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

Percentage Accurate: 100.0% → 100.0%

Time: 10.6s

Alternatives: 9

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 99.9%

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

Alternative 2: 69.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ t_1 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_1 \leq -1000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;-\frac{y}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))) (t_1 (/ (- x y) (- z y))))
   (if (<= t_1 -5e+178)
     (/ x z)
     (if (<= t_1 -1000000000.0)
       t_0
       (if (<= t_1 0.1) (- (/ y z)) (if (<= t_1 1e+32) t_0 (/ x z)))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= -5e+178) {
		tmp = x / z;
	} else if (t_1 <= -1000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.1) {
		tmp = -(y / z);
	} else if (t_1 <= 1e+32) {
		tmp = t_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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    t_1 = (x - y) / (z - y)
    if (t_1 <= (-5d+178)) then
        tmp = x / z
    else if (t_1 <= (-1000000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 0.1d0) then
        tmp = -(y / z)
    else if (t_1 <= 1d+32) then
        tmp = t_0
    else
        tmp = x / z
    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 t_1 = (x - y) / (z - y);
	double tmp;
	if (t_1 <= -5e+178) {
		tmp = x / z;
	} else if (t_1 <= -1000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.1) {
		tmp = -(y / z);
	} else if (t_1 <= 1e+32) {
		tmp = t_0;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	t_1 = (x - y) / (z - y)
	tmp = 0
	if t_1 <= -5e+178:
		tmp = x / z
	elif t_1 <= -1000000000.0:
		tmp = t_0
	elif t_1 <= 0.1:
		tmp = -(y / z)
	elif t_1 <= 1e+32:
		tmp = t_0
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(x / y))
	t_1 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_1 <= -5e+178)
		tmp = Float64(x / z);
	elseif (t_1 <= -1000000000.0)
		tmp = t_0;
	elseif (t_1 <= 0.1)
		tmp = Float64(-Float64(y / z));
	elseif (t_1 <= 1e+32)
		tmp = t_0;
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (x / y);
	t_1 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_1 <= -5e+178)
		tmp = x / z;
	elseif (t_1 <= -1000000000.0)
		tmp = t_0;
	elseif (t_1 <= 0.1)
		tmp = -(y / z);
	elseif (t_1 <= 1e+32)
		tmp = t_0;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+178], N[(x / z), $MachinePrecision], If[LessEqual[t$95$1, -1000000000.0], t$95$0, If[LessEqual[t$95$1, 0.1], (-N[(y / z), $MachinePrecision]), If[LessEqual[t$95$1, 1e+32], t$95$0, N[(x / z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.9999999999999999e178 or 1.00000000000000005e32 < (/.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. /-lowering-/.f64 70.6

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

   5. Simplified 70.6%

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

   if -4.9999999999999999e178 < (/.f64 (-.f64 x y) (-.f64 z y)) < -1e9 or 0.10000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000005e32

   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. --lowering--.f64 N/A

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

      11. /-lowering-/.f64 91.9

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

   5. Simplified 91.9%

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

   if -1e9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.10000000000000001

   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. /-lowering-/.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. --lowering--.f64 57.9

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

   5. Simplified 57.9%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 56.4

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

   8. Simplified 56.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{+178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq -1000000000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.1:\\ \;\;\;\;-\frac{y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{+32}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq -1000000000:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;t\_0 \leq 10^{-11}:\\ \;\;\;\;-\frac{y}{z}\\ \mathbf{elif}\;t\_0 \leq 200000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (<= t_0 -5e+178)
     (/ x z)
     (if (<= t_0 -1000000000.0)
       (/ x (- y))
       (if (<= t_0 1e-11)
         (- (/ y z))
         (if (<= t_0 200000000000.0) 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 <= -5e+178) {
		tmp = x / z;
	} else if (t_0 <= -1000000000.0) {
		tmp = x / -y;
	} else if (t_0 <= 1e-11) {
		tmp = -(y / z);
	} else if (t_0 <= 200000000000.0) {
		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 = (x - y) / (z - y)
    if (t_0 <= (-5d+178)) then
        tmp = x / z
    else if (t_0 <= (-1000000000.0d0)) then
        tmp = x / -y
    else if (t_0 <= 1d-11) then
        tmp = -(y / z)
    else if (t_0 <= 200000000000.0d0) 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 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= -5e+178) {
		tmp = x / z;
	} else if (t_0 <= -1000000000.0) {
		tmp = x / -y;
	} else if (t_0 <= 1e-11) {
		tmp = -(y / z);
	} else if (t_0 <= 200000000000.0) {
		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 <= -5e+178:
		tmp = x / z
	elif t_0 <= -1000000000.0:
		tmp = x / -y
	elif t_0 <= 1e-11:
		tmp = -(y / z)
	elif t_0 <= 200000000000.0:
		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 <= -5e+178)
		tmp = Float64(x / z);
	elseif (t_0 <= -1000000000.0)
		tmp = Float64(x / Float64(-y));
	elseif (t_0 <= 1e-11)
		tmp = Float64(-Float64(y / z));
	elseif (t_0 <= 200000000000.0)
		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 <= -5e+178)
		tmp = x / z;
	elseif (t_0 <= -1000000000.0)
		tmp = x / -y;
	elseif (t_0 <= 1e-11)
		tmp = -(y / z);
	elseif (t_0 <= 200000000000.0)
		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, -5e+178], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, -1000000000.0], N[(x / (-y)), $MachinePrecision], If[LessEqual[t$95$0, 1e-11], (-N[(y / z), $MachinePrecision]), If[LessEqual[t$95$0, 200000000000.0], 1.0, N[(x / z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.9999999999999999e178 or 2e11 < (/.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. /-lowering-/.f64 68.5

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

   5. Simplified 68.5%

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

   if -4.9999999999999999e178 < (/.f64 (-.f64 x y) (-.f64 z y)) < -1e9

   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. --lowering--.f64 N/A

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

      11. /-lowering-/.f64 75.5

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

   5. Simplified 75.5%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 74.2

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

   8. Simplified 74.2%

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

   if -1e9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.99999999999999939e-12

   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. /-lowering-/.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. --lowering--.f64 59.8

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

   5. Simplified 59.8%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 59.1

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

   8. Simplified 59.1%

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

   if 9.99999999999999939e-12 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e11

   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. Simplified 90.7%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{+178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq -1000000000:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-11}:\\ \;\;\;\;-\frac{y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 200000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+178}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 200000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (<= t_0 -5e+178)
     (/ x z)
     (if (<= t_0 -2e+15)
       (/ x (- y))
       (if (<= t_0 2e-8) (/ x z) (if (<= t_0 200000000000.0) 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 <= -5e+178) {
		tmp = x / z;
	} else if (t_0 <= -2e+15) {
		tmp = x / -y;
	} else if (t_0 <= 2e-8) {
		tmp = x / z;
	} else if (t_0 <= 200000000000.0) {
		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 = (x - y) / (z - y)
    if (t_0 <= (-5d+178)) then
        tmp = x / z
    else if (t_0 <= (-2d+15)) then
        tmp = x / -y
    else if (t_0 <= 2d-8) then
        tmp = x / z
    else if (t_0 <= 200000000000.0d0) 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 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= -5e+178) {
		tmp = x / z;
	} else if (t_0 <= -2e+15) {
		tmp = x / -y;
	} else if (t_0 <= 2e-8) {
		tmp = x / z;
	} else if (t_0 <= 200000000000.0) {
		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 <= -5e+178:
		tmp = x / z
	elif t_0 <= -2e+15:
		tmp = x / -y
	elif t_0 <= 2e-8:
		tmp = x / z
	elif t_0 <= 200000000000.0:
		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 <= -5e+178)
		tmp = Float64(x / z);
	elseif (t_0 <= -2e+15)
		tmp = Float64(x / Float64(-y));
	elseif (t_0 <= 2e-8)
		tmp = Float64(x / z);
	elseif (t_0 <= 200000000000.0)
		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 <= -5e+178)
		tmp = x / z;
	elseif (t_0 <= -2e+15)
		tmp = x / -y;
	elseif (t_0 <= 2e-8)
		tmp = x / z;
	elseif (t_0 <= 200000000000.0)
		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, -5e+178], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, -2e+15], N[(x / (-y)), $MachinePrecision], If[LessEqual[t$95$0, 2e-8], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 200000000000.0], 1.0, N[(x / z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.9999999999999999e178 or -2e15 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e-8 or 2e11 < (/.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. /-lowering-/.f64 55.8

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

   5. Simplified 55.8%

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

   if -4.9999999999999999e178 < (/.f64 (-.f64 x y) (-.f64 z y)) < -2e15

   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. --lowering--.f64 N/A

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

      11. /-lowering-/.f64 77.3

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

   5. Simplified 77.3%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 77.3

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

   8. Simplified 77.3%

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

   if 2e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e11

   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. Simplified 94.4%

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

Alternative 5: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -20000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))) (t_1 (/ x (- z y))))
   (if (<= t_0 -20000000.0)
     t_1
     (if (<= t_0 2e-8) (/ (- 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 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -20000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e-8) {
		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 = (x - y) / (z - y)
    t_1 = x / (z - y)
    if (t_0 <= (-20000000.0d0)) then
        tmp = t_1
    else if (t_0 <= 2d-8) 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 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -20000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e-8) {
		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 = (x - y) / (z - y)
	t_1 = x / (z - y)
	tmp = 0
	if t_0 <= -20000000.0:
		tmp = t_1
	elif t_0 <= 2e-8:
		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(x - y) / Float64(z - y))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -20000000.0)
		tmp = t_1;
	elseif (t_0 <= 2e-8)
		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 = (x - y) / (z - y);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -20000000.0)
		tmp = t_1;
	elseif (t_0 <= 2e-8)
		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[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20000000.0], t$95$1, If[LessEqual[t$95$0, 2e-8], 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)) < -2e7 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. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 99.2

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

   5. Simplified 99.2%

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

   if -2e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

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

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

      2. --lowering--.f64 98.9

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

   5. Simplified 98.9%

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

   if 2e-8 < (/.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. /-lowering-/.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. --lowering--.f64 98.2

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

   5. Simplified 98.2%

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

Alternative 6: 83.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;-\frac{y}{z}\\ \mathbf{elif}\;t\_0 \leq 200000000000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))) (t_1 (/ x (- z y))))
   (if (<= t_0 -5e-93)
     t_1
     (if (<= t_0 0.1)
       (- (/ y z))
       (if (<= t_0 200000000000.0) (- 1.0 (/ x y)) 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 <= -5e-93) {
		tmp = t_1;
	} else if (t_0 <= 0.1) {
		tmp = -(y / z);
	} else if (t_0 <= 200000000000.0) {
		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 = (x - y) / (z - y)
    t_1 = x / (z - y)
    if (t_0 <= (-5d-93)) then
        tmp = t_1
    else if (t_0 <= 0.1d0) then
        tmp = -(y / z)
    else if (t_0 <= 200000000000.0d0) 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 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -5e-93) {
		tmp = t_1;
	} else if (t_0 <= 0.1) {
		tmp = -(y / z);
	} else if (t_0 <= 200000000000.0) {
		tmp = 1.0 - (x / y);
	} 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 <= -5e-93:
		tmp = t_1
	elif t_0 <= 0.1:
		tmp = -(y / z)
	elif t_0 <= 200000000000.0:
		tmp = 1.0 - (x / y)
	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 <= -5e-93)
		tmp = t_1;
	elseif (t_0 <= 0.1)
		tmp = Float64(-Float64(y / z));
	elseif (t_0 <= 200000000000.0)
		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 = (x - y) / (z - y);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -5e-93)
		tmp = t_1;
	elseif (t_0 <= 0.1)
		tmp = -(y / z);
	elseif (t_0 <= 200000000000.0)
		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[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-93], t$95$1, If[LessEqual[t$95$0, 0.1], (-N[(y / z), $MachinePrecision]), If[LessEqual[t$95$0, 200000000000.0], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4.99999999999999994e-93 or 2e11 < (/.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. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 90.6

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

   5. Simplified 90.6%

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

   if -4.99999999999999994e-93 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.10000000000000001

   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. /-lowering-/.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. --lowering--.f64 61.5

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

   5. Simplified 61.5%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 60.2

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

   8. Simplified 60.2%

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

   if 0.10000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e11

   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. --lowering--.f64 N/A

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

      11. /-lowering-/.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{-93}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.1:\\ \;\;\;\;-\frac{y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 200000000000:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))) (t_1 (/ x (- z y))))
   (if (<= t_0 -5e-93) t_1 (if (<= t_0 2.0) (/ 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 <= -5e-93) {
		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 = (x - y) / (z - y)
    t_1 = x / (z - y)
    if (t_0 <= (-5d-93)) 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 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -5e-93) {
		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 = (x - y) / (z - y)
	t_1 = x / (z - y)
	tmp = 0
	if t_0 <= -5e-93:
		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(x - y) / Float64(z - y))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -5e-93)
		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 = (x - y) / (z - y);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -5e-93)
		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[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-93], 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)) < -4.99999999999999994e-93 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. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 90.4

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

   5. Simplified 90.4%

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

   if -4.99999999999999994e-93 < (/.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. /-lowering-/.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. --lowering--.f64 81.0

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

   5. Simplified 81.0%

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

Alternative 8: 68.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 200000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (<= t_0 2e-8) (/ x z) (if (<= t_0 200000000000.0) 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 <= 2e-8) {
		tmp = x / z;
	} else if (t_0 <= 200000000000.0) {
		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 = (x - y) / (z - y)
    if (t_0 <= 2d-8) then
        tmp = x / z
    else if (t_0 <= 200000000000.0d0) 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 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= 2e-8) {
		tmp = x / z;
	} else if (t_0 <= 200000000000.0) {
		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 <= 2e-8:
		tmp = x / z
	elif t_0 <= 200000000000.0:
		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 <= 2e-8)
		tmp = Float64(x / z);
	elseif (t_0 <= 200000000000.0)
		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 <= 2e-8)
		tmp = x / z;
	elseif (t_0 <= 200000000000.0)
		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, 2e-8], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 200000000000.0], 1.0, N[(x / z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 2e-8 or 2e11 < (/.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 0

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

      1. /-lowering-/.f64 51.1

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

   5. Simplified 51.1%

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

   if 2e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e11

   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. Simplified 94.4%

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

Alternative 9: 34.2% 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 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. Simplified 34.5%

   \[\leadsto \color{blue}{1} \]
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 2024204 
(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)))