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

Percentage Accurate: 100.0% → 100.0%

Time: 8.2s

Alternatives: 10

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: 97.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{z - x}{y} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))) (t_1 (/ x (- z y))))
   (if (<= t_0 -1e+17)
     t_1
     (if (<= t_0 5e-6)
       (/ (- 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 <= -1e+17) {
		tmp = t_1;
	} else if (t_0 <= 5e-6) {
		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 <= (-1d+17)) then
        tmp = t_1
    else if (t_0 <= 5d-6) 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 <= -1e+17) {
		tmp = t_1;
	} else if (t_0 <= 5e-6) {
		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 <= -1e+17:
		tmp = t_1
	elif t_0 <= 5e-6:
		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 <= -1e+17)
		tmp = t_1;
	elseif (t_0 <= 5e-6)
		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 <= -1e+17)
		tmp = t_1;
	elseif (t_0 <= 5e-6)
		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, -1e+17], t$95$1, If[LessEqual[t$95$0, 5e-6], 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)) < -1e17 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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1e17 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000041e-6

   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 5.00000000000000041e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   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. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

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

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

      15. lower-/.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. +-commutative N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\frac{z - x}{y} - -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))) (t_1 (/ x (- z y))))
   (if (<= t_0 -1e+17)
     t_1
     (if (<= t_0 5e-6) (/ (- 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 <= -1e+17) {
		tmp = t_1;
	} else if (t_0 <= 5e-6) {
		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 <= (-1d+17)) then
        tmp = t_1
    else if (t_0 <= 5d-6) 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 <= -1e+17) {
		tmp = t_1;
	} else if (t_0 <= 5e-6) {
		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 <= -1e+17:
		tmp = t_1
	elif t_0 <= 5e-6:
		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 <= -1e+17)
		tmp = t_1;
	elseif (t_0 <= 5e-6)
		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 <= -1e+17)
		tmp = t_1;
	elseif (t_0 <= 5e-6)
		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, -1e+17], t$95$1, If[LessEqual[t$95$0, 5e-6], 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)) < -1e17 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 100.0

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

   5. Applied rewrites 100.0%

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

   if -1e17 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000041e-6

   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 5.00000000000000041e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -1 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+66}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))))
   (if (<= t_0 -2e+66)
     (/ x (- y))
     (if (<= t_0 5e-6) (/ 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+66) {
		tmp = x / -y;
	} else if (t_0 <= 5e-6) {
		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+66)) then
        tmp = x / -y
    else if (t_0 <= 5d-6) 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+66) {
		tmp = x / -y;
	} else if (t_0 <= 5e-6) {
		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+66:
		tmp = x / -y
	elif t_0 <= 5e-6:
		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+66)
		tmp = Float64(x / Float64(-y));
	elseif (t_0 <= 5e-6)
		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+66)
		tmp = x / -y;
	elseif (t_0 <= 5e-6)
		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+66], N[(x / (-y)), $MachinePrecision], If[LessEqual[t$95$0, 5e-6], 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)) < -1.99999999999999989e66

   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 65.9%

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

   if -1.99999999999999989e66 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000041e-6 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 57.7

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

   5. Applied rewrites 57.7%

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

   if 5.00000000000000041e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 97.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq -2 \cdot 10^{+66}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))) (t_1 (/ x (- z y))))
   (if (<= t_0 5e-40) 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 <= 5e-40) {
		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 <= 5d-40) 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 <= 5e-40) {
		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 <= 5e-40:
		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 <= 5e-40)
		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 <= 5e-40)
		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, 5e-40], 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)) < 4.99999999999999965e-40 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 77.1

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

   5. Applied rewrites 77.1%

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

   if 4.99999999999999965e-40 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   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 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))) (t_1 (/ x (- z y))))
   (if (<= t_0 5e-6) t_1 (if (<= t_0 2.0) (- 1.0 (/ 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 <= 5e-6) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (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 <= 5d-6) then
        tmp = t_1
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 - (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 <= 5e-6) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - (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 <= 5e-6:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = 1.0 - (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 <= 5e-6)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - Float64(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 <= 5e-6)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = 1.0 - (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, 5e-6], t$95$1, If[LessEqual[t$95$0, 2.0], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000041e-6 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 75.5

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

   5. Applied rewrites 75.5%

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

   if 5.00000000000000041e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   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. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

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

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

      15. lower-/.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. +-commutative N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 98.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - z}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y z))))
   (if (<= t_0 5e-6) (/ 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 <= 5e-6) {
		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 <= 5d-6) 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 <= 5e-6) {
		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 <= 5e-6:
		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 <= 5e-6)
		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 <= 5e-6)
		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, 5e-6], 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)) < 5.00000000000000041e-6 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 55.5

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

   5. Applied rewrites 55.5%

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

   if 5.00000000000000041e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 97.8%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - z} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{y - x}{y - z} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -9.2e+24) t_0 (if (<= y 1.9e-7) (/ x z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -9.2e+24) {
		tmp = t_0;
	} else if (y <= 1.9e-7) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    if (y <= (-9.2d+24)) then
        tmp = t_0
    else if (y <= 1.9d-7) then
        tmp = x / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -9.2e+24) {
		tmp = t_0;
	} else if (y <= 1.9e-7) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -9.2e+24:
		tmp = t_0
	elif y <= 1.9e-7:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -9.2e+24)
		tmp = t_0;
	elseif (y <= 1.9e-7)
		tmp = Float64(x / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -9.2e+24)
		tmp = t_0;
	elseif (y <= 1.9e-7)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.2e+24], t$95$0, If[LessEqual[y, 1.9e-7], N[(x / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.1999999999999996e24 or 1.90000000000000007e-7 < y

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   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. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate--r+ N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

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

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

      15. lower-/.f64 N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lft-identity N/A

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

      19. +-commutative N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 78.5

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.1%

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

   if -9.1999999999999996e24 < y < 1.90000000000000007e-7

   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 65.4

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

   5. Applied rewrites 65.4%

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

Alternative 9: 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 10: 34.9% 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 37.4%

   \[\leadsto \color{blue}{1} \]
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 2024276 
(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)))