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

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. div-sub 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 59.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{z}{y}\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{-130}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.48 \cdot 10^{+141}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ z y))))
   (if (<= y -2.6e+56)
     t_0
     (if (<= y -1.42e-130)
       (/ x z)
       (if (<= y -4.5e-165)
         (/ x (- y))
         (if (<= y 2.4e-8) (/ x z) (if (<= y 1.48e+141) (/ (- y) z) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (z / y);
	double tmp;
	if (y <= -2.6e+56) {
		tmp = t_0;
	} else if (y <= -1.42e-130) {
		tmp = x / z;
	} else if (y <= -4.5e-165) {
		tmp = x / -y;
	} else if (y <= 2.4e-8) {
		tmp = x / z;
	} else if (y <= 1.48e+141) {
		tmp = -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 + (z / y)
    if (y <= (-2.6d+56)) then
        tmp = t_0
    else if (y <= (-1.42d-130)) then
        tmp = x / z
    else if (y <= (-4.5d-165)) then
        tmp = x / -y
    else if (y <= 2.4d-8) then
        tmp = x / z
    else if (y <= 1.48d+141) then
        tmp = -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 + (z / y);
	double tmp;
	if (y <= -2.6e+56) {
		tmp = t_0;
	} else if (y <= -1.42e-130) {
		tmp = x / z;
	} else if (y <= -4.5e-165) {
		tmp = x / -y;
	} else if (y <= 2.4e-8) {
		tmp = x / z;
	} else if (y <= 1.48e+141) {
		tmp = -y / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (z / y)
	tmp = 0
	if y <= -2.6e+56:
		tmp = t_0
	elif y <= -1.42e-130:
		tmp = x / z
	elif y <= -4.5e-165:
		tmp = x / -y
	elif y <= 2.4e-8:
		tmp = x / z
	elif y <= 1.48e+141:
		tmp = -y / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(z / y))
	tmp = 0.0
	if (y <= -2.6e+56)
		tmp = t_0;
	elseif (y <= -1.42e-130)
		tmp = Float64(x / z);
	elseif (y <= -4.5e-165)
		tmp = Float64(x / Float64(-y));
	elseif (y <= 2.4e-8)
		tmp = Float64(x / z);
	elseif (y <= 1.48e+141)
		tmp = Float64(Float64(-y) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (z / y);
	tmp = 0.0;
	if (y <= -2.6e+56)
		tmp = t_0;
	elseif (y <= -1.42e-130)
		tmp = x / z;
	elseif (y <= -4.5e-165)
		tmp = x / -y;
	elseif (y <= 2.4e-8)
		tmp = x / z;
	elseif (y <= 1.48e+141)
		tmp = -y / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.6e+56], t$95$0, If[LessEqual[y, -1.42e-130], N[(x / z), $MachinePrecision], If[LessEqual[y, -4.5e-165], N[(x / (-y)), $MachinePrecision], If[LessEqual[y, 2.4e-8], N[(x / z), $MachinePrecision], If[LessEqual[y, 1.48e+141], N[((-y) / z), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.60000000000000011e56 or 1.48000000000000001e141 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 82.3%

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

      1. neg-mul-1 82.3%

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

      2. distribute-neg-frac 82.3%

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

   5. Simplified 82.3%

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

   6. Taylor expanded in y around inf 74.7%

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

   if -2.60000000000000011e56 < y < -1.4199999999999999e-130 or -4.49999999999999992e-165 < y < 2.39999999999999998e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 61.6%

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

   if -1.4199999999999999e-130 < y < -4.49999999999999992e-165

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.5%

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

   4. Taylor expanded in z around 0 79.1%

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

      1. mul-1-neg 79.1%

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

      2. distribute-frac-neg 79.1%

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

   6. Simplified 79.1%

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

   if 2.39999999999999998e-8 < y < 1.48000000000000001e141

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 68.0%

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

   4. Taylor expanded in x around 0 45.8%

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

      1. neg-mul-1 45.8%

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

      2. distribute-neg-frac2 45.8%

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

   6. Simplified 45.8%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+56}:\\ \;\;\;\;1 + \frac{z}{y}\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{-130}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.48 \cdot 10^{+141}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+54}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-130}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+142}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8e+54)
   1.0
   (if (<= y -2.1e-130)
     (/ x z)
     (if (<= y -4.5e-165)
       (/ x (- y))
       (if (<= y 1.8e-7) (/ x z) (if (<= y 2.8e+142) (/ (- y) z) 1.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8e+54) {
		tmp = 1.0;
	} else if (y <= -2.1e-130) {
		tmp = x / z;
	} else if (y <= -4.5e-165) {
		tmp = x / -y;
	} else if (y <= 1.8e-7) {
		tmp = x / z;
	} else if (y <= 2.8e+142) {
		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 <= (-8d+54)) then
        tmp = 1.0d0
    else if (y <= (-2.1d-130)) then
        tmp = x / z
    else if (y <= (-4.5d-165)) then
        tmp = x / -y
    else if (y <= 1.8d-7) then
        tmp = x / z
    else if (y <= 2.8d+142) 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 <= -8e+54) {
		tmp = 1.0;
	} else if (y <= -2.1e-130) {
		tmp = x / z;
	} else if (y <= -4.5e-165) {
		tmp = x / -y;
	} else if (y <= 1.8e-7) {
		tmp = x / z;
	} else if (y <= 2.8e+142) {
		tmp = -y / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8e+54:
		tmp = 1.0
	elif y <= -2.1e-130:
		tmp = x / z
	elif y <= -4.5e-165:
		tmp = x / -y
	elif y <= 1.8e-7:
		tmp = x / z
	elif y <= 2.8e+142:
		tmp = -y / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8e+54)
		tmp = 1.0;
	elseif (y <= -2.1e-130)
		tmp = Float64(x / z);
	elseif (y <= -4.5e-165)
		tmp = Float64(x / Float64(-y));
	elseif (y <= 1.8e-7)
		tmp = Float64(x / z);
	elseif (y <= 2.8e+142)
		tmp = Float64(Float64(-y) / z);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8e+54)
		tmp = 1.0;
	elseif (y <= -2.1e-130)
		tmp = x / z;
	elseif (y <= -4.5e-165)
		tmp = x / -y;
	elseif (y <= 1.8e-7)
		tmp = x / z;
	elseif (y <= 2.8e+142)
		tmp = -y / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8e+54], 1.0, If[LessEqual[y, -2.1e-130], N[(x / z), $MachinePrecision], If[LessEqual[y, -4.5e-165], N[(x / (-y)), $MachinePrecision], If[LessEqual[y, 1.8e-7], N[(x / z), $MachinePrecision], If[LessEqual[y, 2.8e+142], N[((-y) / z), $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.0000000000000006e54 or 2.8e142 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 73.5%

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

   if -8.0000000000000006e54 < y < -2.10000000000000002e-130 or -4.49999999999999992e-165 < y < 1.79999999999999997e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 62.2%

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

   if -2.10000000000000002e-130 < y < -4.49999999999999992e-165

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.5%

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

   4. Taylor expanded in z around 0 79.1%

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

      1. mul-1-neg 79.1%

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

      2. distribute-frac-neg 79.1%

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

   6. Simplified 79.1%

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

   if 1.79999999999999997e-7 < y < 2.8e142

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 68.0%

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

   4. Taylor expanded in x around 0 45.8%

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

      1. neg-mul-1 45.8%

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

      2. distribute-neg-frac2 45.8%

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

   6. Simplified 45.8%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+54}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-130}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+142}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.05 \cdot 10^{+43} \lor \neg \left(y \leq -4.8 \cdot 10^{-63} \lor \neg \left(y \leq -4.5 \cdot 10^{-165}\right) \land y \leq 8.5 \cdot 10^{-36}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.05e+43)
         (not
          (or (<= y -4.8e-63) (and (not (<= y -4.5e-165)) (<= y 8.5e-36)))))
   (- 1.0 (/ x y))
   (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.05e+43) || !((y <= -4.8e-63) || (!(y <= -4.5e-165) && (y <= 8.5e-36)))) {
		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 <= (-3.05d+43)) .or. (.not. (y <= (-4.8d-63)) .or. (.not. (y <= (-4.5d-165))) .and. (y <= 8.5d-36))) 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 <= -3.05e+43) || !((y <= -4.8e-63) || (!(y <= -4.5e-165) && (y <= 8.5e-36)))) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.05e+43) or not ((y <= -4.8e-63) or (not (y <= -4.5e-165) and (y <= 8.5e-36))):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.05e+43) || !((y <= -4.8e-63) || (!(y <= -4.5e-165) && (y <= 8.5e-36))))
		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 <= -3.05e+43) || ~(((y <= -4.8e-63) || (~((y <= -4.5e-165)) && (y <= 8.5e-36)))))
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.05e+43], N[Not[Or[LessEqual[y, -4.8e-63], And[N[Not[LessEqual[y, -4.5e-165]], $MachinePrecision], LessEqual[y, 8.5e-36]]]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.0499999999999999e43 or -4.8000000000000001e-63 < y < -4.49999999999999992e-165 or 8.5000000000000007e-36 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 76.2%

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

      1. div-sub 76.2%

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

      2. sub-neg 76.2%

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

      3. *-inverses 76.2%

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

      4. metadata-eval 76.2%

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

      5. distribute-lft-in 76.2%

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

      6. metadata-eval 76.2%

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

      7. +-commutative 76.2%

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

      8. mul-1-neg 76.2%

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

      9. unsub-neg 76.2%

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

   5. Simplified 76.2%

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

   if -3.0499999999999999e43 < y < -4.8000000000000001e-63 or -4.49999999999999992e-165 < y < 8.5000000000000007e-36

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 67.7%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.05 \cdot 10^{+43} \lor \neg \left(y \leq -4.8 \cdot 10^{-63} \lor \neg \left(y \leq -4.5 \cdot 10^{-165}\right) \land y \leq 8.5 \cdot 10^{-36}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4400:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+141}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -3.2e+43)
     t_0
     (if (<= y 4400.0)
       (/ x (- z y))
       (if (<= y 1.42e+141) (/ (- 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 <= -3.2e+43) {
		tmp = t_0;
	} else if (y <= 4400.0) {
		tmp = x / (z - y);
	} else if (y <= 1.42e+141) {
		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 <= (-3.2d+43)) then
        tmp = t_0
    else if (y <= 4400.0d0) then
        tmp = x / (z - y)
    else if (y <= 1.42d+141) 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 <= -3.2e+43) {
		tmp = t_0;
	} else if (y <= 4400.0) {
		tmp = x / (z - y);
	} else if (y <= 1.42e+141) {
		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 <= -3.2e+43:
		tmp = t_0
	elif y <= 4400.0:
		tmp = x / (z - y)
	elif y <= 1.42e+141:
		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 <= -3.2e+43)
		tmp = t_0;
	elseif (y <= 4400.0)
		tmp = Float64(x / Float64(z - y));
	elseif (y <= 1.42e+141)
		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 <= -3.2e+43)
		tmp = t_0;
	elseif (y <= 4400.0)
		tmp = x / (z - y);
	elseif (y <= 1.42e+141)
		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, -3.2e+43], t$95$0, If[LessEqual[y, 4400.0], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.42e+141], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.20000000000000014e43 or 1.42000000000000005e141 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 90.0%

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

      1. div-sub 90.0%

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

      2. sub-neg 90.0%

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

      3. *-inverses 90.0%

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

      4. metadata-eval 90.0%

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

      5. distribute-lft-in 90.0%

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

      6. metadata-eval 90.0%

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

      7. +-commutative 90.0%

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

      8. mul-1-neg 90.0%

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

      9. unsub-neg 90.0%

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

   5. Simplified 90.0%

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

   if -3.20000000000000014e43 < y < 4400

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 77.7%

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

   if 4400 < y < 1.42000000000000005e141

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 71.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+43}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 4400:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+141}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+55}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+143}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.1e+55)
   1.0
   (if (<= y 1.2e-7) (/ x z) (if (<= y 5.8e+143) (/ (- y) z) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.1e+55) {
		tmp = 1.0;
	} else if (y <= 1.2e-7) {
		tmp = x / z;
	} else if (y <= 5.8e+143) {
		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 <= (-2.1d+55)) then
        tmp = 1.0d0
    else if (y <= 1.2d-7) then
        tmp = x / z
    else if (y <= 5.8d+143) 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 <= -2.1e+55) {
		tmp = 1.0;
	} else if (y <= 1.2e-7) {
		tmp = x / z;
	} else if (y <= 5.8e+143) {
		tmp = -y / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.1e+55:
		tmp = 1.0
	elif y <= 1.2e-7:
		tmp = x / z
	elif y <= 5.8e+143:
		tmp = -y / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.1e+55)
		tmp = 1.0;
	elseif (y <= 1.2e-7)
		tmp = Float64(x / z);
	elseif (y <= 5.8e+143)
		tmp = Float64(Float64(-y) / z);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.1e+55)
		tmp = 1.0;
	elseif (y <= 1.2e-7)
		tmp = x / z;
	elseif (y <= 5.8e+143)
		tmp = -y / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.1e+55], 1.0, If[LessEqual[y, 1.2e-7], N[(x / z), $MachinePrecision], If[LessEqual[y, 5.8e+143], N[((-y) / z), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1000000000000001e55 or 5.7999999999999996e143 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 73.5%

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

   if -2.1000000000000001e55 < y < 1.19999999999999989e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 57.8%

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

   if 1.19999999999999989e-7 < y < 5.7999999999999996e143

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 68.0%

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

   4. Taylor expanded in x around 0 45.8%

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

      1. neg-mul-1 45.8%

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

      2. distribute-neg-frac2 45.8%

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

   6. Simplified 45.8%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+55}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+143}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+44} \lor \neg \left(y \leq 6 \cdot 10^{+67}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.65e+44) (not (<= y 6e+67))) (- 1.0 (/ x y)) (/ x (- z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.65e+44) || !(y <= 6e+67)) {
		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 <= (-1.65d+44)) .or. (.not. (y <= 6d+67))) 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 <= -1.65e+44) || !(y <= 6e+67)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.65e+44) or not (y <= 6e+67):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (z - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.65e+44) || !(y <= 6e+67))
		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 <= -1.65e+44) || ~((y <= 6e+67)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.65e+44], N[Not[LessEqual[y, 6e+67]], $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 < -1.65000000000000007e44 or 6.0000000000000002e67 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 83.0%

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

      1. div-sub 83.0%

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

      2. sub-neg 83.0%

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

      3. *-inverses 83.0%

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

      4. metadata-eval 83.0%

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

      5. distribute-lft-in 83.0%

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

      6. metadata-eval 83.0%

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

      7. +-commutative 83.0%

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

      8. mul-1-neg 83.0%

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

      9. unsub-neg 83.0%

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

   5. Simplified 83.0%

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

   if -1.65000000000000007e44 < y < 6.0000000000000002e67

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 75.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+44} \lor \neg \left(y \leq 6 \cdot 10^{+67}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+44}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.3e+44)
   (- 1.0 (/ x y))
   (if (<= y 6.6e+43) (/ x (- z y)) (/ y (- y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.3e+44) {
		tmp = 1.0 - (x / y);
	} else if (y <= 6.6e+43) {
		tmp = x / (z - y);
	} else {
		tmp = y / (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) :: tmp
    if (y <= (-2.3d+44)) then
        tmp = 1.0d0 - (x / y)
    else if (y <= 6.6d+43) then
        tmp = x / (z - y)
    else
        tmp = y / (y - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.3e+44) {
		tmp = 1.0 - (x / y);
	} else if (y <= 6.6e+43) {
		tmp = x / (z - y);
	} else {
		tmp = y / (y - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.3e+44:
		tmp = 1.0 - (x / y)
	elif y <= 6.6e+43:
		tmp = x / (z - y)
	else:
		tmp = y / (y - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.3e+44)
		tmp = Float64(1.0 - Float64(x / y));
	elseif (y <= 6.6e+43)
		tmp = Float64(x / Float64(z - y));
	else
		tmp = Float64(y / Float64(y - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.3e+44)
		tmp = 1.0 - (x / y);
	elseif (y <= 6.6e+43)
		tmp = x / (z - y);
	else
		tmp = y / (y - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.3e+44], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.6e+43], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.30000000000000004e44

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 92.2%

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

      1. div-sub 92.3%

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

      2. sub-neg 92.3%

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

      3. *-inverses 92.3%

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

      4. metadata-eval 92.3%

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

      5. distribute-lft-in 92.3%

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

      6. metadata-eval 92.3%

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

      7. +-commutative 92.3%

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

      8. mul-1-neg 92.3%

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

      9. unsub-neg 92.3%

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

   5. Simplified 92.3%

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

   if -2.30000000000000004e44 < y < 6.6000000000000003e43

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 76.7%

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

   if 6.6000000000000003e43 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 82.7%

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

      1. neg-mul-1 82.7%

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

      2. distribute-neg-frac 82.7%

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

   5. Simplified 82.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+44}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+54}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.5e+54) 1.0 (if (<= y 1.55e+67) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.5e+54) {
		tmp = 1.0;
	} else if (y <= 1.55e+67) {
		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 <= (-7.5d+54)) then
        tmp = 1.0d0
    else if (y <= 1.55d+67) 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 <= -7.5e+54) {
		tmp = 1.0;
	} else if (y <= 1.55e+67) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.5e+54:
		tmp = 1.0
	elif y <= 1.55e+67:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.5e+54)
		tmp = 1.0;
	elseif (y <= 1.55e+67)
		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 <= -7.5e+54)
		tmp = 1.0;
	elseif (y <= 1.55e+67)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.5e+54], 1.0, If[LessEqual[y, 1.55e+67], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -7.50000000000000042e54 or 1.54999999999999998e67 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 67.1%

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

   if -7.50000000000000042e54 < y < 1.54999999999999998e67

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 55.3%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+54}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 11: 35.0% 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 99.9%

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

3. Taylor expanded in y around inf 38.9%

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

4. Final simplification 38.9%

   \[\leadsto 1 \]
5. 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 2024096 
(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)))