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

Percentage Accurate: 88.2% → 99.0%

Time: 7.9s

Alternatives: 8

Speedup: 0.8×

• 20.7% 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: 88.2% 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: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -130:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -130.0)
   (* y (- 1.0 (/ x z)))
   (if (<= y 1.0) (+ y (/ x z)) (* y (/ (- z x) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -130.0) {
		tmp = y * (1.0 - (x / z));
	} else if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = 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 <= (-130.0d0)) then
        tmp = y * (1.0d0 - (x / z))
    else if (y <= 1.0d0) then
        tmp = y + (x / z)
    else
        tmp = 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 <= -130.0) {
		tmp = y * (1.0 - (x / z));
	} else if (y <= 1.0) {
		tmp = y + (x / z);
	} else {
		tmp = y * ((z - x) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -130.0:
		tmp = y * (1.0 - (x / z))
	elif y <= 1.0:
		tmp = y + (x / z)
	else:
		tmp = y * ((z - x) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -130.0)
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	elseif (y <= 1.0)
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(y * Float64(Float64(z - x) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -130.0)
		tmp = y * (1.0 - (x / z));
	elseif (y <= 1.0)
		tmp = y + (x / z);
	else
		tmp = y * ((z - x) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -130.0], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -130

   1. Initial program 80.0%

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

   2. Taylor expanded in x around inf 94.4%

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

   3. Taylor expanded in y around 0 94.4%

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

      1. +-commutative 94.4%

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

      2. mul-1-neg 94.4%

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

      3. unsub-neg 94.4%

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

      4. *-commutative 94.4%

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

   5. Simplified 94.4%

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

   6. Taylor expanded in y around inf 99.2%

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

   if -130 < y < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 98.5%

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

   3. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

   5. Simplified 98.5%

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

   if 1 < y

   1. Initial program 70.6%

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

   2. Taylor expanded in y around inf 70.0%

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

      1. associate-/l* 98.8%

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

   4. Simplified 98.8%

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

      1. clear-num 98.7%

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

      2. associate-/r/ 98.8%

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

      3. clear-num 98.8%

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

   6. Applied egg-rr 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -130:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \end{array} \]

Alternative 2: 75.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{-x}{z}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+32}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+104} \lor \neg \left(y \leq 1.12 \cdot 10^{+286}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ (- x) z))))
   (if (<= y -2.2e+120)
     t_0
     (if (<= y 1.45e+32)
       (+ y (/ x z))
       (if (or (<= y 7.6e+104) (not (<= y 1.12e+286))) t_0 y)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (-x / z);
	double tmp;
	if (y <= -2.2e+120) {
		tmp = t_0;
	} else if (y <= 1.45e+32) {
		tmp = y + (x / z);
	} else if ((y <= 7.6e+104) || !(y <= 1.12e+286)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y * (-x / z)
    if (y <= (-2.2d+120)) then
        tmp = t_0
    else if (y <= 1.45d+32) then
        tmp = y + (x / z)
    else if ((y <= 7.6d+104) .or. (.not. (y <= 1.12d+286))) then
        tmp = t_0
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (-x / z);
	double tmp;
	if (y <= -2.2e+120) {
		tmp = t_0;
	} else if (y <= 1.45e+32) {
		tmp = y + (x / z);
	} else if ((y <= 7.6e+104) || !(y <= 1.12e+286)) {
		tmp = t_0;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (-x / z)
	tmp = 0
	if y <= -2.2e+120:
		tmp = t_0
	elif y <= 1.45e+32:
		tmp = y + (x / z)
	elif (y <= 7.6e+104) or not (y <= 1.12e+286):
		tmp = t_0
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(Float64(-x) / z))
	tmp = 0.0
	if (y <= -2.2e+120)
		tmp = t_0;
	elseif (y <= 1.45e+32)
		tmp = Float64(y + Float64(x / z));
	elseif ((y <= 7.6e+104) || !(y <= 1.12e+286))
		tmp = t_0;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (-x / z);
	tmp = 0.0;
	if (y <= -2.2e+120)
		tmp = t_0;
	elseif (y <= 1.45e+32)
		tmp = y + (x / z);
	elseif ((y <= 7.6e+104) || ~((y <= 1.12e+286)))
		tmp = t_0;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.2e+120], t$95$0, If[LessEqual[y, 1.45e+32], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 7.6e+104], N[Not[LessEqual[y, 1.12e+286]], $MachinePrecision]], t$95$0, y]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2000000000000001e120 or 1.45000000000000001e32 < y < 7.59999999999999938e104 or 1.12000000000000006e286 < y

   1. Initial program 86.9%

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

   2. Taylor expanded in x around inf 95.0%

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

   3. Taylor expanded in y around 0 95.0%

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

      1. +-commutative 95.0%

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

      2. mul-1-neg 95.0%

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

      3. unsub-neg 95.0%

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

      4. *-commutative 95.0%

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

   5. Simplified 95.0%

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

   6. Taylor expanded in y around inf 99.9%

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

   7. Taylor expanded in x around inf 65.9%

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

      1. associate-*r/ 65.9%

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

      2. mul-1-neg 65.9%

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

      3. distribute-rgt-neg-in 65.9%

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

      4. associate-*r/ 70.0%

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

   9. Simplified 70.0%

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

   if -2.2000000000000001e120 < y < 1.45000000000000001e32

   1. Initial program 94.7%

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

   2. Taylor expanded in z around inf 90.1%

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

   3. Taylor expanded in x around 0 94.2%

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

      1. +-commutative 94.2%

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

   5. Simplified 94.2%

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

   if 7.59999999999999938e104 < y < 1.12000000000000006e286

   1. Initial program 58.4%

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

   2. Taylor expanded in x around 0 66.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+32}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+104} \lor \neg \left(y \leq 1.12 \cdot 10^{+286}\right):\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 3: 76.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{-x}{z}\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+119}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+54}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+290}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ (- x) z))))
   (if (<= y -2.5e+119)
     t_0
     (if (<= y 1.15e+54)
       (+ y (/ x z))
       (if (<= y 1.02e+104) (/ y (/ (- z) x)) (if (<= y 4.1e+290) y t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (-x / z);
	double tmp;
	if (y <= -2.5e+119) {
		tmp = t_0;
	} else if (y <= 1.15e+54) {
		tmp = y + (x / z);
	} else if (y <= 1.02e+104) {
		tmp = y / (-z / x);
	} else if (y <= 4.1e+290) {
		tmp = y;
	} 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 = y * (-x / z)
    if (y <= (-2.5d+119)) then
        tmp = t_0
    else if (y <= 1.15d+54) then
        tmp = y + (x / z)
    else if (y <= 1.02d+104) then
        tmp = y / (-z / x)
    else if (y <= 4.1d+290) then
        tmp = y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (-x / z);
	double tmp;
	if (y <= -2.5e+119) {
		tmp = t_0;
	} else if (y <= 1.15e+54) {
		tmp = y + (x / z);
	} else if (y <= 1.02e+104) {
		tmp = y / (-z / x);
	} else if (y <= 4.1e+290) {
		tmp = y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (-x / z)
	tmp = 0
	if y <= -2.5e+119:
		tmp = t_0
	elif y <= 1.15e+54:
		tmp = y + (x / z)
	elif y <= 1.02e+104:
		tmp = y / (-z / x)
	elif y <= 4.1e+290:
		tmp = y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(Float64(-x) / z))
	tmp = 0.0
	if (y <= -2.5e+119)
		tmp = t_0;
	elseif (y <= 1.15e+54)
		tmp = Float64(y + Float64(x / z));
	elseif (y <= 1.02e+104)
		tmp = Float64(y / Float64(Float64(-z) / x));
	elseif (y <= 4.1e+290)
		tmp = y;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (-x / z);
	tmp = 0.0;
	if (y <= -2.5e+119)
		tmp = t_0;
	elseif (y <= 1.15e+54)
		tmp = y + (x / z);
	elseif (y <= 1.02e+104)
		tmp = y / (-z / x);
	elseif (y <= 4.1e+290)
		tmp = y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.5e+119], t$95$0, If[LessEqual[y, 1.15e+54], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.02e+104], N[(y / N[((-z) / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.1e+290], y, t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.5e119 or 4.09999999999999999e290 < y

   1. Initial program 87.9%

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

   2. Taylor expanded in x around inf 96.0%

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

   3. Taylor expanded in y around 0 96.0%

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

      1. +-commutative 96.0%

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

      2. mul-1-neg 96.0%

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

      3. unsub-neg 96.0%

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

      4. *-commutative 96.0%

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

   5. Simplified 96.0%

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

   6. Taylor expanded in y around inf 99.9%

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

   7. Taylor expanded in x around inf 64.2%

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

      1. associate-*r/ 64.2%

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

      2. mul-1-neg 64.2%

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

      3. distribute-rgt-neg-in 64.2%

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

      4. associate-*r/ 67.2%

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

   9. Simplified 67.2%

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

   if -2.5e119 < y < 1.14999999999999997e54

   1. Initial program 94.8%

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

   2. Taylor expanded in z around inf 89.6%

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

   3. Taylor expanded in x around 0 93.7%

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

      1. +-commutative 93.7%

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

   5. Simplified 93.7%

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

   if 1.14999999999999997e54 < y < 1.02e104

   1. Initial program 78.6%

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

   2. Taylor expanded in y around inf 78.6%

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

      1. associate-/l* 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in z around 0 89.0%

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

      1. associate-*r/ 89.0%

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

      2. neg-mul-1 89.0%

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

   7. Simplified 89.0%

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

   if 1.02e104 < y < 4.09999999999999999e290

   1. Initial program 58.4%

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

   2. Taylor expanded in x around 0 66.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+119}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+54}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+290}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \end{array} \]

Alternative 4: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -130 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -130.0) (not (<= y 1.0))) (* y (- 1.0 (/ x z))) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -130.0) || !(y <= 1.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = 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 <= (-130.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -130.0) || !(y <= 1.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -130.0) or not (y <= 1.0):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -130.0) || !(y <= 1.0))
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -130.0) || ~((y <= 1.0)))
		tmp = y * (1.0 - (x / z));
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -130.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -130 or 1 < y

   1. Initial program 75.6%

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

   2. Taylor expanded in x around inf 90.9%

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

   3. Taylor expanded in y around 0 90.9%

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

      1. +-commutative 90.9%

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

      2. mul-1-neg 90.9%

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

      3. unsub-neg 90.9%

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

      4. *-commutative 90.9%

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

   5. Simplified 90.9%

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

   6. Taylor expanded in y around inf 99.0%

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

   if -130 < y < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 98.5%

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

   3. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

   5. Simplified 98.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -130 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 5: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 72000000000:\\ \;\;\;\;y + \frac{x - y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 72000000000.0) (+ y (/ (- x (* y x)) z)) (* y (- 1.0 (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 72000000000.0) {
		tmp = y + ((x - (y * x)) / z);
	} else {
		tmp = y * (1.0 - (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 <= 72000000000.0d0) then
        tmp = y + ((x - (y * x)) / z)
    else
        tmp = y * (1.0d0 - (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 72000000000.0) {
		tmp = y + ((x - (y * x)) / z);
	} else {
		tmp = y * (1.0 - (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 72000000000.0:
		tmp = y + ((x - (y * x)) / z)
	else:
		tmp = y * (1.0 - (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 72000000000.0)
		tmp = Float64(y + Float64(Float64(x - Float64(y * x)) / z));
	else
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 72000000000.0)
		tmp = y + ((x - (y * x)) / z);
	else
		tmp = y * (1.0 - (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 72000000000.0], N[(y + N[(N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.2e10

   1. Initial program 93.3%

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

   2. Taylor expanded in x around inf 98.1%

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

   3. Taylor expanded in y around 0 98.1%

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

      1. +-commutative 98.1%

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

      2. mul-1-neg 98.1%

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

      3. unsub-neg 98.1%

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

      4. *-commutative 98.1%

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

   5. Simplified 98.1%

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

   if 7.2e10 < y

   1. Initial program 70.1%

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

   2. Taylor expanded in x around inf 86.8%

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

   3. Taylor expanded in y around 0 86.8%

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

      1. +-commutative 86.8%

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

      2. mul-1-neg 86.8%

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

      3. unsub-neg 86.8%

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

      4. *-commutative 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in y around inf 99.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 72000000000:\\ \;\;\;\;y + \frac{x - y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \]

Alternative 6: 61.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-15}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.5e-15) y (if (<= y 3.25e-40) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.5e-15) {
		tmp = y;
	} else if (y <= 3.25e-40) {
		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 <= (-7.5d-15)) then
        tmp = y
    else if (y <= 3.25d-40) 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 <= -7.5e-15) {
		tmp = y;
	} else if (y <= 3.25e-40) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.5e-15:
		tmp = y
	elif y <= 3.25e-40:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.5e-15)
		tmp = y;
	elseif (y <= 3.25e-40)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7.5e-15)
		tmp = y;
	elseif (y <= 3.25e-40)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.5e-15], y, If[LessEqual[y, 3.25e-40], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -7.4999999999999996e-15 or 3.25e-40 < y

   1. Initial program 79.0%

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

   2. Taylor expanded in x around 0 53.9%

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

   if -7.4999999999999996e-15 < y < 3.25e-40

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 74.9%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-15}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7: 78.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

2. Taylor expanded in z around inf 70.0%

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

3. Taylor expanded in x around 0 78.2%

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

   1. +-commutative 78.2%

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

5. Simplified 78.2%

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

6. Final simplification 78.2%

   \[\leadsto y + \frac{x}{z} \]

Alternative 8: 40.0% 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 88.1%

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

2. Taylor expanded in x around 0 42.1%

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

3. Final simplification 42.1%

   \[\leadsto y \]

Developer target: 93.5% 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 2023230 
(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))