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

Percentage Accurate: 88.5% → 99.9%

Time: 8.4s

Alternatives: 9

Speedup: 0.8×

• 20.9% 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: 88.5% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+33} \lor \neg \left(z \leq 10^{-10}\right):\\ \;\;\;\;\frac{x}{z} + \left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1e+33) (not (<= z 1e-10)))
   (+ (/ x z) (* (- 1.0 (/ x z)) y))
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e+33) || !(z <= 1e-10)) {
		tmp = (x / z) + ((1.0 - (x / z)) * y);
	} 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 ((z <= (-1d+33)) .or. (.not. (z <= 1d-10))) then
        tmp = (x / z) + ((1.0d0 - (x / z)) * y)
    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 ((z <= -1e+33) || !(z <= 1e-10)) {
		tmp = (x / z) + ((1.0 - (x / z)) * y);
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1e+33) or not (z <= 1e-10):
		tmp = (x / z) + ((1.0 - (x / z)) * y)
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1e+33) || !(z <= 1e-10))
		tmp = Float64(Float64(x / z) + Float64(Float64(1.0 - Float64(x / z)) * y));
	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 ((z <= -1e+33) || ~((z <= 1e-10)))
		tmp = (x / z) + ((1.0 - (x / z)) * y);
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1e+33], N[Not[LessEqual[z, 1e-10]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] + N[(N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.9999999999999995e32 or 1.00000000000000004e-10 < z

   1. Initial program 70.5%

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

   2. Taylor expanded in y around 0 99.9%

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

   if -9.9999999999999995e32 < z < 1.00000000000000004e-10

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+33} \lor \neg \left(z \leq 10^{-10}\right):\\ \;\;\;\;\frac{x}{z} + \left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]

Alternative 2: 78.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{z} + y\\ t_1 := \frac{x}{z} \cdot \left(-y\right)\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ x z) y)) (t_1 (* (/ x z) (- y))))
   (if (<= y -8.5e+181)
     t_0
     (if (<= y -3.8e+95)
       t_1
       (if (<= y 2.5e+37) t_0 (if (<= y 2.25e+136) t_1 (* z (/ y z))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x / z) + y;
	double t_1 = (x / z) * -y;
	double tmp;
	if (y <= -8.5e+181) {
		tmp = t_0;
	} else if (y <= -3.8e+95) {
		tmp = t_1;
	} else if (y <= 2.5e+37) {
		tmp = t_0;
	} else if (y <= 2.25e+136) {
		tmp = t_1;
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x / z) + y
    t_1 = (x / z) * -y
    if (y <= (-8.5d+181)) then
        tmp = t_0
    else if (y <= (-3.8d+95)) then
        tmp = t_1
    else if (y <= 2.5d+37) then
        tmp = t_0
    else if (y <= 2.25d+136) then
        tmp = t_1
    else
        tmp = z * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x / z) + y;
	double t_1 = (x / z) * -y;
	double tmp;
	if (y <= -8.5e+181) {
		tmp = t_0;
	} else if (y <= -3.8e+95) {
		tmp = t_1;
	} else if (y <= 2.5e+37) {
		tmp = t_0;
	} else if (y <= 2.25e+136) {
		tmp = t_1;
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / z) + y
	t_1 = (x / z) * -y
	tmp = 0
	if y <= -8.5e+181:
		tmp = t_0
	elif y <= -3.8e+95:
		tmp = t_1
	elif y <= 2.5e+37:
		tmp = t_0
	elif y <= 2.25e+136:
		tmp = t_1
	else:
		tmp = z * (y / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / z) + y)
	t_1 = Float64(Float64(x / z) * Float64(-y))
	tmp = 0.0
	if (y <= -8.5e+181)
		tmp = t_0;
	elseif (y <= -3.8e+95)
		tmp = t_1;
	elseif (y <= 2.5e+37)
		tmp = t_0;
	elseif (y <= 2.25e+136)
		tmp = t_1;
	else
		tmp = Float64(z * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / z) + y;
	t_1 = (x / z) * -y;
	tmp = 0.0;
	if (y <= -8.5e+181)
		tmp = t_0;
	elseif (y <= -3.8e+95)
		tmp = t_1;
	elseif (y <= 2.5e+37)
		tmp = t_0;
	elseif (y <= 2.25e+136)
		tmp = t_1;
	else
		tmp = z * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * (-y)), $MachinePrecision]}, If[LessEqual[y, -8.5e+181], t$95$0, If[LessEqual[y, -3.8e+95], t$95$1, If[LessEqual[y, 2.5e+37], t$95$0, If[LessEqual[y, 2.25e+136], t$95$1, N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.49999999999999966e181 or -3.7999999999999999e95 < y < 2.49999999999999994e37

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0 97.7%

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

   3. Taylor expanded in y around 0 91.3%

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

   if -8.49999999999999966e181 < y < -3.7999999999999999e95 or 2.49999999999999994e37 < y < 2.25e136

   1. Initial program 73.3%

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

   2. Taylor expanded in y around 0 92.7%

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

   3. Taylor expanded in y around inf 99.7%

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

   4. Taylor expanded in x around inf 72.5%

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

      1. mul-1-neg 72.5%

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

      2. distribute-frac-neg 72.5%

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

   6. Simplified 72.5%

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

   if 2.25e136 < y

   1. Initial program 62.0%

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

   2. Taylor expanded in y around inf 62.0%

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

   3. Taylor expanded in z around inf 27.5%

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

      1. associate-/l* 54.6%

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

      2. associate-/r/ 59.9%

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

   5. Applied egg-rr 59.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{+95}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]

Alternative 3: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{z} + y\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+182}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{+95}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+135}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ x z) y)))
   (if (<= y -7.5e+182)
     t_0
     (if (<= y -1.95e+95)
       (/ (- y) (/ z x))
       (if (<= y 2.3e+38)
         t_0
         (if (<= y 1.75e+135) (* (/ x z) (- y)) (* z (/ y z))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x / z) + y;
	double tmp;
	if (y <= -7.5e+182) {
		tmp = t_0;
	} else if (y <= -1.95e+95) {
		tmp = -y / (z / x);
	} else if (y <= 2.3e+38) {
		tmp = t_0;
	} else if (y <= 1.75e+135) {
		tmp = (x / z) * -y;
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / z) + y
    if (y <= (-7.5d+182)) then
        tmp = t_0
    else if (y <= (-1.95d+95)) then
        tmp = -y / (z / x)
    else if (y <= 2.3d+38) then
        tmp = t_0
    else if (y <= 1.75d+135) then
        tmp = (x / z) * -y
    else
        tmp = z * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x / z) + y;
	double tmp;
	if (y <= -7.5e+182) {
		tmp = t_0;
	} else if (y <= -1.95e+95) {
		tmp = -y / (z / x);
	} else if (y <= 2.3e+38) {
		tmp = t_0;
	} else if (y <= 1.75e+135) {
		tmp = (x / z) * -y;
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x / z) + y
	tmp = 0
	if y <= -7.5e+182:
		tmp = t_0
	elif y <= -1.95e+95:
		tmp = -y / (z / x)
	elif y <= 2.3e+38:
		tmp = t_0
	elif y <= 1.75e+135:
		tmp = (x / z) * -y
	else:
		tmp = z * (y / z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x / z) + y)
	tmp = 0.0
	if (y <= -7.5e+182)
		tmp = t_0;
	elseif (y <= -1.95e+95)
		tmp = Float64(Float64(-y) / Float64(z / x));
	elseif (y <= 2.3e+38)
		tmp = t_0;
	elseif (y <= 1.75e+135)
		tmp = Float64(Float64(x / z) * Float64(-y));
	else
		tmp = Float64(z * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x / z) + y;
	tmp = 0.0;
	if (y <= -7.5e+182)
		tmp = t_0;
	elseif (y <= -1.95e+95)
		tmp = -y / (z / x);
	elseif (y <= 2.3e+38)
		tmp = t_0;
	elseif (y <= 1.75e+135)
		tmp = (x / z) * -y;
	else
		tmp = z * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[y, -7.5e+182], t$95$0, If[LessEqual[y, -1.95e+95], N[((-y) / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e+38], t$95$0, If[LessEqual[y, 1.75e+135], N[(N[(x / z), $MachinePrecision] * (-y)), $MachinePrecision], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.49999999999999989e182 or -1.9499999999999999e95 < y < 2.3000000000000001e38

   1. Initial program 91.5%

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

   2. Taylor expanded in x around 0 97.7%

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

   3. Taylor expanded in y around 0 91.3%

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

   if -7.49999999999999989e182 < y < -1.9499999999999999e95

   1. Initial program 66.5%

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

   2. Taylor expanded in y around inf 66.5%

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

      1. flip-- 56.0%

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

      2. associate-*r/ 51.3%

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

      3. +-commutative 51.3%

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

   4. Applied egg-rr 51.3%

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

      1. associate-/l* 55.9%

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

      2. difference-of-squares 56.6%

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

      3. +-commutative 56.6%

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

      4. associate-/r* 66.5%

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

      5. *-inverses 66.5%

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

   6. Simplified 66.5%

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

   7. Taylor expanded in z around 0 48.5%

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

      1. associate-/l* 72.2%

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

      2. associate-*r/ 72.2%

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

      3. neg-mul-1 72.2%

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

   9. Simplified 72.2%

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

   if 2.3000000000000001e38 < y < 1.7500000000000001e135

   1. Initial program 79.2%

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

   2. Taylor expanded in y around 0 86.8%

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

   3. Taylor expanded in y around inf 99.8%

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

   4. Taylor expanded in x around inf 73.0%

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

      1. mul-1-neg 73.0%

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

      2. distribute-frac-neg 73.0%

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

   6. Simplified 73.0%

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

   if 1.7500000000000001e135 < y

   1. Initial program 62.0%

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

   2. Taylor expanded in y around inf 62.0%

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

   3. Taylor expanded in z around inf 27.5%

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

      1. associate-/l* 54.6%

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

      2. associate-/r/ 59.9%

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

   5. Applied egg-rr 59.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+182}:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{+95}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+135}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -10000000000000 \lor \neg \left(y \leq 135000000000\right):\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -10000000000000.0) (not (<= y 135000000000.0)))
   (* (- 1.0 (/ x z)) y)
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -10000000000000.0) || !(y <= 135000000000.0)) {
		tmp = (1.0 - (x / z)) * y;
	} 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 <= (-10000000000000.0d0)) .or. (.not. (y <= 135000000000.0d0))) then
        tmp = (1.0d0 - (x / z)) * y
    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 <= -10000000000000.0) || !(y <= 135000000000.0)) {
		tmp = (1.0 - (x / z)) * y;
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -10000000000000.0) or not (y <= 135000000000.0):
		tmp = (1.0 - (x / z)) * y
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -10000000000000.0) || !(y <= 135000000000.0))
		tmp = Float64(Float64(1.0 - Float64(x / z)) * y);
	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 <= -10000000000000.0) || ~((y <= 135000000000.0)))
		tmp = (1.0 - (x / z)) * y;
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -10000000000000.0], N[Not[LessEqual[y, 135000000000.0]], $MachinePrecision]], N[(N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e13 or 1.35e11 < y

   1. Initial program 68.1%

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

   2. Taylor expanded in y around 0 95.8%

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

   3. Taylor expanded in y around inf 99.8%

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

   if -1e13 < y < 1.35e11

   1. Initial program 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10000000000000 \lor \neg \left(y \leq 135000000000\right):\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]

Alternative 5: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -11000000000000 \lor \neg \left(y \leq 2.9 \cdot 10^{-9}\right):\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -11000000000000.0) (not (<= y 2.9e-9)))
   (* (- 1.0 (/ x z)) y)
   (+ (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -11000000000000.0) || !(y <= 2.9e-9)) {
		tmp = (1.0 - (x / z)) * 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 <= (-11000000000000.0d0)) .or. (.not. (y <= 2.9d-9))) then
        tmp = (1.0d0 - (x / z)) * 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 <= -11000000000000.0) || !(y <= 2.9e-9)) {
		tmp = (1.0 - (x / z)) * y;
	} else {
		tmp = (x / z) + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -11000000000000.0) or not (y <= 2.9e-9):
		tmp = (1.0 - (x / z)) * y
	else:
		tmp = (x / z) + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -11000000000000.0) || !(y <= 2.9e-9))
		tmp = Float64(Float64(1.0 - Float64(x / z)) * y);
	else
		tmp = Float64(Float64(x / z) + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -11000000000000.0) || ~((y <= 2.9e-9)))
		tmp = (1.0 - (x / z)) * y;
	else
		tmp = (x / z) + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -11000000000000.0], N[Not[LessEqual[y, 2.9e-9]], $MachinePrecision]], N[(N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(x / z), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e13 or 2.89999999999999991e-9 < y

   1. Initial program 69.9%

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

   2. Taylor expanded in y around 0 95.9%

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

   3. Taylor expanded in y around inf 99.4%

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

   if -1.1e13 < y < 2.89999999999999991e-9

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 99.8%

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

   3. Taylor expanded in y around 0 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -11000000000000 \lor \neg \left(y \leq 2.9 \cdot 10^{-9}\right):\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y\\ \end{array} \]

Alternative 6: 62.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-33} \lor \neg \left(y \leq 1.6 \cdot 10^{-41}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.8e-33) (not (<= y 1.6e-41))) (* z (/ y z)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.8e-33) || !(y <= 1.6e-41)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-3.8d-33)) .or. (.not. (y <= 1.6d-41))) then
        tmp = z * (y / z)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.8e-33) || !(y <= 1.6e-41)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.8e-33) or not (y <= 1.6e-41):
		tmp = z * (y / z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.8e-33) || !(y <= 1.6e-41))
		tmp = Float64(z * Float64(y / z));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.8e-33) || ~((y <= 1.6e-41)))
		tmp = z * (y / z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.8e-33], N[Not[LessEqual[y, 1.6e-41]], $MachinePrecision]], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.79999999999999994e-33 or 1.60000000000000006e-41 < y

   1. Initial program 72.0%

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

   2. Taylor expanded in y around inf 69.6%

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

   3. Taylor expanded in z around inf 32.1%

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

      1. associate-/l* 51.2%

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

      2. associate-/r/ 53.0%

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

   5. Applied egg-rr 53.0%

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

   if -3.79999999999999994e-33 < y < 1.60000000000000006e-41

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 74.2%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-33} \lor \neg \left(y \leq 1.6 \cdot 10^{-41}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 7: 59.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-36}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.5e-36) y (if (<= y 6.2e-46) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.5e-36) {
		tmp = y;
	} else if (y <= 6.2e-46) {
		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-36)) then
        tmp = y
    else if (y <= 6.2d-46) 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-36) {
		tmp = y;
	} else if (y <= 6.2e-46) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.5e-36:
		tmp = y
	elif y <= 6.2e-46:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.5e-36)
		tmp = y;
	elseif (y <= 6.2e-46)
		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-36)
		tmp = y;
	elseif (y <= 6.2e-46)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.5e-36], y, If[LessEqual[y, 6.2e-46], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -7.49999999999999972e-36 or 6.2000000000000002e-46 < y

   1. Initial program 72.0%

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

   2. Taylor expanded in x around 0 51.2%

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

   if -7.49999999999999972e-36 < y < 6.2000000000000002e-46

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 74.2%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-36}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 8: 77.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

2. Taylor expanded in x around 0 95.8%

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

3. Taylor expanded in y around 0 75.4%

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

4. Final simplification 75.4%

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

Alternative 9: 39.6% 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 84.5%

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

2. Taylor expanded in x around 0 40.7%

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

3. Final simplification 40.7%

   \[\leadsto y \]

Developer target: 93.8% 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 2023181 
(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))