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

Percentage Accurate: 100.0% → 100.0%

Time: 8.8s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 68.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{y}\\ t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{y}{-z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -10000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-135}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_1 \leq 0.0005:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1 + \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ x y))) (t_1 (/ (- x y) (- z y))) (t_2 (/ y (- z))))
   (if (<= t_1 -5e+119)
     t_0
     (if (<= t_1 -10000000.0)
       (/ x z)
       (if (<= t_1 -1e-135)
         t_2
         (if (<= t_1 1e-178)
           (/ x z)
           (if (<= t_1 0.0005) t_2 (if (<= t_1 2.0) (+ 1.0 (/ z y)) t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = -(x / y);
	double t_1 = (x - y) / (z - y);
	double t_2 = y / -z;
	double tmp;
	if (t_1 <= -5e+119) {
		tmp = t_0;
	} else if (t_1 <= -10000000.0) {
		tmp = x / z;
	} else if (t_1 <= -1e-135) {
		tmp = t_2;
	} else if (t_1 <= 1e-178) {
		tmp = x / z;
	} else if (t_1 <= 0.0005) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = 1.0 + (z / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = -(x / y)
    t_1 = (x - y) / (z - y)
    t_2 = y / -z
    if (t_1 <= (-5d+119)) then
        tmp = t_0
    else if (t_1 <= (-10000000.0d0)) then
        tmp = x / z
    else if (t_1 <= (-1d-135)) then
        tmp = t_2
    else if (t_1 <= 1d-178) then
        tmp = x / z
    else if (t_1 <= 0.0005d0) then
        tmp = t_2
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0 + (z / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -(x / y);
	double t_1 = (x - y) / (z - y);
	double t_2 = y / -z;
	double tmp;
	if (t_1 <= -5e+119) {
		tmp = t_0;
	} else if (t_1 <= -10000000.0) {
		tmp = x / z;
	} else if (t_1 <= -1e-135) {
		tmp = t_2;
	} else if (t_1 <= 1e-178) {
		tmp = x / z;
	} else if (t_1 <= 0.0005) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = 1.0 + (z / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(x / y)
	t_1 = (x - y) / (z - y)
	t_2 = y / -z
	tmp = 0
	if t_1 <= -5e+119:
		tmp = t_0
	elif t_1 <= -10000000.0:
		tmp = x / z
	elif t_1 <= -1e-135:
		tmp = t_2
	elif t_1 <= 1e-178:
		tmp = x / z
	elif t_1 <= 0.0005:
		tmp = t_2
	elif t_1 <= 2.0:
		tmp = 1.0 + (z / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(x / y))
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(y / Float64(-z))
	tmp = 0.0
	if (t_1 <= -5e+119)
		tmp = t_0;
	elseif (t_1 <= -10000000.0)
		tmp = Float64(x / z);
	elseif (t_1 <= -1e-135)
		tmp = t_2;
	elseif (t_1 <= 1e-178)
		tmp = Float64(x / z);
	elseif (t_1 <= 0.0005)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = Float64(1.0 + Float64(z / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(x / y);
	t_1 = (x - y) / (z - y);
	t_2 = y / -z;
	tmp = 0.0;
	if (t_1 <= -5e+119)
		tmp = t_0;
	elseif (t_1 <= -10000000.0)
		tmp = x / z;
	elseif (t_1 <= -1e-135)
		tmp = t_2;
	elseif (t_1 <= 1e-178)
		tmp = x / z;
	elseif (t_1 <= 0.0005)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = 1.0 + (z / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(x / y), $MachinePrecision])}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y / (-z)), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+119], t$95$0, If[LessEqual[t$95$1, -10000000.0], N[(x / z), $MachinePrecision], If[LessEqual[t$95$1, -1e-135], t$95$2, If[LessEqual[t$95$1, 1e-178], N[(x / z), $MachinePrecision], If[LessEqual[t$95$1, 0.0005], t$95$2, If[LessEqual[t$95$1, 2.0], N[(1.0 + N[(z / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.9999999999999999e119 or 2 < (/.f64 (-.f64 x y) (-.f64 z 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. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 65.7

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

   5. Applied rewrites 65.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 64.5%

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

   if -4.9999999999999999e119 < (/.f64 (-.f64 x y) (-.f64 z y)) < -1e7 or -1e-135 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-179

   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 75.8

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

   5. Applied rewrites 75.8%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < -1e-135 or 9.9999999999999995e-179 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.0000000000000001e-4

   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 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.3%

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

   if 5.0000000000000001e-4 < (/.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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 97.5%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{+119}:\\ \;\;\;\;-\frac{x}{y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq -10000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq -1 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.0005:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;1 + \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{y}\\ t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{y}{-z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -10000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-135}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_1 \leq 0.0005:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (/ x y))) (t_1 (/ (- x y) (- z y))) (t_2 (/ y (- z))))
   (if (<= t_1 -5e+119)
     t_0
     (if (<= t_1 -10000000.0)
       (/ x z)
       (if (<= t_1 -1e-135)
         t_2
         (if (<= t_1 1e-178)
           (/ x z)
           (if (<= t_1 0.0005) t_2 (if (<= t_1 2.0) 1.0 t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = -(x / y);
	double t_1 = (x - y) / (z - y);
	double t_2 = y / -z;
	double tmp;
	if (t_1 <= -5e+119) {
		tmp = t_0;
	} else if (t_1 <= -10000000.0) {
		tmp = x / z;
	} else if (t_1 <= -1e-135) {
		tmp = t_2;
	} else if (t_1 <= 1e-178) {
		tmp = x / z;
	} else if (t_1 <= 0.0005) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = -(x / y)
    t_1 = (x - y) / (z - y)
    t_2 = y / -z
    if (t_1 <= (-5d+119)) then
        tmp = t_0
    else if (t_1 <= (-10000000.0d0)) then
        tmp = x / z
    else if (t_1 <= (-1d-135)) then
        tmp = t_2
    else if (t_1 <= 1d-178) then
        tmp = x / z
    else if (t_1 <= 0.0005d0) then
        tmp = t_2
    else if (t_1 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -(x / y);
	double t_1 = (x - y) / (z - y);
	double t_2 = y / -z;
	double tmp;
	if (t_1 <= -5e+119) {
		tmp = t_0;
	} else if (t_1 <= -10000000.0) {
		tmp = x / z;
	} else if (t_1 <= -1e-135) {
		tmp = t_2;
	} else if (t_1 <= 1e-178) {
		tmp = x / z;
	} else if (t_1 <= 0.0005) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(x / y)
	t_1 = (x - y) / (z - y)
	t_2 = y / -z
	tmp = 0
	if t_1 <= -5e+119:
		tmp = t_0
	elif t_1 <= -10000000.0:
		tmp = x / z
	elif t_1 <= -1e-135:
		tmp = t_2
	elif t_1 <= 1e-178:
		tmp = x / z
	elif t_1 <= 0.0005:
		tmp = t_2
	elif t_1 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(x / y))
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(y / Float64(-z))
	tmp = 0.0
	if (t_1 <= -5e+119)
		tmp = t_0;
	elseif (t_1 <= -10000000.0)
		tmp = Float64(x / z);
	elseif (t_1 <= -1e-135)
		tmp = t_2;
	elseif (t_1 <= 1e-178)
		tmp = Float64(x / z);
	elseif (t_1 <= 0.0005)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(x / y);
	t_1 = (x - y) / (z - y);
	t_2 = y / -z;
	tmp = 0.0;
	if (t_1 <= -5e+119)
		tmp = t_0;
	elseif (t_1 <= -10000000.0)
		tmp = x / z;
	elseif (t_1 <= -1e-135)
		tmp = t_2;
	elseif (t_1 <= 1e-178)
		tmp = x / z;
	elseif (t_1 <= 0.0005)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(x / y), $MachinePrecision])}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y / (-z)), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+119], t$95$0, If[LessEqual[t$95$1, -10000000.0], N[(x / z), $MachinePrecision], If[LessEqual[t$95$1, -1e-135], t$95$2, If[LessEqual[t$95$1, 1e-178], N[(x / z), $MachinePrecision], If[LessEqual[t$95$1, 0.0005], t$95$2, If[LessEqual[t$95$1, 2.0], 1.0, t$95$0]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.9999999999999999e119 or 2 < (/.f64 (-.f64 x y) (-.f64 z 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. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 65.7

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

   5. Applied rewrites 65.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 64.5%

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

   if -4.9999999999999999e119 < (/.f64 (-.f64 x y) (-.f64 z y)) < -1e7 or -1e-135 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-179

   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 75.8

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

   5. Applied rewrites 75.8%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < -1e-135 or 9.9999999999999995e-179 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.0000000000000001e-4

   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 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.3%

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

   if 5.0000000000000001e-4 < (/.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 97.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{+119}:\\ \;\;\;\;-\frac{x}{y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq -10000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq -1 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.0005:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{-z}\\ t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z - y}\\ \mathbf{if}\;t\_1 \leq -10000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-135}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_1 \leq 0.0005:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1.5:\\ \;\;\;\;1 + \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (- z))) (t_1 (/ (- x y) (- z y))) (t_2 (/ x (- z y))))
   (if (<= t_1 -10000000.0)
     t_2
     (if (<= t_1 -1e-135)
       t_0
       (if (<= t_1 1e-178)
         (/ x z)
         (if (<= t_1 0.0005) t_0 (if (<= t_1 1.5) (+ 1.0 (/ z y)) t_2)))))))

C

double code(double x, double y, double z) {
	double t_0 = y / -z;
	double t_1 = (x - y) / (z - y);
	double t_2 = x / (z - y);
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = t_2;
	} else if (t_1 <= -1e-135) {
		tmp = t_0;
	} else if (t_1 <= 1e-178) {
		tmp = x / z;
	} else if (t_1 <= 0.0005) {
		tmp = t_0;
	} else if (t_1 <= 1.5) {
		tmp = 1.0 + (z / y);
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = y / -z
    t_1 = (x - y) / (z - y)
    t_2 = x / (z - y)
    if (t_1 <= (-10000000.0d0)) then
        tmp = t_2
    else if (t_1 <= (-1d-135)) then
        tmp = t_0
    else if (t_1 <= 1d-178) then
        tmp = x / z
    else if (t_1 <= 0.0005d0) then
        tmp = t_0
    else if (t_1 <= 1.5d0) then
        tmp = 1.0d0 + (z / y)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / -z;
	double t_1 = (x - y) / (z - y);
	double t_2 = x / (z - y);
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = t_2;
	} else if (t_1 <= -1e-135) {
		tmp = t_0;
	} else if (t_1 <= 1e-178) {
		tmp = x / z;
	} else if (t_1 <= 0.0005) {
		tmp = t_0;
	} else if (t_1 <= 1.5) {
		tmp = 1.0 + (z / y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / -z
	t_1 = (x - y) / (z - y)
	t_2 = x / (z - y)
	tmp = 0
	if t_1 <= -10000000.0:
		tmp = t_2
	elif t_1 <= -1e-135:
		tmp = t_0
	elif t_1 <= 1e-178:
		tmp = x / z
	elif t_1 <= 0.0005:
		tmp = t_0
	elif t_1 <= 1.5:
		tmp = 1.0 + (z / y)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(-z))
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -10000000.0)
		tmp = t_2;
	elseif (t_1 <= -1e-135)
		tmp = t_0;
	elseif (t_1 <= 1e-178)
		tmp = Float64(x / z);
	elseif (t_1 <= 0.0005)
		tmp = t_0;
	elseif (t_1 <= 1.5)
		tmp = Float64(1.0 + Float64(z / y));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / -z;
	t_1 = (x - y) / (z - y);
	t_2 = x / (z - y);
	tmp = 0.0;
	if (t_1 <= -10000000.0)
		tmp = t_2;
	elseif (t_1 <= -1e-135)
		tmp = t_0;
	elseif (t_1 <= 1e-178)
		tmp = x / z;
	elseif (t_1 <= 0.0005)
		tmp = t_0;
	elseif (t_1 <= 1.5)
		tmp = 1.0 + (z / y);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / (-z)), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -10000000.0], t$95$2, If[LessEqual[t$95$1, -1e-135], t$95$0, If[LessEqual[t$95$1, 1e-178], N[(x / z), $MachinePrecision], If[LessEqual[t$95$1, 0.0005], t$95$0, If[LessEqual[t$95$1, 1.5], N[(1.0 + N[(z / y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e7 or 1.5 < (/.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 98.6

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

   5. Applied rewrites 98.6%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < -1e-135 or 9.9999999999999995e-179 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.0000000000000001e-4

   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 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.3%

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

   if -1e-135 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-179

   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 77.1

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

   5. Applied rewrites 77.1%

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

   if 5.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.5

   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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 98.7%

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

Alternative 5: 68.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ t_1 := \frac{y}{-z}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+119}:\\ \;\;\;\;-\frac{x}{y}\\ \mathbf{elif}\;t\_0 \leq -10000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 0.0005:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))) (t_1 (/ y (- z))))
   (if (<= t_0 -5e+119)
     (- (/ x y))
     (if (<= t_0 -10000000.0)
       (/ x z)
       (if (<= t_0 -1e-135)
         t_1
         (if (<= t_0 1e-178)
           (/ x z)
           (if (<= t_0 0.0005) t_1 (- 1.0 (/ x y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = y / -z;
	double tmp;
	if (t_0 <= -5e+119) {
		tmp = -(x / y);
	} else if (t_0 <= -10000000.0) {
		tmp = x / z;
	} else if (t_0 <= -1e-135) {
		tmp = t_1;
	} else if (t_0 <= 1e-178) {
		tmp = x / z;
	} else if (t_0 <= 0.0005) {
		tmp = t_1;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x - y) / (z - y)
    t_1 = y / -z
    if (t_0 <= (-5d+119)) then
        tmp = -(x / y)
    else if (t_0 <= (-10000000.0d0)) then
        tmp = x / z
    else if (t_0 <= (-1d-135)) then
        tmp = t_1
    else if (t_0 <= 1d-178) then
        tmp = x / z
    else if (t_0 <= 0.0005d0) then
        tmp = t_1
    else
        tmp = 1.0d0 - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = y / -z;
	double tmp;
	if (t_0 <= -5e+119) {
		tmp = -(x / y);
	} else if (t_0 <= -10000000.0) {
		tmp = x / z;
	} else if (t_0 <= -1e-135) {
		tmp = t_1;
	} else if (t_0 <= 1e-178) {
		tmp = x / z;
	} else if (t_0 <= 0.0005) {
		tmp = t_1;
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	t_1 = y / -z
	tmp = 0
	if t_0 <= -5e+119:
		tmp = -(x / y)
	elif t_0 <= -10000000.0:
		tmp = x / z
	elif t_0 <= -1e-135:
		tmp = t_1
	elif t_0 <= 1e-178:
		tmp = x / z
	elif t_0 <= 0.0005:
		tmp = t_1
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	t_1 = Float64(y / Float64(-z))
	tmp = 0.0
	if (t_0 <= -5e+119)
		tmp = Float64(-Float64(x / y));
	elseif (t_0 <= -10000000.0)
		tmp = Float64(x / z);
	elseif (t_0 <= -1e-135)
		tmp = t_1;
	elseif (t_0 <= 1e-178)
		tmp = Float64(x / z);
	elseif (t_0 <= 0.0005)
		tmp = t_1;
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	t_1 = y / -z;
	tmp = 0.0;
	if (t_0 <= -5e+119)
		tmp = -(x / y);
	elseif (t_0 <= -10000000.0)
		tmp = x / z;
	elseif (t_0 <= -1e-135)
		tmp = t_1;
	elseif (t_0 <= 1e-178)
		tmp = x / z;
	elseif (t_0 <= 0.0005)
		tmp = t_1;
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / (-z)), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+119], (-N[(x / y), $MachinePrecision]), If[LessEqual[t$95$0, -10000000.0], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, -1e-135], t$95$1, If[LessEqual[t$95$0, 1e-178], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 0.0005], t$95$1, N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.9999999999999999e119

   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. 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. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.2%

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

   if -4.9999999999999999e119 < (/.f64 (-.f64 x y) (-.f64 z y)) < -1e7 or -1e-135 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-179

   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 75.8

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

   5. Applied rewrites 75.8%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < -1e-135 or 9.9999999999999995e-179 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.0000000000000001e-4

   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 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.3%

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

   if 5.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.9%

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

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \frac{x - y}{y}} \]

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

         \[\leadsto 0 - \left(\frac{x}{y} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto 0 - \left(\frac{x}{y} + \color{blue}{-1}\right) \]

      7. +-commutative N/A

         \[\leadsto 0 - \color{blue}{\left(-1 + \frac{x}{y}\right)} \]

      8. associate--r+ N/A

         \[\leadsto \color{blue}{\left(0 - -1\right) - \frac{x}{y}} \]

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{+119}:\\ \;\;\;\;-\frac{x}{y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq -10000000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq -1 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.0005:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y - z}\\ t_1 := \frac{x - y}{z - y}\\ t_2 := \frac{x}{z - y}\\ \mathbf{if}\;t\_1 \leq -10000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-135}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_1 \leq 1.5:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (- y z))) (t_1 (/ (- x y) (- z y))) (t_2 (/ x (- z y))))
   (if (<= t_1 -10000000.0)
     t_2
     (if (<= t_1 -1e-135)
       t_0
       (if (<= t_1 1e-178) (/ x z) (if (<= t_1 1.5) t_0 t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double t_1 = (x - y) / (z - y);
	double t_2 = x / (z - y);
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = t_2;
	} else if (t_1 <= -1e-135) {
		tmp = t_0;
	} else if (t_1 <= 1e-178) {
		tmp = x / z;
	} else if (t_1 <= 1.5) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = y / (y - z)
    t_1 = (x - y) / (z - y)
    t_2 = x / (z - y)
    if (t_1 <= (-10000000.0d0)) then
        tmp = t_2
    else if (t_1 <= (-1d-135)) then
        tmp = t_0
    else if (t_1 <= 1d-178) then
        tmp = x / z
    else if (t_1 <= 1.5d0) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double t_1 = (x - y) / (z - y);
	double t_2 = x / (z - y);
	double tmp;
	if (t_1 <= -10000000.0) {
		tmp = t_2;
	} else if (t_1 <= -1e-135) {
		tmp = t_0;
	} else if (t_1 <= 1e-178) {
		tmp = x / z;
	} else if (t_1 <= 1.5) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (y - z)
	t_1 = (x - y) / (z - y)
	t_2 = x / (z - y)
	tmp = 0
	if t_1 <= -10000000.0:
		tmp = t_2
	elif t_1 <= -1e-135:
		tmp = t_0
	elif t_1 <= 1e-178:
		tmp = x / z
	elif t_1 <= 1.5:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(y - z))
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	t_2 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -10000000.0)
		tmp = t_2;
	elseif (t_1 <= -1e-135)
		tmp = t_0;
	elseif (t_1 <= 1e-178)
		tmp = Float64(x / z);
	elseif (t_1 <= 1.5)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (y - z);
	t_1 = (x - y) / (z - y);
	t_2 = x / (z - y);
	tmp = 0.0;
	if (t_1 <= -10000000.0)
		tmp = t_2;
	elseif (t_1 <= -1e-135)
		tmp = t_0;
	elseif (t_1 <= 1e-178)
		tmp = x / z;
	elseif (t_1 <= 1.5)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -10000000.0], t$95$2, If[LessEqual[t$95$1, -1e-135], t$95$0, If[LessEqual[t$95$1, 1e-178], N[(x / z), $MachinePrecision], If[LessEqual[t$95$1, 1.5], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e7 or 1.5 < (/.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 98.6

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

   5. Applied rewrites 98.6%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < -1e-135 or 9.9999999999999995e-179 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.5

   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 88.6

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

   5. Applied rewrites 88.6%

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

   if -1e-135 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999995e-179

   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 77.1

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

   5. Applied rewrites 77.1%

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

Alternative 7: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -10000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-23}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;t\_0 \leq 1.5:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))) (t_1 (/ x (- z y))))
   (if (<= t_0 -10000000.0)
     t_1
     (if (<= t_0 1e-23) (/ (- x y) z) (if (<= t_0 1.5) (/ y (- y z)) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -10000000.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-23) {
		tmp = (x - y) / z;
	} else if (t_0 <= 1.5) {
		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 = (x - y) / (z - y)
    t_1 = x / (z - y)
    if (t_0 <= (-10000000.0d0)) then
        tmp = t_1
    else if (t_0 <= 1d-23) then
        tmp = (x - y) / z
    else if (t_0 <= 1.5d0) 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 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -10000000.0) {
		tmp = t_1;
	} else if (t_0 <= 1e-23) {
		tmp = (x - y) / z;
	} else if (t_0 <= 1.5) {
		tmp = y / (y - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	t_1 = x / (z - y)
	tmp = 0
	if t_0 <= -10000000.0:
		tmp = t_1
	elif t_0 <= 1e-23:
		tmp = (x - y) / z
	elif t_0 <= 1.5:
		tmp = y / (y - z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -10000000.0)
		tmp = t_1;
	elseif (t_0 <= 1e-23)
		tmp = Float64(Float64(x - y) / z);
	elseif (t_0 <= 1.5)
		tmp = Float64(y / Float64(y - z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -10000000.0)
		tmp = t_1;
	elseif (t_0 <= 1e-23)
		tmp = (x - y) / z;
	elseif (t_0 <= 1.5)
		tmp = y / (y - z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10000000.0], t$95$1, If[LessEqual[t$95$0, 1e-23], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 1.5], 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)) < -1e7 or 1.5 < (/.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 98.6

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

   5. Applied rewrites 98.6%

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

   if -1e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999996e-24

   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 99.2

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

   5. Applied rewrites 99.2%

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

   if 9.9999999999999996e-24 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.5

   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 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 8: 69.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ t_1 := -\frac{x}{y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-23}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))) (t_1 (- (/ x y))))
   (if (<= t_0 -5e+119)
     t_1
     (if (<= t_0 1e-23) (/ x z) (if (<= t_0 2.0) 1.0 t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = -(x / y);
	double tmp;
	if (t_0 <= -5e+119) {
		tmp = t_1;
	} else if (t_0 <= 1e-23) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 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 = (x - y) / (z - y)
    t_1 = -(x / y)
    if (t_0 <= (-5d+119)) then
        tmp = t_1
    else if (t_0 <= 1d-23) then
        tmp = x / z
    else if (t_0 <= 2.0d0) then
        tmp = 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 = (x - y) / (z - y);
	double t_1 = -(x / y);
	double tmp;
	if (t_0 <= -5e+119) {
		tmp = t_1;
	} else if (t_0 <= 1e-23) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	t_1 = -(x / y)
	tmp = 0
	if t_0 <= -5e+119:
		tmp = t_1
	elif t_0 <= 1e-23:
		tmp = x / z
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	t_1 = Float64(-Float64(x / y))
	tmp = 0.0
	if (t_0 <= -5e+119)
		tmp = t_1;
	elseif (t_0 <= 1e-23)
		tmp = Float64(x / z);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	t_1 = -(x / y);
	tmp = 0.0;
	if (t_0 <= -5e+119)
		tmp = t_1;
	elseif (t_0 <= 1e-23)
		tmp = x / z;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x / y), $MachinePrecision])}, If[LessEqual[t$95$0, -5e+119], t$95$1, If[LessEqual[t$95$0, 1e-23], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.9999999999999999e119 or 2 < (/.f64 (-.f64 x y) (-.f64 z 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. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 65.7

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

   5. Applied rewrites 65.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 64.5%

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

   if -4.9999999999999999e119 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999996e-24

   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 59.9

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

   5. Applied rewrites 59.9%

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

   if 9.9999999999999996e-24 < (/.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 93.9%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{+119}:\\ \;\;\;\;-\frac{x}{y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-23}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq 10^{-23}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 1.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (<= t_0 1e-23) (/ x z) (if (<= t_0 1.5) 1.0 (/ x z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= 1e-23) {
		tmp = x / z;
	} else if (t_0 <= 1.5) {
		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 = (x - y) / (z - y)
    if (t_0 <= 1d-23) then
        tmp = x / z
    else if (t_0 <= 1.5d0) 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 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= 1e-23) {
		tmp = x / z;
	} else if (t_0 <= 1.5) {
		tmp = 1.0;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	tmp = 0
	if t_0 <= 1e-23:
		tmp = x / z
	elif t_0 <= 1.5:
		tmp = 1.0
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_0 <= 1e-23)
		tmp = Float64(x / z);
	elseif (t_0 <= 1.5)
		tmp = 1.0;
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_0 <= 1e-23)
		tmp = x / z;
	elseif (t_0 <= 1.5)
		tmp = 1.0;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-23], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 1.5], 1.0, N[(x / z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999996e-24 or 1.5 < (/.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 53.8

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

   5. Applied rewrites 53.8%

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

   if 9.9999999999999996e-24 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.5

   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 94.9%

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

Alternative 10: 35.2% 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 32.4%

   \[\leadsto \color{blue}{1} \]
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 2024219 
(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)))