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

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

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: 68.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -7.4 \cdot 10^{-111}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-203}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{-203}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-142}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-114}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-30} \lor \neg \left(y \leq 100000000000\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))))
   (if (<= y -7.4e-111)
     t_0
     (if (<= y 1.25e-203)
       (/ x z)
       (if (<= y 1.26e-203)
         (/ x (- y))
         (if (<= y 2.3e-142)
           (/ x z)
           (if (<= y 5e-114)
             (/ (- y) z)
             (if (or (<= y 2.5e-30) (not (<= y 100000000000.0)))
               t_0
               (/ x z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double tmp;
	if (y <= -7.4e-111) {
		tmp = t_0;
	} else if (y <= 1.25e-203) {
		tmp = x / z;
	} else if (y <= 1.26e-203) {
		tmp = x / -y;
	} else if (y <= 2.3e-142) {
		tmp = x / z;
	} else if (y <= 5e-114) {
		tmp = -y / z;
	} else if ((y <= 2.5e-30) || !(y <= 100000000000.0)) {
		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) :: tmp
    t_0 = 1.0d0 - (x / y)
    if (y <= (-7.4d-111)) then
        tmp = t_0
    else if (y <= 1.25d-203) then
        tmp = x / z
    else if (y <= 1.26d-203) then
        tmp = x / -y
    else if (y <= 2.3d-142) then
        tmp = x / z
    else if (y <= 5d-114) then
        tmp = -y / z
    else if ((y <= 2.5d-30) .or. (.not. (y <= 100000000000.0d0))) 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 tmp;
	if (y <= -7.4e-111) {
		tmp = t_0;
	} else if (y <= 1.25e-203) {
		tmp = x / z;
	} else if (y <= 1.26e-203) {
		tmp = x / -y;
	} else if (y <= 2.3e-142) {
		tmp = x / z;
	} else if (y <= 5e-114) {
		tmp = -y / z;
	} else if ((y <= 2.5e-30) || !(y <= 100000000000.0)) {
		tmp = t_0;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	tmp = 0
	if y <= -7.4e-111:
		tmp = t_0
	elif y <= 1.25e-203:
		tmp = x / z
	elif y <= 1.26e-203:
		tmp = x / -y
	elif y <= 2.3e-142:
		tmp = x / z
	elif y <= 5e-114:
		tmp = -y / z
	elif (y <= 2.5e-30) or not (y <= 100000000000.0):
		tmp = t_0
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 - Float64(x / y))
	tmp = 0.0
	if (y <= -7.4e-111)
		tmp = t_0;
	elseif (y <= 1.25e-203)
		tmp = Float64(x / z);
	elseif (y <= 1.26e-203)
		tmp = Float64(x / Float64(-y));
	elseif (y <= 2.3e-142)
		tmp = Float64(x / z);
	elseif (y <= 5e-114)
		tmp = Float64(Float64(-y) / z);
	elseif ((y <= 2.5e-30) || !(y <= 100000000000.0))
		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);
	tmp = 0.0;
	if (y <= -7.4e-111)
		tmp = t_0;
	elseif (y <= 1.25e-203)
		tmp = x / z;
	elseif (y <= 1.26e-203)
		tmp = x / -y;
	elseif (y <= 2.3e-142)
		tmp = x / z;
	elseif (y <= 5e-114)
		tmp = -y / z;
	elseif ((y <= 2.5e-30) || ~((y <= 100000000000.0)))
		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]}, If[LessEqual[y, -7.4e-111], t$95$0, If[LessEqual[y, 1.25e-203], N[(x / z), $MachinePrecision], If[LessEqual[y, 1.26e-203], N[(x / (-y)), $MachinePrecision], If[LessEqual[y, 2.3e-142], N[(x / z), $MachinePrecision], If[LessEqual[y, 5e-114], N[((-y) / z), $MachinePrecision], If[Or[LessEqual[y, 2.5e-30], N[Not[LessEqual[y, 100000000000.0]], $MachinePrecision]], t$95$0, N[(x / z), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.4000000000000003e-111 or 4.99999999999999989e-114 < y < 2.49999999999999986e-30 or 1e11 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 71.9%

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

      1. div-sub 71.9%

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

      2. sub-neg 71.9%

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

      3. *-inverses 71.9%

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

      4. metadata-eval 71.9%

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

      5. distribute-lft-in 71.9%

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

      6. metadata-eval 71.9%

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

      7. +-commutative 71.9%

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

      8. mul-1-neg 71.9%

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

      9. unsub-neg 71.9%

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

   5. Simplified 71.9%

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

   if -7.4000000000000003e-111 < y < 1.25e-203 or 1.25999999999999998e-203 < y < 2.30000000000000002e-142 or 2.49999999999999986e-30 < y < 1e11

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 74.9%

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

   if 1.25e-203 < y < 1.25999999999999998e-203

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

   4. Taylor expanded in z around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

   if 2.30000000000000002e-142 < y < 4.99999999999999989e-114

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 95.6%

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

   4. Taylor expanded in x around 0 59.4%

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

      1. neg-mul-1 59.4%

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

      2. distribute-neg-frac2 59.4%

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

   6. Simplified 59.4%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-111}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-203}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{-203}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-142}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-114}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-30} \lor \neg \left(y \leq 100000000000\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z}\\ t_1 := 1 - \frac{x}{y}\\ t_2 := \frac{x}{z - y}\\ \mathbf{if}\;y \leq -7 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-158}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 2700000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+137}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) z)) (t_1 (- 1.0 (/ x y))) (t_2 (/ x (- z y))))
   (if (<= y -7e-19)
     t_1
     (if (<= y 1.25e-158)
       t_2
       (if (<= y 1.6e-113)
         t_0
         (if (<= y 1.05e-30)
           t_2
           (if (<= y 2700000000.0) t_0 (if (<= y 2.9e+137) t_2 t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / z;
	double t_1 = 1.0 - (x / y);
	double t_2 = x / (z - y);
	double tmp;
	if (y <= -7e-19) {
		tmp = t_1;
	} else if (y <= 1.25e-158) {
		tmp = t_2;
	} else if (y <= 1.6e-113) {
		tmp = t_0;
	} else if (y <= 1.05e-30) {
		tmp = t_2;
	} else if (y <= 2700000000.0) {
		tmp = t_0;
	} else if (y <= 2.9e+137) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = (x - y) / z
    t_1 = 1.0d0 - (x / y)
    t_2 = x / (z - y)
    if (y <= (-7d-19)) then
        tmp = t_1
    else if (y <= 1.25d-158) then
        tmp = t_2
    else if (y <= 1.6d-113) then
        tmp = t_0
    else if (y <= 1.05d-30) then
        tmp = t_2
    else if (y <= 2700000000.0d0) then
        tmp = t_0
    else if (y <= 2.9d+137) then
        tmp = t_2
    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;
	double t_1 = 1.0 - (x / y);
	double t_2 = x / (z - y);
	double tmp;
	if (y <= -7e-19) {
		tmp = t_1;
	} else if (y <= 1.25e-158) {
		tmp = t_2;
	} else if (y <= 1.6e-113) {
		tmp = t_0;
	} else if (y <= 1.05e-30) {
		tmp = t_2;
	} else if (y <= 2700000000.0) {
		tmp = t_0;
	} else if (y <= 2.9e+137) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / z
	t_1 = 1.0 - (x / y)
	t_2 = x / (z - y)
	tmp = 0
	if y <= -7e-19:
		tmp = t_1
	elif y <= 1.25e-158:
		tmp = t_2
	elif y <= 1.6e-113:
		tmp = t_0
	elif y <= 1.05e-30:
		tmp = t_2
	elif y <= 2700000000.0:
		tmp = t_0
	elif y <= 2.9e+137:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / z)
	t_1 = Float64(1.0 - Float64(x / y))
	t_2 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (y <= -7e-19)
		tmp = t_1;
	elseif (y <= 1.25e-158)
		tmp = t_2;
	elseif (y <= 1.6e-113)
		tmp = t_0;
	elseif (y <= 1.05e-30)
		tmp = t_2;
	elseif (y <= 2700000000.0)
		tmp = t_0;
	elseif (y <= 2.9e+137)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / z;
	t_1 = 1.0 - (x / y);
	t_2 = x / (z - y);
	tmp = 0.0;
	if (y <= -7e-19)
		tmp = t_1;
	elseif (y <= 1.25e-158)
		tmp = t_2;
	elseif (y <= 1.6e-113)
		tmp = t_0;
	elseif (y <= 1.05e-30)
		tmp = t_2;
	elseif (y <= 2700000000.0)
		tmp = t_0;
	elseif (y <= 2.9e+137)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e-19], t$95$1, If[LessEqual[y, 1.25e-158], t$95$2, If[LessEqual[y, 1.6e-113], t$95$0, If[LessEqual[y, 1.05e-30], t$95$2, If[LessEqual[y, 2700000000.0], t$95$0, If[LessEqual[y, 2.9e+137], t$95$2, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.00000000000000031e-19 or 2.89999999999999985e137 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 84.3%

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

      1. div-sub 84.4%

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

      2. sub-neg 84.4%

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

      3. *-inverses 84.4%

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

      4. metadata-eval 84.4%

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

      5. distribute-lft-in 84.4%

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

      6. metadata-eval 84.4%

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

      7. +-commutative 84.4%

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

      8. mul-1-neg 84.4%

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

      9. unsub-neg 84.4%

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

   5. Simplified 84.4%

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

   if -7.00000000000000031e-19 < y < 1.24999999999999993e-158 or 1.6000000000000001e-113 < y < 1.0500000000000001e-30 or 2.7e9 < y < 2.89999999999999985e137

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 80.4%

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

   if 1.24999999999999993e-158 < y < 1.6000000000000001e-113 or 1.0500000000000001e-30 < y < 2.7e9

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 82.6%

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

Alternative 4: 58.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-y}\\ \mathbf{if}\;y \leq -8.2 \cdot 10^{+151}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -36000000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+152}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (- y))))
   (if (<= y -8.2e+151)
     1.0
     (if (<= y -1.15e+106)
       t_0
       (if (<= y -36000000000000.0)
         1.0
         (if (<= y 5.2e-5)
           (/ x z)
           (if (<= y 4.2e+95) (/ (- y) z) (if (<= y 7.5e+152) t_0 1.0))))))))

C

double code(double x, double y, double z) {
	double t_0 = x / -y;
	double tmp;
	if (y <= -8.2e+151) {
		tmp = 1.0;
	} else if (y <= -1.15e+106) {
		tmp = t_0;
	} else if (y <= -36000000000000.0) {
		tmp = 1.0;
	} else if (y <= 5.2e-5) {
		tmp = x / z;
	} else if (y <= 4.2e+95) {
		tmp = -y / z;
	} else if (y <= 7.5e+152) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x / -y
    if (y <= (-8.2d+151)) then
        tmp = 1.0d0
    else if (y <= (-1.15d+106)) then
        tmp = t_0
    else if (y <= (-36000000000000.0d0)) then
        tmp = 1.0d0
    else if (y <= 5.2d-5) then
        tmp = x / z
    else if (y <= 4.2d+95) then
        tmp = -y / z
    else if (y <= 7.5d+152) then
        tmp = t_0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x / -y;
	double tmp;
	if (y <= -8.2e+151) {
		tmp = 1.0;
	} else if (y <= -1.15e+106) {
		tmp = t_0;
	} else if (y <= -36000000000000.0) {
		tmp = 1.0;
	} else if (y <= 5.2e-5) {
		tmp = x / z;
	} else if (y <= 4.2e+95) {
		tmp = -y / z;
	} else if (y <= 7.5e+152) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / -y
	tmp = 0
	if y <= -8.2e+151:
		tmp = 1.0
	elif y <= -1.15e+106:
		tmp = t_0
	elif y <= -36000000000000.0:
		tmp = 1.0
	elif y <= 5.2e-5:
		tmp = x / z
	elif y <= 4.2e+95:
		tmp = -y / z
	elif y <= 7.5e+152:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(-y))
	tmp = 0.0
	if (y <= -8.2e+151)
		tmp = 1.0;
	elseif (y <= -1.15e+106)
		tmp = t_0;
	elseif (y <= -36000000000000.0)
		tmp = 1.0;
	elseif (y <= 5.2e-5)
		tmp = Float64(x / z);
	elseif (y <= 4.2e+95)
		tmp = Float64(Float64(-y) / z);
	elseif (y <= 7.5e+152)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / -y;
	tmp = 0.0;
	if (y <= -8.2e+151)
		tmp = 1.0;
	elseif (y <= -1.15e+106)
		tmp = t_0;
	elseif (y <= -36000000000000.0)
		tmp = 1.0;
	elseif (y <= 5.2e-5)
		tmp = x / z;
	elseif (y <= 4.2e+95)
		tmp = -y / z;
	elseif (y <= 7.5e+152)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / (-y)), $MachinePrecision]}, If[LessEqual[y, -8.2e+151], 1.0, If[LessEqual[y, -1.15e+106], t$95$0, If[LessEqual[y, -36000000000000.0], 1.0, If[LessEqual[y, 5.2e-5], N[(x / z), $MachinePrecision], If[LessEqual[y, 4.2e+95], N[((-y) / z), $MachinePrecision], If[LessEqual[y, 7.5e+152], t$95$0, 1.0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.1999999999999996e151 or -1.1500000000000001e106 < y < -3.6e13 or 7.50000000000000046e152 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 75.0%

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

   if -8.1999999999999996e151 < y < -1.1500000000000001e106 or 4.2e95 < y < 7.50000000000000046e152

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.1%

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

   4. Taylor expanded in z around 0 57.0%

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

      1. associate-*r/ 57.0%

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

      2. neg-mul-1 57.0%

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

   6. Simplified 57.0%

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

   if -3.6e13 < y < 5.19999999999999968e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 59.3%

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

   if 5.19999999999999968e-5 < y < 4.2e95

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 66.7%

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

   4. Taylor expanded in x around 0 46.8%

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

      1. neg-mul-1 46.8%

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

      2. distribute-neg-frac2 46.8%

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

   6. Simplified 46.8%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+151}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+106}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{elif}\;y \leq -36000000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y - z}\\ t_1 := 1 - \frac{x}{y}\\ t_2 := \frac{x}{z - y}\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-49}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+152}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+249}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (- y z))) (t_1 (- 1.0 (/ x y))) (t_2 (/ x (- z y))))
   (if (<= y -7.8e-19)
     t_1
     (if (<= y 3.3e-49)
       t_2
       (if (<= y 8.5e+54)
         t_0
         (if (<= y 7.5e+152) t_2 (if (<= y 2.5e+249) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double t_1 = 1.0 - (x / y);
	double t_2 = x / (z - y);
	double tmp;
	if (y <= -7.8e-19) {
		tmp = t_1;
	} else if (y <= 3.3e-49) {
		tmp = t_2;
	} else if (y <= 8.5e+54) {
		tmp = t_0;
	} else if (y <= 7.5e+152) {
		tmp = t_2;
	} else if (y <= 2.5e+249) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y / (y - z)
    t_1 = 1.0d0 - (x / y)
    t_2 = x / (z - y)
    if (y <= (-7.8d-19)) then
        tmp = t_1
    else if (y <= 3.3d-49) then
        tmp = t_2
    else if (y <= 8.5d+54) then
        tmp = t_0
    else if (y <= 7.5d+152) then
        tmp = t_2
    else if (y <= 2.5d+249) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / (y - z);
	double t_1 = 1.0 - (x / y);
	double t_2 = x / (z - y);
	double tmp;
	if (y <= -7.8e-19) {
		tmp = t_1;
	} else if (y <= 3.3e-49) {
		tmp = t_2;
	} else if (y <= 8.5e+54) {
		tmp = t_0;
	} else if (y <= 7.5e+152) {
		tmp = t_2;
	} else if (y <= 2.5e+249) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (y - z)
	t_1 = 1.0 - (x / y)
	t_2 = x / (z - y)
	tmp = 0
	if y <= -7.8e-19:
		tmp = t_1
	elif y <= 3.3e-49:
		tmp = t_2
	elif y <= 8.5e+54:
		tmp = t_0
	elif y <= 7.5e+152:
		tmp = t_2
	elif y <= 2.5e+249:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(y - z))
	t_1 = Float64(1.0 - Float64(x / y))
	t_2 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (y <= -7.8e-19)
		tmp = t_1;
	elseif (y <= 3.3e-49)
		tmp = t_2;
	elseif (y <= 8.5e+54)
		tmp = t_0;
	elseif (y <= 7.5e+152)
		tmp = t_2;
	elseif (y <= 2.5e+249)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (y - z);
	t_1 = 1.0 - (x / y);
	t_2 = x / (z - y);
	tmp = 0.0;
	if (y <= -7.8e-19)
		tmp = t_1;
	elseif (y <= 3.3e-49)
		tmp = t_2;
	elseif (y <= 8.5e+54)
		tmp = t_0;
	elseif (y <= 7.5e+152)
		tmp = t_2;
	elseif (y <= 2.5e+249)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.8e-19], t$95$1, If[LessEqual[y, 3.3e-49], t$95$2, If[LessEqual[y, 8.5e+54], t$95$0, If[LessEqual[y, 7.5e+152], t$95$2, If[LessEqual[y, 2.5e+249], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.7999999999999999e-19 or 2.4999999999999998e249 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 84.5%

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

      1. div-sub 84.6%

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

      2. sub-neg 84.6%

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

      3. *-inverses 84.6%

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

      4. metadata-eval 84.6%

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

      5. distribute-lft-in 84.6%

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

      6. metadata-eval 84.6%

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

      7. +-commutative 84.6%

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

      8. mul-1-neg 84.6%

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

      9. unsub-neg 84.6%

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

   5. Simplified 84.6%

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

   if -7.7999999999999999e-19 < y < 3.3e-49 or 8.4999999999999995e54 < y < 7.50000000000000046e152

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 80.1%

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

   if 3.3e-49 < y < 8.4999999999999995e54 or 7.50000000000000046e152 < y < 2.4999999999999998e249

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 73.0%

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

      1. neg-mul-1 73.0%

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

      2. distribute-neg-frac 73.0%

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

   5. Simplified 73.0%

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

      1. frac-2neg 73.0%

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

      2. div-inv 72.9%

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

      3. remove-double-neg 72.9%

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

      4. sub-neg 72.9%

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

      5. distribute-neg-in 72.9%

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

      6. remove-double-neg 72.9%

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

   7. Applied egg-rr 72.9%

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

      1. associate-*r/ 73.0%

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

      2. *-rgt-identity 73.0%

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

      3. +-commutative 73.0%

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

      4. unsub-neg 73.0%

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

   9. Simplified 73.0%

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

Alternative 6: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{y}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 16000000000:\\ \;\;\;\;\frac{x}{z} - \frac{y}{z}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+137}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ x y))) (t_1 (/ x (- z y))))
   (if (<= y -5.5e-19)
     t_0
     (if (<= y 2.7e-31)
       t_1
       (if (<= y 16000000000.0)
         (- (/ x z) (/ y z))
         (if (<= y 2.9e+137) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 - (x / y);
	double t_1 = x / (z - y);
	double tmp;
	if (y <= -5.5e-19) {
		tmp = t_0;
	} else if (y <= 2.7e-31) {
		tmp = t_1;
	} else if (y <= 16000000000.0) {
		tmp = (x / z) - (y / z);
	} else if (y <= 2.9e+137) {
		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 = x / (z - y)
    if (y <= (-5.5d-19)) then
        tmp = t_0
    else if (y <= 2.7d-31) then
        tmp = t_1
    else if (y <= 16000000000.0d0) then
        tmp = (x / z) - (y / z)
    else if (y <= 2.9d+137) 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 = x / (z - y);
	double tmp;
	if (y <= -5.5e-19) {
		tmp = t_0;
	} else if (y <= 2.7e-31) {
		tmp = t_1;
	} else if (y <= 16000000000.0) {
		tmp = (x / z) - (y / z);
	} else if (y <= 2.9e+137) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 - (x / y)
	t_1 = x / (z - y)
	tmp = 0
	if y <= -5.5e-19:
		tmp = t_0
	elif y <= 2.7e-31:
		tmp = t_1
	elif y <= 16000000000.0:
		tmp = (x / z) - (y / z)
	elif y <= 2.9e+137:
		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(x / Float64(z - y))
	tmp = 0.0
	if (y <= -5.5e-19)
		tmp = t_0;
	elseif (y <= 2.7e-31)
		tmp = t_1;
	elseif (y <= 16000000000.0)
		tmp = Float64(Float64(x / z) - Float64(y / z));
	elseif (y <= 2.9e+137)
		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 = x / (z - y);
	tmp = 0.0;
	if (y <= -5.5e-19)
		tmp = t_0;
	elseif (y <= 2.7e-31)
		tmp = t_1;
	elseif (y <= 16000000000.0)
		tmp = (x / z) - (y / z);
	elseif (y <= 2.9e+137)
		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[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e-19], t$95$0, If[LessEqual[y, 2.7e-31], t$95$1, If[LessEqual[y, 16000000000.0], N[(N[(x / z), $MachinePrecision] - N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+137], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.4999999999999996e-19 or 2.89999999999999985e137 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 84.3%

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

      1. div-sub 84.4%

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

      2. sub-neg 84.4%

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

      3. *-inverses 84.4%

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

      4. metadata-eval 84.4%

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

      5. distribute-lft-in 84.4%

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

      6. metadata-eval 84.4%

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

      7. +-commutative 84.4%

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

      8. mul-1-neg 84.4%

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

      9. unsub-neg 84.4%

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

   5. Simplified 84.4%

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

   if -5.4999999999999996e-19 < y < 2.70000000000000014e-31 or 1.6e10 < y < 2.89999999999999985e137

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 78.0%

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

   if 2.70000000000000014e-31 < y < 1.6e10

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf 76.3%

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

      1. div-sub 76.4%

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

   5. Applied egg-rr 76.4%

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

Alternative 7: 60.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+23}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.85e+23)
   1.0
   (if (<= y 2.9e-5) (/ x z) (if (<= y 3.2e+86) (/ (- y) z) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.85e+23) {
		tmp = 1.0;
	} else if (y <= 2.9e-5) {
		tmp = x / z;
	} else if (y <= 3.2e+86) {
		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 <= (-2.85d+23)) then
        tmp = 1.0d0
    else if (y <= 2.9d-5) then
        tmp = x / z
    else if (y <= 3.2d+86) 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 <= -2.85e+23) {
		tmp = 1.0;
	} else if (y <= 2.9e-5) {
		tmp = x / z;
	} else if (y <= 3.2e+86) {
		tmp = -y / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.85e+23:
		tmp = 1.0
	elif y <= 2.9e-5:
		tmp = x / z
	elif y <= 3.2e+86:
		tmp = -y / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.85e+23)
		tmp = 1.0;
	elseif (y <= 2.9e-5)
		tmp = Float64(x / z);
	elseif (y <= 3.2e+86)
		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 <= -2.85e+23)
		tmp = 1.0;
	elseif (y <= 2.9e-5)
		tmp = x / z;
	elseif (y <= 3.2e+86)
		tmp = -y / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.85e+23], 1.0, If[LessEqual[y, 2.9e-5], N[(x / z), $MachinePrecision], If[LessEqual[y, 3.2e+86], N[((-y) / z), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.85e23 or 3.2e86 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 64.1%

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

   if -2.85e23 < y < 2.9e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 58.1%

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

   if 2.9e-5 < y < 3.2e86

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 66.7%

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

   4. Taylor expanded in x around 0 46.8%

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

      1. neg-mul-1 46.8%

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

      2. distribute-neg-frac2 46.8%

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

   6. Simplified 46.8%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.85 \cdot 10^{+23}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-19} \lor \neg \left(y \leq 2 \cdot 10^{+112}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.05e-19) (not (<= y 2e+112))) (- 1.0 (/ x y)) (/ x (- z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.05e-19) || !(y <= 2e+112)) {
		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 <= (-2.05d-19)) .or. (.not. (y <= 2d+112))) 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 <= -2.05e-19) || !(y <= 2e+112)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / (z - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.05e-19) or not (y <= 2e+112):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / (z - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.05e-19) || !(y <= 2e+112))
		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 <= -2.05e-19) || ~((y <= 2e+112)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / (z - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.05e-19], N[Not[LessEqual[y, 2e+112]], $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 < -2.04999999999999993e-19 or 1.9999999999999999e112 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 83.1%

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

      1. div-sub 83.1%

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

      2. sub-neg 83.1%

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

      3. *-inverses 83.1%

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

      4. metadata-eval 83.1%

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

      5. distribute-lft-in 83.1%

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

      6. metadata-eval 83.1%

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

      7. +-commutative 83.1%

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

      8. mul-1-neg 83.1%

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

      9. unsub-neg 83.1%

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

   5. Simplified 83.1%

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

   if -2.04999999999999993e-19 < y < 1.9999999999999999e112

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 74.9%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-19} \lor \neg \left(y \leq 2 \cdot 10^{+112}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+30}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -6e+30) 1.0 (if (<= y 2e+112) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -6e+30) {
		tmp = 1.0;
	} else if (y <= 2e+112) {
		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 <= (-6d+30)) then
        tmp = 1.0d0
    else if (y <= 2d+112) 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 <= -6e+30) {
		tmp = 1.0;
	} else if (y <= 2e+112) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -6e+30:
		tmp = 1.0
	elif y <= 2e+112:
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -6e+30)
		tmp = 1.0;
	elseif (y <= 2e+112)
		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 <= -6e+30)
		tmp = 1.0;
	elseif (y <= 2e+112)
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -6e+30], 1.0, If[LessEqual[y, 2e+112], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < -5.99999999999999956e30 or 1.9999999999999999e112 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 66.0%

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

   if -5.99999999999999956e30 < y < 1.9999999999999999e112

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 54.2%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
3. Recombined 2 regimes into one program.
4. 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: 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 99.9%

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

3. Taylor expanded in y around inf 32.1%

   \[\leadsto \color{blue}{1} \]
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 2024107 
(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)))