%

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

   if -4.8e8 < y < 2.1e-7

   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 63.2%

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

   7. Simplified 63.2%

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

Alternative 7: 34.7% 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 34.7%

   \[\leadsto \color{blue}{1} \]
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 2024158 
(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)))