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

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

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: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+111}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq -2.55 \cdot 10^{-80} \lor \neg \left(y \leq 1.32 \cdot 10^{+29}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -3.1e+111)
     t_0
     (if (<= y -2.4e-27)
       (/ y (- y z))
       (if (or (<= y -2.55e-80) (not (<= y 1.32e+29))) t_0 (/ (- x y) z))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -3.1e+111) {
		tmp = t_0;
	} else if (y <= -2.4e-27) {
		tmp = y / (y - z);
	} else if ((y <= -2.55e-80) || !(y <= 1.32e+29)) {
		tmp = t_0;
	} else {
		tmp = (x - y) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    if (y <= (-3.1d+111)) then
        tmp = t_0
    else if (y <= (-2.4d-27)) then
        tmp = y / (y - z)
    else if ((y <= (-2.55d-80)) .or. (.not. (y <= 1.32d+29))) then
        tmp = t_0
    else
        tmp = (x - y) / 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 tmp;
	if (y <= -3.1e+111) {
		tmp = t_0;
	} else if (y <= -2.4e-27) {
		tmp = y / (y - z);
	} else if ((y <= -2.55e-80) || !(y <= 1.32e+29)) {
		tmp = t_0;
	} else {
		tmp = (x - y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -3.1e+111:
		tmp = t_0
	elif y <= -2.4e-27:
		tmp = y / (y - z)
	elif (y <= -2.55e-80) or not (y <= 1.32e+29):
		tmp = t_0
	else:
		tmp = (x - y) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -3.1e+111)
		tmp = t_0;
	elseif (y <= -2.4e-27)
		tmp = Float64(y / Float64(y - z));
	elseif ((y <= -2.55e-80) || !(y <= 1.32e+29))
		tmp = t_0;
	else
		tmp = Float64(Float64(x - y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -3.1e+111)
		tmp = t_0;
	elseif (y <= -2.4e-27)
		tmp = y / (y - z);
	elseif ((y <= -2.55e-80) || ~((y <= 1.32e+29)))
		tmp = t_0;
	else
		tmp = (x - y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.1e+111], t$95$0, If[LessEqual[y, -2.4e-27], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -2.55e-80], N[Not[LessEqual[y, 1.32e+29]], $MachinePrecision]], t$95$0, N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.1e111 or -2.40000000000000002e-27 < y < -2.55000000000000004e-80 or 1.32e29 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 83.2%

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

      1. div-sub 83.2%

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

      2. sub-neg 83.2%

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

      3. *-inverses 83.2%

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

      4. metadata-eval 83.2%

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

      5. distribute-lft-in 83.2%

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

      6. metadata-eval 83.2%

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

      7. +-commutative 83.2%

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

      8. mul-1-neg 83.2%

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

      9. unsub-neg 83.2%

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

   5. Simplified 83.2%

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

   if -3.1e111 < y < -2.40000000000000002e-27

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 79.0%

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

      1. neg-mul-1 79.0%

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

      2. distribute-neg-frac 79.0%

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

   5. Simplified 79.0%

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

   if -2.55000000000000004e-80 < y < 1.32e29

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 83.3%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+111}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq -2.55 \cdot 10^{-80} \lor \neg \left(y \leq 1.32 \cdot 10^{+29}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+110}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-81} \lor \neg \left(y \leq 1.9 \cdot 10^{+30}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -4.6e+110)
     t_0
     (if (<= y -3.6e-28)
       (/ y (- y z))
       (if (or (<= y -1.45e-81) (not (<= y 1.9e+30))) t_0 (/ (- x y) z))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -4.6e+110) {
		tmp = t_0;
	} else if (y <= -3.6e-28) {
		tmp = y / (y - z);
	} else if ((y <= -1.45e-81) || !(y <= 1.9e+30)) {
		tmp = t_0;
	} else {
		tmp = (x - y) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    if (y <= (-4.6d+110)) then
        tmp = t_0
    else if (y <= (-3.6d-28)) then
        tmp = y / (y - z)
    else if ((y <= (-1.45d-81)) .or. (.not. (y <= 1.9d+30))) then
        tmp = t_0
    else
        tmp = (x - y) / 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 tmp;
	if (y <= -4.6e+110) {
		tmp = t_0;
	} else if (y <= -3.6e-28) {
		tmp = y / (y - z);
	} else if ((y <= -1.45e-81) || !(y <= 1.9e+30)) {
		tmp = t_0;
	} else {
		tmp = (x - y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -4.6e+110:
		tmp = t_0
	elif y <= -3.6e-28:
		tmp = y / (y - z)
	elif (y <= -1.45e-81) or not (y <= 1.9e+30):
		tmp = t_0
	else:
		tmp = (x - y) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -4.6e+110)
		tmp = t_0;
	elseif (y <= -3.6e-28)
		tmp = Float64(y / Float64(y - z));
	elseif ((y <= -1.45e-81) || !(y <= 1.9e+30))
		tmp = t_0;
	else
		tmp = Float64(Float64(x - y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -4.6e+110)
		tmp = t_0;
	elseif (y <= -3.6e-28)
		tmp = y / (y - z);
	elseif ((y <= -1.45e-81) || ~((y <= 1.9e+30)))
		tmp = t_0;
	else
		tmp = (x - y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.6e+110], t$95$0, If[LessEqual[y, -3.6e-28], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -1.45e-81], N[Not[LessEqual[y, 1.9e+30]], $MachinePrecision]], t$95$0, N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.6e110 or -3.5999999999999999e-28 < y < -1.44999999999999994e-81 or 1.9000000000000001e30 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 83.2%

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

      1. div-sub 83.2%

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

      2. sub-neg 83.2%

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

      3. *-inverses 83.2%

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

      4. metadata-eval 83.2%

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

      5. distribute-lft-in 83.2%

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

      6. metadata-eval 83.2%

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

      7. +-commutative 83.2%

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

      8. mul-1-neg 83.2%

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

      9. unsub-neg 83.2%

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

   5. Simplified 83.2%

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

   if -4.6e110 < y < -3.5999999999999999e-28

   1. Initial program 99.9%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. associate-*l/ 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. clear-num 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0 79.0%

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

      1. mul-1-neg 79.0%

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

      2. distribute-neg-frac2 79.0%

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

      3. neg-sub0 79.0%

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

      4. associate-+l- 79.0%

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

      5. neg-sub0 79.0%

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

      6. +-commutative 79.0%

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

      7. unsub-neg 79.0%

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

   9. Simplified 79.0%

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

   if -1.44999999999999994e-81 < y < 1.9000000000000001e30

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 83.3%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+110}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-81} \lor \neg \left(y \leq 1.9 \cdot 10^{+30}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -1.16 \cdot 10^{+110}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq -2.55 \cdot 10^{-80}:\\ \;\;\;\;1 - \frac{x - z}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -1.16e+110)
     t_0
     (if (<= y -3.2e-27)
       (/ y (- y z))
       (if (<= y -2.55e-80)
         (- 1.0 (/ (- x z) y))
         (if (<= y 1.1e+35) (/ (- x y) z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -1.16e+110) {
		tmp = t_0;
	} else if (y <= -3.2e-27) {
		tmp = y / (y - z);
	} else if (y <= -2.55e-80) {
		tmp = 1.0 - ((x - z) / y);
	} else if (y <= 1.1e+35) {
		tmp = (x - y) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x / y)
    if (y <= (-1.16d+110)) then
        tmp = t_0
    else if (y <= (-3.2d-27)) then
        tmp = y / (y - z)
    else if (y <= (-2.55d-80)) then
        tmp = 1.0d0 - ((x - z) / y)
    else if (y <= 1.1d+35) then
        tmp = (x - y) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -1.16e+110) {
		tmp = t_0;
	} else if (y <= -3.2e-27) {
		tmp = y / (y - z);
	} else if (y <= -2.55e-80) {
		tmp = 1.0 - ((x - z) / y);
	} else if (y <= 1.1e+35) {
		tmp = (x - y) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -1.16e+110:
		tmp = t_0
	elif y <= -3.2e-27:
		tmp = y / (y - z)
	elif y <= -2.55e-80:
		tmp = 1.0 - ((x - z) / y)
	elif y <= 1.1e+35:
		tmp = (x - y) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -1.16e+110)
		tmp = t_0;
	elseif (y <= -3.2e-27)
		tmp = Float64(y / Float64(y - z));
	elseif (y <= -2.55e-80)
		tmp = Float64(1.0 - Float64(Float64(x - z) / y));
	elseif (y <= 1.1e+35)
		tmp = Float64(Float64(x - y) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (x / y);
	tmp = 0.0;
	if (y <= -1.16e+110)
		tmp = t_0;
	elseif (y <= -3.2e-27)
		tmp = y / (y - z);
	elseif (y <= -2.55e-80)
		tmp = 1.0 - ((x - z) / y);
	elseif (y <= 1.1e+35)
		tmp = (x - y) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.16e+110], t$95$0, If[LessEqual[y, -3.2e-27], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.55e-80], N[(1.0 - N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+35], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.16e110 or 1.0999999999999999e35 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 84.7%

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

      1. div-sub 84.7%

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

      2. sub-neg 84.7%

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

      3. *-inverses 84.7%

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

      4. metadata-eval 84.7%

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

      5. distribute-lft-in 84.7%

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

      6. metadata-eval 84.7%

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

      7. +-commutative 84.7%

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

      8. mul-1-neg 84.7%

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

      9. unsub-neg 84.7%

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

   5. Simplified 84.7%

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

   if -1.16e110 < y < -3.19999999999999991e-27

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 79.0%

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

      1. neg-mul-1 79.0%

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

      2. distribute-neg-frac 79.0%

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

   5. Simplified 79.0%

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

   if -3.19999999999999991e-27 < y < -2.55000000000000004e-80

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 72.9%

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

      1. associate-+r- 72.9%

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

      2. distribute-lft-out-- 72.9%

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

      3. div-sub 72.9%

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

      4. mul-1-neg 72.9%

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

      5. unsub-neg 72.9%

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

   5. Simplified 72.9%

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

   if -2.55000000000000004e-80 < y < 1.0999999999999999e35

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 83.3%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+110}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq -2.55 \cdot 10^{-80}:\\ \;\;\;\;1 - \frac{x - z}{y}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ t_1 := \frac{y}{y - z}\\ \mathbf{if}\;y \leq -3.15 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.12 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+117}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))) (t_1 (/ y (- y z))))
   (if (<= y -3.15e+109)
     t_0
     (if (<= y -1.12e-44)
       t_1
       (if (<= y 4.2e-50) (/ x (- z y)) (if (<= y 7.2e+117) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double t_1 = y / (y - z);
	double tmp;
	if (y <= -3.15e+109) {
		tmp = t_0;
	} else if (y <= -1.12e-44) {
		tmp = t_1;
	} else if (y <= 4.2e-50) {
		tmp = x / (z - y);
	} else if (y <= 7.2e+117) {
		tmp = t_1;
	} 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) :: tmp
    t_0 = 1.0d0 - (x / y)
    t_1 = y / (y - z)
    if (y <= (-3.15d+109)) then
        tmp = t_0
    else if (y <= (-1.12d-44)) then
        tmp = t_1
    else if (y <= 4.2d-50) then
        tmp = x / (z - y)
    else if (y <= 7.2d+117) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double t_1 = y / (y - z);
	double tmp;
	if (y <= -3.15e+109) {
		tmp = t_0;
	} else if (y <= -1.12e-44) {
		tmp = t_1;
	} else if (y <= 4.2e-50) {
		tmp = x / (z - y);
	} else if (y <= 7.2e+117) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	t_1 = y / (y - z)
	tmp = 0
	if y <= -3.15e+109:
		tmp = t_0
	elif y <= -1.12e-44:
		tmp = t_1
	elif y <= 4.2e-50:
		tmp = x / (z - y)
	elif y <= 7.2e+117:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(x / y))
	t_1 = Float64(y / Float64(y - z))
	tmp = 0.0
	if (y <= -3.15e+109)
		tmp = t_0;
	elseif (y <= -1.12e-44)
		tmp = t_1;
	elseif (y <= 4.2e-50)
		tmp = Float64(x / Float64(z - y));
	elseif (y <= 7.2e+117)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 - (x / y);
	t_1 = y / (y - z);
	tmp = 0.0;
	if (y <= -3.15e+109)
		tmp = t_0;
	elseif (y <= -1.12e-44)
		tmp = t_1;
	elseif (y <= 4.2e-50)
		tmp = x / (z - y);
	elseif (y <= 7.2e+117)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.15e+109], t$95$0, If[LessEqual[y, -1.12e-44], t$95$1, If[LessEqual[y, 4.2e-50], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e+117], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.14999999999999981e109 or 7.20000000000000025e117 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 89.1%

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

      1. div-sub 89.1%

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

      2. sub-neg 89.1%

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

      3. *-inverses 89.1%

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

      4. metadata-eval 89.1%

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

      5. distribute-lft-in 89.1%

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

      6. metadata-eval 89.1%

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

      7. +-commutative 89.1%

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

      8. mul-1-neg 89.1%

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

      9. unsub-neg 89.1%

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

   5. Simplified 89.1%

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

   if -3.14999999999999981e109 < y < -1.1200000000000001e-44 or 4.2000000000000002e-50 < y < 7.20000000000000025e117

   1. Initial program 99.9%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. clear-num 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 75.2%

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

      1. mul-1-neg 75.2%

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

      2. distribute-neg-frac2 75.2%

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

      3. neg-sub0 75.2%

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

      4. associate-+l- 75.2%

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

      5. neg-sub0 75.2%

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

      6. +-commutative 75.2%

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

      7. unsub-neg 75.2%

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

   9. Simplified 75.2%

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

   if -1.1200000000000001e-44 < y < 4.2000000000000002e-50

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.9%

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

Alternative 6: 61.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{-44}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.12e-44)
   1.0
   (if (<= y 3.4e-38) (/ x z) (if (<= y 2.5e+88) (/ y (- z)) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.12e-44) {
		tmp = 1.0;
	} else if (y <= 3.4e-38) {
		tmp = x / z;
	} else if (y <= 2.5e+88) {
		tmp = y / -z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.12d-44)) then
        tmp = 1.0d0
    else if (y <= 3.4d-38) then
        tmp = x / z
    else if (y <= 2.5d+88) then
        tmp = y / -z
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.12e-44) {
		tmp = 1.0;
	} else if (y <= 3.4e-38) {
		tmp = x / z;
	} else if (y <= 2.5e+88) {
		tmp = y / -z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.12e-44:
		tmp = 1.0
	elif y <= 3.4e-38:
		tmp = x / z
	elif y <= 2.5e+88:
		tmp = y / -z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.12e-44)
		tmp = 1.0;
	elseif (y <= 3.4e-38)
		tmp = Float64(x / z);
	elseif (y <= 2.5e+88)
		tmp = Float64(y / Float64(-z));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.12e-44)
		tmp = 1.0;
	elseif (y <= 3.4e-38)
		tmp = x / z;
	elseif (y <= 2.5e+88)
		tmp = y / -z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.12e-44], 1.0, If[LessEqual[y, 3.4e-38], N[(x / z), $MachinePrecision], If[LessEqual[y, 2.5e+88], N[(y / (-z)), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1200000000000001e-44 or 2.49999999999999999e88 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 61.6%

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

   if -1.1200000000000001e-44 < y < 3.4000000000000002e-38

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 69.2%

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

   if 3.4000000000000002e-38 < y < 2.49999999999999999e88

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 58.6%

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

   4. Taylor expanded in x around 0 47.1%

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

      1. neg-mul-1 47.1%

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

      2. distribute-neg-frac 47.1%

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

   6. Simplified 47.1%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{-44}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-44} \lor \neg \left(y \leq 5.9 \cdot 10^{+32}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.95e-44) (not (<= y 5.9e+32))) (- 1.0 (/ x y)) (/ x (- z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.95e-44) || !(y <= 5.9e+32)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (z - 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) :: tmp
    if ((y <= (-1.95d-44)) .or. (.not. (y <= 5.9d+32))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / (z - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.95e-44) || !(y <= 5.9e+32)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.95e-44) or not (y <= 5.9e+32):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (z - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.95e-44) || !(y <= 5.9e+32))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / Float64(z - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.95e-44) || ~((y <= 5.9e+32)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.95e-44], N[Not[LessEqual[y, 5.9e+32]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9500000000000001e-44 or 5.89999999999999965e32 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 79.3%

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

      1. div-sub 79.3%

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

      2. sub-neg 79.3%

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

      3. *-inverses 79.3%

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

      4. metadata-eval 79.3%

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

      5. distribute-lft-in 79.3%

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

      6. metadata-eval 79.3%

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

      7. +-commutative 79.3%

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

      8. mul-1-neg 79.3%

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

      9. unsub-neg 79.3%

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

   5. Simplified 79.3%

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

   if -1.9500000000000001e-44 < y < 5.89999999999999965e32

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 76.6%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-44} \lor \neg \left(y \leq 5.9 \cdot 10^{+32}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-99} \lor \neg \left(y \leq 5.5 \cdot 10^{-74}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.6e-99) (not (<= y 5.5e-74))) (- 1.0 (/ x y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.6e-99) || !(y <= 5.5e-74)) {
		tmp = 1.0 - (x / y);
	} 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) :: tmp
    if ((y <= (-5.6d-99)) .or. (.not. (y <= 5.5d-74))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.6e-99) || !(y <= 5.5e-74)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5.6e-99) or not (y <= 5.5e-74):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.6e-99) || !(y <= 5.5e-74))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5.6e-99) || ~((y <= 5.5e-74)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.6e-99], N[Not[LessEqual[y, 5.5e-74]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.6000000000000001e-99 or 5.5000000000000001e-74 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 73.4%

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

      1. div-sub 73.4%

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

      2. sub-neg 73.4%

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

      3. *-inverses 73.4%

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

      4. metadata-eval 73.4%

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

      5. distribute-lft-in 73.4%

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

      6. metadata-eval 73.4%

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

      7. +-commutative 73.4%

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

      8. mul-1-neg 73.4%

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

      9. unsub-neg 73.4%

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

   5. Simplified 73.4%

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

   if -5.6000000000000001e-99 < y < 5.5000000000000001e-74

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-99} \lor \neg \left(y \leq 5.5 \cdot 10^{-74}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-44}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+88}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.9e-44) 1.0 (if (<= y 1.05e+88) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.9e-44) {
		tmp = 1.0;
	} else if (y <= 1.05e+88) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.9d-44)) then
        tmp = 1.0d0
    else if (y <= 1.05d+88) then
        tmp = x / z
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.9e-44) {
		tmp = 1.0;
	} else if (y <= 1.05e+88) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.9e-44:
		tmp = 1.0
	elif y <= 1.05e+88:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.9e-44)
		tmp = 1.0;
	elseif (y <= 1.05e+88)
		tmp = Float64(x / z);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.9e-44)
		tmp = 1.0;
	elseif (y <= 1.05e+88)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.9e-44], 1.0, If[LessEqual[y, 1.05e+88], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9000000000000001e-44 or 1.05e88 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 61.6%

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

   if -2.9000000000000001e-44 < y < 1.05e88

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 63.4%

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

Alternative 10: 34.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

double code(double x, double y, double z) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0;
}

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf 35.9%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Developer target: 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 2024103 
(FPCore (x y z)
  :name "Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1"
  :precision binary64

  :alt
  (- (/ x (- z y)) (/ y (- z y)))

  (/ (- x y) (- z y)))