Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3

Percentage Accurate: 88.7% → 99.9%

Time: 7.4s

Alternatives: 11

Speedup: 0.7×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x + y \cdot \left(z - x\right)}{z} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / 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 + (y * (z - x))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y \cdot \left(z - x\right)}{z} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (x + (y * (z - x))) / 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 + (y * (z - x))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-60)
   (+ (/ x z) (* y (- 1.0 (/ x z))))
   (if (<= y 2.05e+15) (/ (fma y (- z x) x) z) (- y (/ y (/ z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-60) {
		tmp = (x / z) + (y * (1.0 - (x / z)));
	} else if (y <= 2.05e+15) {
		tmp = fma(y, (z - x), x) / z;
	} else {
		tmp = y - (y / (z / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-60)
		tmp = Float64(Float64(x / z) + Float64(y * Float64(1.0 - Float64(x / z))));
	elseif (y <= 2.05e+15)
		tmp = Float64(fma(y, Float64(z - x), x) / z);
	else
		tmp = Float64(y - Float64(y / Float64(z / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-60], N[(N[(x / z), $MachinePrecision] + N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e+15], N[(N[(y * N[(z - x), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision], N[(y - N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.0000000000000001e-60

   1. Initial program 79.0%

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

   2. Taylor expanded in y around 0 99.9%

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

   if -5.0000000000000001e-60 < y < 2.05e15

   1. Initial program 100.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   if 2.05e15 < y

   1. Initial program 80.4%

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

   2. Taylor expanded in y around 0 81.7%

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

   3. Taylor expanded in y around inf 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. associate-*r/ 91.2%

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

      6. associate-*l/ 88.8%

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

      7. distribute-rgt-neg-out 88.8%

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

      8. unsub-neg 88.8%

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

      9. associate-/r/ 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 2: 58.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{y}{z}\\ t_1 := \frac{y}{\frac{-z}{x}}\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{+269}:\\ \;\;\;\;\frac{z}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{+177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-108}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ y z))) (t_1 (/ y (/ (- z) x))))
   (if (<= y -1.5e+269)
     (/ z (/ z y))
     (if (<= y -1.1e+244)
       t_1
       (if (<= y -7.8e+177)
         t_0
         (if (<= y -3.8e-108)
           y
           (if (<= y 5.3e-84) (/ x z) (if (<= y 4.9e+39) t_0 t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y / z);
	double t_1 = y / (-z / x);
	double tmp;
	if (y <= -1.5e+269) {
		tmp = z / (z / y);
	} else if (y <= -1.1e+244) {
		tmp = t_1;
	} else if (y <= -7.8e+177) {
		tmp = t_0;
	} else if (y <= -3.8e-108) {
		tmp = y;
	} else if (y <= 5.3e-84) {
		tmp = x / z;
	} else if (y <= 4.9e+39) {
		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) :: tmp
    t_0 = z * (y / z)
    t_1 = y / (-z / x)
    if (y <= (-1.5d+269)) then
        tmp = z / (z / y)
    else if (y <= (-1.1d+244)) then
        tmp = t_1
    else if (y <= (-7.8d+177)) then
        tmp = t_0
    else if (y <= (-3.8d-108)) then
        tmp = y
    else if (y <= 5.3d-84) then
        tmp = x / z
    else if (y <= 4.9d+39) 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 = z * (y / z);
	double t_1 = y / (-z / x);
	double tmp;
	if (y <= -1.5e+269) {
		tmp = z / (z / y);
	} else if (y <= -1.1e+244) {
		tmp = t_1;
	} else if (y <= -7.8e+177) {
		tmp = t_0;
	} else if (y <= -3.8e-108) {
		tmp = y;
	} else if (y <= 5.3e-84) {
		tmp = x / z;
	} else if (y <= 4.9e+39) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y / z)
	t_1 = y / (-z / x)
	tmp = 0
	if y <= -1.5e+269:
		tmp = z / (z / y)
	elif y <= -1.1e+244:
		tmp = t_1
	elif y <= -7.8e+177:
		tmp = t_0
	elif y <= -3.8e-108:
		tmp = y
	elif y <= 5.3e-84:
		tmp = x / z
	elif y <= 4.9e+39:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y / z))
	t_1 = Float64(y / Float64(Float64(-z) / x))
	tmp = 0.0
	if (y <= -1.5e+269)
		tmp = Float64(z / Float64(z / y));
	elseif (y <= -1.1e+244)
		tmp = t_1;
	elseif (y <= -7.8e+177)
		tmp = t_0;
	elseif (y <= -3.8e-108)
		tmp = y;
	elseif (y <= 5.3e-84)
		tmp = Float64(x / z);
	elseif (y <= 4.9e+39)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y / z);
	t_1 = y / (-z / x);
	tmp = 0.0;
	if (y <= -1.5e+269)
		tmp = z / (z / y);
	elseif (y <= -1.1e+244)
		tmp = t_1;
	elseif (y <= -7.8e+177)
		tmp = t_0;
	elseif (y <= -3.8e-108)
		tmp = y;
	elseif (y <= 5.3e-84)
		tmp = x / z;
	elseif (y <= 4.9e+39)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[((-z) / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.5e+269], N[(z / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.1e+244], t$95$1, If[LessEqual[y, -7.8e+177], t$95$0, If[LessEqual[y, -3.8e-108], y, If[LessEqual[y, 5.3e-84], N[(x / z), $MachinePrecision], If[LessEqual[y, 4.9e+39], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.5000000000000001e269

   1. Initial program 42.9%

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

   2. Taylor expanded in x around 0 26.5%

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

      1. associate-/l* 67.4%

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

      2. associate-/r/ 83.7%

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

   4. Applied egg-rr 83.7%

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

      1. *-commutative 83.7%

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

      2. clear-num 83.7%

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

      3. un-div-inv 83.8%

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

   6. Applied egg-rr 83.8%

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

   if -1.5000000000000001e269 < y < -1.10000000000000001e244 or 4.89999999999999987e39 < y

   1. Initial program 82.4%

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

   2. Taylor expanded in y around inf 82.4%

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

      1. associate-/l* 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 64.5%

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

      1. associate-*r/ 64.5%

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

      2. neg-mul-1 64.5%

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

   7. Simplified 64.5%

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

   if -1.10000000000000001e244 < y < -7.7999999999999998e177 or 5.3000000000000004e-84 < y < 4.89999999999999987e39

   1. Initial program 79.3%

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

   2. Taylor expanded in x around 0 46.4%

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

      1. associate-/l* 67.0%

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

      2. associate-/r/ 76.3%

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

   4. Applied egg-rr 76.3%

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

   if -7.7999999999999998e177 < y < -3.79999999999999973e-108

   1. Initial program 92.1%

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

   2. Taylor expanded in x around 0 58.0%

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

   if -3.79999999999999973e-108 < y < 5.3000000000000004e-84

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 81.9%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+269}:\\ \;\;\;\;\frac{z}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+244}:\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{+177}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-108}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+39}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+22} \lor \neg \left(y \leq 3.4 \cdot 10^{+15}\right):\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1e+22) (not (<= y 3.4e+15)))
   (- y (/ y (/ z x)))
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e+22) || !(y <= 3.4e+15)) {
		tmp = y - (y / (z / x));
	} else {
		tmp = (x + (y * (z - 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 <= (-1d+22)) .or. (.not. (y <= 3.4d+15))) then
        tmp = y - (y / (z / x))
    else
        tmp = (x + (y * (z - x))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e+22) || !(y <= 3.4e+15)) {
		tmp = y - (y / (z / x));
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1e+22) or not (y <= 3.4e+15):
		tmp = y - (y / (z / x))
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1e+22) || !(y <= 3.4e+15))
		tmp = Float64(y - Float64(y / Float64(z / x)));
	else
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1e+22) || ~((y <= 3.4e+15)))
		tmp = y - (y / (z / x));
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1e+22], N[Not[LessEqual[y, 3.4e+15]], $MachinePrecision]], N[(y - N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e22 or 3.4e15 < y

   1. Initial program 76.2%

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

   2. Taylor expanded in y around 0 90.3%

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

   3. Taylor expanded in y around inf 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. associate-*r/ 92.3%

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

      6. associate-*l/ 88.9%

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

      7. distribute-rgt-neg-out 88.9%

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

      8. unsub-neg 88.9%

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

      9. associate-/r/ 100.0%

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

   5. Simplified 100.0%

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

   if -1e22 < y < 3.4e15

   1. Initial program 99.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+22} \lor \neg \left(y \leq 3.4 \cdot 10^{+15}\right):\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-60)
   (+ (/ x z) (* y (- 1.0 (/ x z))))
   (if (<= y 2.05e+15) (/ (+ x (* y (- z x))) z) (- y (/ y (/ z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-60) {
		tmp = (x / z) + (y * (1.0 - (x / z)));
	} else if (y <= 2.05e+15) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y - (y / (z / x));
	}
	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 <= (-2d-60)) then
        tmp = (x / z) + (y * (1.0d0 - (x / z)))
    else if (y <= 2.05d+15) then
        tmp = (x + (y * (z - x))) / z
    else
        tmp = y - (y / (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-60) {
		tmp = (x / z) + (y * (1.0 - (x / z)));
	} else if (y <= 2.05e+15) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y - (y / (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2e-60:
		tmp = (x / z) + (y * (1.0 - (x / z)))
	elif y <= 2.05e+15:
		tmp = (x + (y * (z - x))) / z
	else:
		tmp = y - (y / (z / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-60)
		tmp = Float64(Float64(x / z) + Float64(y * Float64(1.0 - Float64(x / z))));
	elseif (y <= 2.05e+15)
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	else
		tmp = Float64(y - Float64(y / Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2e-60)
		tmp = (x / z) + (y * (1.0 - (x / z)));
	elseif (y <= 2.05e+15)
		tmp = (x + (y * (z - x))) / z;
	else
		tmp = y - (y / (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-60], N[(N[(x / z), $MachinePrecision] + N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e+15], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y - N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9999999999999999e-60

   1. Initial program 79.0%

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

   2. Taylor expanded in y around 0 99.9%

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

   if -1.9999999999999999e-60 < y < 2.05e15

   1. Initial program 100.0%

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

   if 2.05e15 < y

   1. Initial program 80.4%

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

   2. Taylor expanded in y around 0 81.7%

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

   3. Taylor expanded in y around inf 99.9%

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

      1. sub-neg 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. associate-*r/ 91.2%

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

      6. associate-*l/ 88.8%

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

      7. distribute-rgt-neg-out 88.8%

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

      8. unsub-neg 88.8%

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

      9. associate-/r/ 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 5: 56.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-108}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+165}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+193}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.5e-108)
   y
   (if (<= y 5.2e-84)
     (/ x z)
     (if (<= y 3e+165) y (if (<= y 2.4e+193) (/ (- x) z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.5e-108) {
		tmp = y;
	} else if (y <= 5.2e-84) {
		tmp = x / z;
	} else if (y <= 3e+165) {
		tmp = y;
	} else if (y <= 2.4e+193) {
		tmp = -x / z;
	} else {
		tmp = 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 <= (-2.5d-108)) then
        tmp = y
    else if (y <= 5.2d-84) then
        tmp = x / z
    else if (y <= 3d+165) then
        tmp = y
    else if (y <= 2.4d+193) then
        tmp = -x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.5e-108) {
		tmp = y;
	} else if (y <= 5.2e-84) {
		tmp = x / z;
	} else if (y <= 3e+165) {
		tmp = y;
	} else if (y <= 2.4e+193) {
		tmp = -x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.5e-108:
		tmp = y
	elif y <= 5.2e-84:
		tmp = x / z
	elif y <= 3e+165:
		tmp = y
	elif y <= 2.4e+193:
		tmp = -x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.5e-108)
		tmp = y;
	elseif (y <= 5.2e-84)
		tmp = Float64(x / z);
	elseif (y <= 3e+165)
		tmp = y;
	elseif (y <= 2.4e+193)
		tmp = Float64(Float64(-x) / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.5e-108)
		tmp = y;
	elseif (y <= 5.2e-84)
		tmp = x / z;
	elseif (y <= 3e+165)
		tmp = y;
	elseif (y <= 2.4e+193)
		tmp = -x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.5e-108], y, If[LessEqual[y, 5.2e-84], N[(x / z), $MachinePrecision], If[LessEqual[y, 3e+165], y, If[LessEqual[y, 2.4e+193], N[((-x) / z), $MachinePrecision], y]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5e-108 or 5.2e-84 < y < 2.9999999999999999e165 or 2.4e193 < y

   1. Initial program 82.0%

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

   2. Taylor expanded in x around 0 53.7%

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

   if -2.5e-108 < y < 5.2e-84

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 81.9%

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

   if 2.9999999999999999e165 < y < 2.4e193

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. *-lft-identity 100.0%

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

      3. mul-1-neg 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

   4. Simplified 100.0%

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

      1. frac-2neg 100.0%

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

      2. div-inv 99.8%

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

      3. add-sqr-sqrt 56.7%

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

      4. sqrt-unprod 44.4%

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

      5. sqr-neg 44.4%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-rgt-identity 0.0%

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

      3. neg-sub0 0.0%

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

      4. sub-neg 0.0%

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

      5. distribute-rgt-neg-out 0.0%

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

      6. +-commutative 0.0%

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

      7. associate--r+ 0.0%

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

      8. neg-sub0 0.0%

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

      9. distribute-rgt-neg-out 0.0%

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

      10. remove-double-neg 0.0%

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

      11. *-commutative 0.0%

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

   8. Simplified 0.0%

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

   9. Taylor expanded in y around 0 59.0%

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

       1. neg-mul-1 59.0%

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

       2. distribute-neg-frac 59.0%

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

   11. Simplified 59.0%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-108}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+165}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+193}:\\ \;\;\;\;\frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6: 58.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -8 \cdot 10^{+177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-108}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ y z))))
   (if (<= y -8e+177)
     t_0
     (if (<= y -1.4e-108) y (if (<= y 1.25e-84) (/ x z) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y / z);
	double tmp;
	if (y <= -8e+177) {
		tmp = t_0;
	} else if (y <= -1.4e-108) {
		tmp = y;
	} else if (y <= 1.25e-84) {
		tmp = x / 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 = z * (y / z)
    if (y <= (-8d+177)) then
        tmp = t_0
    else if (y <= (-1.4d-108)) then
        tmp = y
    else if (y <= 1.25d-84) then
        tmp = x / 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 = z * (y / z);
	double tmp;
	if (y <= -8e+177) {
		tmp = t_0;
	} else if (y <= -1.4e-108) {
		tmp = y;
	} else if (y <= 1.25e-84) {
		tmp = x / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y / z)
	tmp = 0
	if y <= -8e+177:
		tmp = t_0
	elif y <= -1.4e-108:
		tmp = y
	elif y <= 1.25e-84:
		tmp = x / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y / z))
	tmp = 0.0
	if (y <= -8e+177)
		tmp = t_0;
	elseif (y <= -1.4e-108)
		tmp = y;
	elseif (y <= 1.25e-84)
		tmp = Float64(x / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y / z);
	tmp = 0.0;
	if (y <= -8e+177)
		tmp = t_0;
	elseif (y <= -1.4e-108)
		tmp = y;
	elseif (y <= 1.25e-84)
		tmp = x / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8e+177], t$95$0, If[LessEqual[y, -1.4e-108], y, If[LessEqual[y, 1.25e-84], N[(x / z), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.0000000000000001e177 or 1.25e-84 < y

   1. Initial program 77.6%

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

   2. Taylor expanded in x around 0 30.6%

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

      1. associate-/l* 48.0%

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

      2. associate-/r/ 59.0%

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

   4. Applied egg-rr 59.0%

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

   if -8.0000000000000001e177 < y < -1.4e-108

   1. Initial program 92.1%

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

   2. Taylor expanded in x around 0 58.0%

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

   if -1.4e-108 < y < 1.25e-84

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 81.9%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+177}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-108}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]

Alternative 7: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-94} \lor \neg \left(x \leq 1.6 \cdot 10^{-111}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.4e-94) (not (<= x 1.6e-111))) (* (/ x z) (- 1.0 y)) y))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.4e-94) || !(x <= 1.6e-111)) {
		tmp = (x / z) * (1.0 - y);
	} else {
		tmp = 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 ((x <= (-4.4d-94)) .or. (.not. (x <= 1.6d-111))) then
        tmp = (x / z) * (1.0d0 - y)
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.4e-94) || !(x <= 1.6e-111)) {
		tmp = (x / z) * (1.0 - y);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.4e-94) or not (x <= 1.6e-111):
		tmp = (x / z) * (1.0 - y)
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.4e-94) || !(x <= 1.6e-111))
		tmp = Float64(Float64(x / z) * Float64(1.0 - y));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.4e-94) || ~((x <= 1.6e-111)))
		tmp = (x / z) * (1.0 - y);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.4e-94], N[Not[LessEqual[x, 1.6e-111]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if x < -4.40000000000000002e-94 or 1.5999999999999999e-111 < x

   1. Initial program 90.7%

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

   2. Taylor expanded in x around inf 77.5%

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

      1. associate-/l* 78.2%

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

      2. associate-/r/ 79.2%

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

      3. mul-1-neg 79.2%

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

      4. unsub-neg 79.2%

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

   4. Simplified 79.2%

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

   if -4.40000000000000002e-94 < x < 1.5999999999999999e-111

   1. Initial program 84.3%

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

   2. Taylor expanded in x around 0 74.3%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-94} \lor \neg \left(x \leq 1.6 \cdot 10^{-111}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 8: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-109} \lor \neg \left(y \leq 7.8 \cdot 10^{-85}\right):\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.1e-109) (not (<= y 7.8e-85)))
   (- y (/ y (/ z x)))
   (* (/ x z) (- 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.1e-109) || !(y <= 7.8e-85)) {
		tmp = y - (y / (z / x));
	} else {
		tmp = (x / z) * (1.0 - 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 <= (-4.1d-109)) .or. (.not. (y <= 7.8d-85))) then
        tmp = y - (y / (z / x))
    else
        tmp = (x / z) * (1.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.1e-109) || !(y <= 7.8e-85)) {
		tmp = y - (y / (z / x));
	} else {
		tmp = (x / z) * (1.0 - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.1e-109) or not (y <= 7.8e-85):
		tmp = y - (y / (z / x))
	else:
		tmp = (x / z) * (1.0 - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.1e-109) || !(y <= 7.8e-85))
		tmp = Float64(y - Float64(y / Float64(z / x)));
	else
		tmp = Float64(Float64(x / z) * Float64(1.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.1e-109) || ~((y <= 7.8e-85)))
		tmp = y - (y / (z / x));
	else
		tmp = (x / z) * (1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.1e-109], N[Not[LessEqual[y, 7.8e-85]], $MachinePrecision]], N[(y - N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1000000000000002e-109 or 7.79999999999999977e-85 < y

   1. Initial program 82.8%

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

   2. Taylor expanded in y around 0 92.4%

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

   3. Taylor expanded in y around inf 94.9%

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

      1. sub-neg 94.9%

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

      2. distribute-lft-in 94.9%

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

      3. *-rgt-identity 94.9%

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

      4. distribute-frac-neg 94.9%

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

      5. associate-*r/ 88.9%

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

      6. associate-*l/ 85.9%

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

      7. distribute-rgt-neg-out 85.9%

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

      8. unsub-neg 85.9%

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

      9. associate-/r/ 94.4%

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

   5. Simplified 94.4%

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

   if -4.1000000000000002e-109 < y < 7.79999999999999977e-85

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 81.9%

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

      1. associate-/l* 81.9%

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

      2. associate-/r/ 81.9%

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

      3. mul-1-neg 81.9%

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

      4. unsub-neg 81.9%

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

   4. Simplified 81.9%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \left(1 - y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-109} \lor \neg \left(y \leq 7.8 \cdot 10^{-85}\right):\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 - y\right)\\ \end{array} \]

Alternative 9: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-109} \lor \neg \left(y \leq 3.2 \cdot 10^{-84}\right):\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -9.5e-109) (not (<= y 3.2e-84)))
   (- y (/ y (/ z x)))
   (/ (- x (* y x)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -9.5e-109) || !(y <= 3.2e-84)) {
		tmp = y - (y / (z / x));
	} else {
		tmp = (x - (y * 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 <= (-9.5d-109)) .or. (.not. (y <= 3.2d-84))) then
        tmp = y - (y / (z / x))
    else
        tmp = (x - (y * x)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -9.5e-109) || !(y <= 3.2e-84)) {
		tmp = y - (y / (z / x));
	} else {
		tmp = (x - (y * x)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -9.5e-109) or not (y <= 3.2e-84):
		tmp = y - (y / (z / x))
	else:
		tmp = (x - (y * x)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -9.5e-109) || !(y <= 3.2e-84))
		tmp = Float64(y - Float64(y / Float64(z / x)));
	else
		tmp = Float64(Float64(x - Float64(y * x)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -9.5e-109) || ~((y <= 3.2e-84)))
		tmp = y - (y / (z / x));
	else
		tmp = (x - (y * x)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -9.5e-109], N[Not[LessEqual[y, 3.2e-84]], $MachinePrecision]], N[(y - N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.49999999999999933e-109 or 3.1999999999999999e-84 < y

   1. Initial program 82.8%

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

   2. Taylor expanded in y around 0 92.4%

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

   3. Taylor expanded in y around inf 94.9%

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

      1. sub-neg 94.9%

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

      2. distribute-lft-in 94.9%

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

      3. *-rgt-identity 94.9%

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

      4. distribute-frac-neg 94.9%

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

      5. associate-*r/ 88.9%

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

      6. associate-*l/ 85.9%

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

      7. distribute-rgt-neg-out 85.9%

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

      8. unsub-neg 85.9%

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

      9. associate-/r/ 94.4%

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

   5. Simplified 94.4%

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

   if -9.49999999999999933e-109 < y < 3.1999999999999999e-84

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 81.9%

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

      1. distribute-rgt-in 81.9%

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

      2. *-lft-identity 81.9%

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

      3. mul-1-neg 81.9%

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

      4. cancel-sign-sub-inv 81.9%

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

   4. Simplified 81.9%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-109} \lor \neg \left(y \leq 3.2 \cdot 10^{-84}\right):\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot x}{z}\\ \end{array} \]

Alternative 10: 57.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-109}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.8e-109) y (if (<= y 7.8e-85) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e-109) {
		tmp = y;
	} else if (y <= 7.8e-85) {
		tmp = x / z;
	} else {
		tmp = 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 <= (-4.8d-109)) then
        tmp = y
    else if (y <= 7.8d-85) then
        tmp = x / z
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.8e-109) {
		tmp = y;
	} else if (y <= 7.8e-85) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.8e-109:
		tmp = y
	elif y <= 7.8e-85:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.8e-109)
		tmp = y;
	elseif (y <= 7.8e-85)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.8e-109)
		tmp = y;
	elseif (y <= 7.8e-85)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.8e-109], y, If[LessEqual[y, 7.8e-85], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -4.79999999999999977e-109 or 7.79999999999999977e-85 < y

   1. Initial program 82.8%

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

   2. Taylor expanded in x around 0 51.6%

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

   if -4.79999999999999977e-109 < y < 7.79999999999999977e-85

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 81.9%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-109}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 11: 40.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 88.4%

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

2. Taylor expanded in x around 0 41.9%

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

3. Final simplification 41.9%

   \[\leadsto y \]

Developer target: 93.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}} \end{array} \]

FPCore

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

C

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

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 / z)) - (y / (z / x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023290 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

  (/ (+ x (* y (- z x))) z))