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

Percentage Accurate: 100.0% → 100.0%

Time: 8.0s

Alternatives: 12

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

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

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]

Derivation

1. Initial program 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 69.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{z}{y} - -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))))
   (if (<= t_0 -2e+38)
     (/ x (- y))
     (if (<= t_0 -2e-60)
       (/ x z)
       (if (<= t_0 1e-5)
         (/ (- y) z)
         (if (<= t_0 2.0) (- (/ z y) -1.0) (/ x z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double tmp;
	if (t_0 <= -2e+38) {
		tmp = x / -y;
	} else if (t_0 <= -2e-60) {
		tmp = x / z;
	} else if (t_0 <= 1e-5) {
		tmp = -y / z;
	} else if (t_0 <= 2.0) {
		tmp = (z / y) - -1.0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (y - x) / (y - z)
    if (t_0 <= (-2d+38)) then
        tmp = x / -y
    else if (t_0 <= (-2d-60)) then
        tmp = x / z
    else if (t_0 <= 1d-5) then
        tmp = -y / z
    else if (t_0 <= 2.0d0) then
        tmp = (z / y) - (-1.0d0)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double tmp;
	if (t_0 <= -2e+38) {
		tmp = x / -y;
	} else if (t_0 <= -2e-60) {
		tmp = x / z;
	} else if (t_0 <= 1e-5) {
		tmp = -y / z;
	} else if (t_0 <= 2.0) {
		tmp = (z / y) - -1.0;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - x) / (y - z)
	tmp = 0
	if t_0 <= -2e+38:
		tmp = x / -y
	elif t_0 <= -2e-60:
		tmp = x / z
	elif t_0 <= 1e-5:
		tmp = -y / z
	elif t_0 <= 2.0:
		tmp = (z / y) - -1.0
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - x) / Float64(y - z))
	tmp = 0.0
	if (t_0 <= -2e+38)
		tmp = Float64(x / Float64(-y));
	elseif (t_0 <= -2e-60)
		tmp = Float64(x / z);
	elseif (t_0 <= 1e-5)
		tmp = Float64(Float64(-y) / z);
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(z / y) - -1.0);
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - x) / (y - z);
	tmp = 0.0;
	if (t_0 <= -2e+38)
		tmp = x / -y;
	elseif (t_0 <= -2e-60)
		tmp = x / z;
	elseif (t_0 <= 1e-5)
		tmp = -y / z;
	elseif (t_0 <= 2.0)
		tmp = (z / y) - -1.0;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+38], N[(x / (-y)), $MachinePrecision], If[LessEqual[t$95$0, -2e-60], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 1e-5], N[((-y) / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(z / y), $MachinePrecision] - -1.0), $MachinePrecision], N[(x / z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.99999999999999995e38

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 69.0%

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

   if -1.99999999999999995e38 < (/.f64 (-.f64 x y) (-.f64 z y)) < -1.9999999999999999e-60 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if -1.9999999999999999e-60 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 67.8%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. distribute-lft-out-- N/A

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

      10. lower-/.f64 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. +-commutative N/A

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

      15. mul-1-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.9%

      \[\leadsto \frac{z}{y} - -1 \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 10^{-5}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\frac{z}{y} - -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))))
   (if (<= t_0 -2e+38)
     (/ x (- y))
     (if (<= t_0 -2e-60)
       (/ x z)
       (if (<= t_0 2e-13) (/ (- y) z) (if (<= t_0 2.0) 1.0 (/ x z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double tmp;
	if (t_0 <= -2e+38) {
		tmp = x / -y;
	} else if (t_0 <= -2e-60) {
		tmp = x / z;
	} else if (t_0 <= 2e-13) {
		tmp = -y / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (y - x) / (y - z)
    if (t_0 <= (-2d+38)) then
        tmp = x / -y
    else if (t_0 <= (-2d-60)) then
        tmp = x / z
    else if (t_0 <= 2d-13) then
        tmp = -y / z
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double tmp;
	if (t_0 <= -2e+38) {
		tmp = x / -y;
	} else if (t_0 <= -2e-60) {
		tmp = x / z;
	} else if (t_0 <= 2e-13) {
		tmp = -y / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - x) / (y - z)
	tmp = 0
	if t_0 <= -2e+38:
		tmp = x / -y
	elif t_0 <= -2e-60:
		tmp = x / z
	elif t_0 <= 2e-13:
		tmp = -y / z
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - x) / Float64(y - z))
	tmp = 0.0
	if (t_0 <= -2e+38)
		tmp = Float64(x / Float64(-y));
	elseif (t_0 <= -2e-60)
		tmp = Float64(x / z);
	elseif (t_0 <= 2e-13)
		tmp = Float64(Float64(-y) / z);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - x) / (y - z);
	tmp = 0.0;
	if (t_0 <= -2e+38)
		tmp = x / -y;
	elseif (t_0 <= -2e-60)
		tmp = x / z;
	elseif (t_0 <= 2e-13)
		tmp = -y / z;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+38], N[(x / (-y)), $MachinePrecision], If[LessEqual[t$95$0, -2e-60], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 2e-13], N[((-y) / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(x / z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.99999999999999995e38

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 69.0%

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

   if -1.99999999999999995e38 < (/.f64 (-.f64 x y) (-.f64 z y)) < -1.9999999999999999e-60 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if -1.9999999999999999e-60 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000001e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 69.5%

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

   if 2.0000000000000001e-13 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 96.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{z - x}{y} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))) (t_1 (/ x (- z y))))
   (if (<= t_0 -1000.0)
     t_1
     (if (<= t_0 1e-5)
       (/ (- x y) z)
       (if (<= t_0 2.0) (- (/ (- z x) y) -1.0) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = ((z - x) / y) - -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y - x) / (y - z)
    t_1 = x / (z - y)
    if (t_0 <= (-1000.0d0)) then
        tmp = t_1
    else if (t_0 <= 1d-5) then
        tmp = (x - y) / z
    else if (t_0 <= 2.0d0) then
        tmp = ((z - x) / y) - (-1.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = ((z - x) / y) - -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - x) / (y - z)
	t_1 = x / (z - y)
	tmp = 0
	if t_0 <= -1000.0:
		tmp = t_1
	elif t_0 <= 1e-5:
		tmp = (x - y) / z
	elif t_0 <= 2.0:
		tmp = ((z - x) / y) - -1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - x) / Float64(y - z))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = Float64(Float64(x - y) / z);
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(Float64(z - x) / y) - -1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - x) / (y - z);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = (x - y) / z;
	elseif (t_0 <= 2.0)
		tmp = ((z - x) / y) - -1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], t$95$1, If[LessEqual[t$95$0, 1e-5], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] - -1.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e3 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1e3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. distribute-lft-out-- N/A

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

      10. lower-/.f64 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. +-commutative N/A

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

      15. mul-1-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -1000:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 10^{-5}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\frac{z - x}{y} - -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y - x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))) (t_1 (/ x (- z y))))
   (if (<= t_0 -1000.0)
     t_1
     (if (<= t_0 1e-5) (/ (- x y) z) (if (<= t_0 2.0) (/ (- y x) y) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = (y - x) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y - x) / (y - z)
    t_1 = x / (z - y)
    if (t_0 <= (-1000.0d0)) then
        tmp = t_1
    else if (t_0 <= 1d-5) then
        tmp = (x - y) / z
    else if (t_0 <= 2.0d0) then
        tmp = (y - x) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = (y - x) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - x) / (y - z)
	t_1 = x / (z - y)
	tmp = 0
	if t_0 <= -1000.0:
		tmp = t_1
	elif t_0 <= 1e-5:
		tmp = (x - y) / z
	elif t_0 <= 2.0:
		tmp = (y - x) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - x) / Float64(y - z))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = Float64(Float64(x - y) / z);
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(y - x) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - x) / (y - z);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = (x - y) / z;
	elseif (t_0 <= 2.0)
		tmp = (y - x) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], t$95$1, If[LessEqual[t$95$0, 1e-5], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e3 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1e3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. unsub-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower--.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -1000:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 10^{-5}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\frac{y - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))) (t_1 (/ x (- z y))))
   (if (<= t_0 -1000.0)
     t_1
     (if (<= t_0 1e-5) (/ (- x y) z) (if (<= t_0 2.0) (/ y (- y z)) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = y / (y - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y - x) / (y - z)
    t_1 = x / (z - y)
    if (t_0 <= (-1000.0d0)) then
        tmp = t_1
    else if (t_0 <= 1d-5) then
        tmp = (x - y) / z
    else if (t_0 <= 2.0d0) then
        tmp = y / (y - z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = y / (y - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - x) / (y - z)
	t_1 = x / (z - y)
	tmp = 0
	if t_0 <= -1000.0:
		tmp = t_1
	elif t_0 <= 1e-5:
		tmp = (x - y) / z
	elif t_0 <= 2.0:
		tmp = y / (y - z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - x) / Float64(y - z))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = Float64(Float64(x - y) / z);
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(y - z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - x) / (y - z);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -1000.0)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = (x - y) / z;
	elseif (t_0 <= 2.0)
		tmp = y / (y - z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], t$95$1, If[LessEqual[t$95$0, 1e-5], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e3 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1e3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -1000:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 10^{-5}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{z}{y} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))) (t_1 (/ x (- z y))))
   (if (<= t_0 -2e-60)
     t_1
     (if (<= t_0 1e-5) (/ (- y) z) (if (<= t_0 2.0) (- (/ z y) -1.0) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -2e-60) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = -y / z;
	} else if (t_0 <= 2.0) {
		tmp = (z / y) - -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y - x) / (y - z)
    t_1 = x / (z - y)
    if (t_0 <= (-2d-60)) then
        tmp = t_1
    else if (t_0 <= 1d-5) then
        tmp = -y / z
    else if (t_0 <= 2.0d0) then
        tmp = (z / y) - (-1.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -2e-60) {
		tmp = t_1;
	} else if (t_0 <= 1e-5) {
		tmp = -y / z;
	} else if (t_0 <= 2.0) {
		tmp = (z / y) - -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - x) / (y - z)
	t_1 = x / (z - y)
	tmp = 0
	if t_0 <= -2e-60:
		tmp = t_1
	elif t_0 <= 1e-5:
		tmp = -y / z
	elif t_0 <= 2.0:
		tmp = (z / y) - -1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - x) / Float64(y - z))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -2e-60)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = Float64(Float64(-y) / z);
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(z / y) - -1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - x) / (y - z);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -2e-60)
		tmp = t_1;
	elseif (t_0 <= 1e-5)
		tmp = -y / z;
	elseif (t_0 <= 2.0)
		tmp = (z / y) - -1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-60], t$95$1, If[LessEqual[t$95$0, 1e-5], N[((-y) / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(z / y), $MachinePrecision] - -1.0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.9999999999999999e-60 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -1.9999999999999999e-60 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 67.8%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-out-- N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. distribute-lft-out-- N/A

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

      10. lower-/.f64 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. +-commutative N/A

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

      15. mul-1-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.9%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 10^{-5}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\frac{z}{y} - -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))))
   (if (<= t_0 -2e+38)
     (/ x (- y))
     (if (<= t_0 1e-5) (/ x z) (if (<= t_0 2.0) 1.0 (/ x z))))))

C

double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double tmp;
	if (t_0 <= -2e+38) {
		tmp = x / -y;
	} else if (t_0 <= 1e-5) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (y - x) / (y - z)
    if (t_0 <= (-2d+38)) then
        tmp = x / -y
    else if (t_0 <= 1d-5) then
        tmp = x / z
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double tmp;
	if (t_0 <= -2e+38) {
		tmp = x / -y;
	} else if (t_0 <= 1e-5) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - x) / (y - z)
	tmp = 0
	if t_0 <= -2e+38:
		tmp = x / -y
	elif t_0 <= 1e-5:
		tmp = x / z
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - x) / Float64(y - z))
	tmp = 0.0
	if (t_0 <= -2e+38)
		tmp = Float64(x / Float64(-y));
	elseif (t_0 <= 1e-5)
		tmp = Float64(x / z);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - x) / (y - z);
	tmp = 0.0;
	if (t_0 <= -2e+38)
		tmp = x / -y;
	elseif (t_0 <= 1e-5)
		tmp = x / z;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+38], N[(x / (-y)), $MachinePrecision], If[LessEqual[t$95$0, 1e-5], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(x / z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.99999999999999995e38

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 69.0%

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

   if -1.99999999999999995e38 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 57.4

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

   5. Applied rewrites 57.4%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 98.5%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))) (t_1 (/ x (- z y))))
   (if (<= t_0 -2e-60) t_1 (if (<= t_0 2.0) (/ y (- y z)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -2e-60) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = y / (y - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y - x) / (y - z)
    t_1 = x / (z - y)
    if (t_0 <= (-2d-60)) then
        tmp = t_1
    else if (t_0 <= 2.0d0) then
        tmp = y / (y - z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -2e-60) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = y / (y - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - x) / (y - z)
	t_1 = x / (z - y)
	tmp = 0
	if t_0 <= -2e-60:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = y / (y - z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - x) / Float64(y - z))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -2e-60)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(y - z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - x) / (y - z);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -2e-60)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = y / (y - z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-60], t$95$1, If[LessEqual[t$95$0, 2.0], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.9999999999999999e-60 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -1.9999999999999999e-60 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. sub-neg N/A

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

      9. lower--.f64 84.0

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

   5. Applied rewrites 84.0%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ \mathbf{if}\;t\_0 \leq 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))))
   (if (<= t_0 1e-5) (/ x z) (if (<= t_0 2.0) 1.0 (/ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (y - x) / (y - z)
    if (t_0 <= 1d-5) then
        tmp = x / z
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y - x) / (y - z);
	double tmp;
	if (t_0 <= 1e-5) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y - x) / (y - z)
	tmp = 0
	if t_0 <= 1e-5:
		tmp = x / z
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y - x) / Float64(y - z))
	tmp = 0.0
	if (t_0 <= 1e-5)
		tmp = Float64(x / z);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y - x) / (y - z);
	tmp = 0.0;
	if (t_0 <= 1e-5)
		tmp = x / z;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-5], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(x / z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 54.3

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

   5. Applied rewrites 54.3%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 98.5%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 12: 35.9% accurate, 18.0× 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. Add Preprocessing

3. Taylor expanded in y around inf

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

5. Applied rewrites 34.0%

   \[\leadsto \color{blue}{1} \]
6. 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 2024249 
(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)))