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

Percentage Accurate: 88.0% → 99.5%

Time: 10.4s

Alternatives: 11

Speedup: 0.8×

• 20.1% 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: 88.0% 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: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-270}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{+16}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.5e-270)
   (+ (/ x z) (* y (- 1.0 (/ x z))))
   (if (<= y 1.56e+16) (/ (+ x (* y (- z x))) z) (/ y (/ z (- z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.5e-270) {
		tmp = (x / z) + (y * (1.0 - (x / z)));
	} else if (y <= 1.56e+16) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y / (z / (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 <= 5.5d-270) then
        tmp = (x / z) + (y * (1.0d0 - (x / z)))
    else if (y <= 1.56d+16) then
        tmp = (x + (y * (z - x))) / z
    else
        tmp = y / (z / (z - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.5e-270) {
		tmp = (x / z) + (y * (1.0 - (x / z)));
	} else if (y <= 1.56e+16) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y / (z / (z - x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.5e-270:
		tmp = (x / z) + (y * (1.0 - (x / z)))
	elif y <= 1.56e+16:
		tmp = (x + (y * (z - x))) / z
	else:
		tmp = y / (z / (z - x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.5e-270)
		tmp = Float64(Float64(x / z) + Float64(y * Float64(1.0 - Float64(x / z))));
	elseif (y <= 1.56e+16)
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	else
		tmp = Float64(y / Float64(z / Float64(z - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.5e-270)
		tmp = (x / z) + (y * (1.0 - (x / z)));
	elseif (y <= 1.56e+16)
		tmp = (x + (y * (z - x))) / z;
	else
		tmp = y / (z / (z - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.5e-270], N[(N[(x / z), $MachinePrecision] + N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.56e+16], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.4999999999999996e-270

   1. Initial program 88.8%

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

   2. Taylor expanded in y around 0 100.0%

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

   if 5.4999999999999996e-270 < y < 1.56e16

   1. Initial program 99.9%

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

   if 1.56e16 < y

   1. Initial program 66.0%

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

   2. Taylor expanded in y around inf 66.0%

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

      1. associate-/l* 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-270}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{+16}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+31}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.7e+31)
   (- y (* y (/ x z)))
   (if (<= y 8e+14) (/ (+ x (* y (- z x))) z) (/ y (/ z (- z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.7e+31) {
		tmp = y - (y * (x / z));
	} else if (y <= 8e+14) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y / (z / (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 <= (-1.7d+31)) then
        tmp = y - (y * (x / z))
    else if (y <= 8d+14) then
        tmp = (x + (y * (z - x))) / z
    else
        tmp = y / (z / (z - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.7e+31) {
		tmp = y - (y * (x / z));
	} else if (y <= 8e+14) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y / (z / (z - x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.7e+31:
		tmp = y - (y * (x / z))
	elif y <= 8e+14:
		tmp = (x + (y * (z - x))) / z
	else:
		tmp = y / (z / (z - x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.7e+31)
		tmp = Float64(y - Float64(y * Float64(x / z)));
	elseif (y <= 8e+14)
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	else
		tmp = Float64(y / Float64(z / Float64(z - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.7e+31)
		tmp = y - (y * (x / z));
	elseif (y <= 8e+14)
		tmp = (x + (y * (z - x))) / z;
	else
		tmp = y / (z / (z - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.7e+31], N[(y - N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+14], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6999999999999999e31

   1. Initial program 76.2%

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

   2. Taylor expanded in y around inf 76.2%

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

   3. Taylor expanded in z around 0 97.0%

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

      1. +-commutative 97.0%

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

      2. mul-1-neg 97.0%

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

      3. associate-*r/ 100.0%

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

      4. unsub-neg 100.0%

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

      5. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -1.6999999999999999e31 < y < 8e14

   1. Initial program 99.9%

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

   if 8e14 < y

   1. Initial program 66.0%

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

   2. Taylor expanded in y around inf 66.0%

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

      1. associate-/l* 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+31}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \]

Alternative 3: 99.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (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 <= -1.0) || !(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 <= (-1.0d0)) .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 <= -1.0) || !(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 <= -1.0) 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 <= -1.0) || !(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 <= -1.0) || ~((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, -1.0], 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 < -1 or 1 < y

   1. Initial program 73.3%

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

   2. Taylor expanded in y around 0 95.3%

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

   3. Taylor expanded in y around inf 99.5%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 89.6%

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

   3. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.4%

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

Alternative 4: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.0)
		tmp = Float64(y - Float64(y * Float64(x / z)));
	elseif (y <= 1.0)
		tmp = Float64(y + Float64(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 <= -1.0)
		tmp = y - (y * (x / z));
	elseif (y <= 1.0)
		tmp = y + (x / z);
	else
		tmp = y * (1.0 - (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 78.0%

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

   2. Taylor expanded in y around inf 77.6%

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

   3. Taylor expanded in z around 0 96.9%

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

      1. +-commutative 96.9%

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

      2. mul-1-neg 96.9%

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

      3. associate-*r/ 99.6%

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

      4. unsub-neg 99.6%

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

      5. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 89.6%

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

   3. Taylor expanded in x around 0 99.3%

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

   if 1 < y

   1. Initial program 68.2%

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

   2. Taylor expanded in y around 0 90.1%

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

   3. Taylor expanded in y around inf 99.4%

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

4. Final simplification 99.4%

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

Alternative 5: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 78.0%

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

   2. Taylor expanded in y around inf 77.6%

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

   3. Taylor expanded in z around 0 96.9%

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

      1. +-commutative 96.9%

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

      2. mul-1-neg 96.9%

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

      3. associate-*r/ 99.6%

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

      4. unsub-neg 99.6%

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

      5. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 89.6%

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

   3. Taylor expanded in x around 0 99.3%

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

   if 1 < y

   1. Initial program 68.2%

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

   2. Taylor expanded in y around inf 67.7%

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

      1. associate-/l* 99.5%

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

   4. Simplified 99.5%

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

4. Final simplification 99.4%

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

Alternative 6: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-44} \lor \neg \left(y \leq 0.00042\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8.2e-44) (not (<= y 0.00042))) (* z (/ y z)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8.2e-44) || !(y <= 0.00042)) {
		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 <= (-8.2d-44)) .or. (.not. (y <= 0.00042d0))) 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 <= -8.2e-44) || !(y <= 0.00042)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -8.2e-44) or not (y <= 0.00042):
		tmp = z * (y / z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -8.2e-44) || !(y <= 0.00042))
		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 <= -8.2e-44) || ~((y <= 0.00042)))
		tmp = z * (y / z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -8.2e-44], N[Not[LessEqual[y, 0.00042]], $MachinePrecision]], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.19999999999999984e-44 or 4.2000000000000002e-4 < y

   1. Initial program 75.5%

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

   2. Taylor expanded in z around inf 46.0%

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

   3. Taylor expanded in x around 0 38.7%

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

      1. associate-/l* 58.5%

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

      2. associate-/r/ 59.4%

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

   5. Applied egg-rr 59.4%

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

   if -8.19999999999999984e-44 < y < 4.2000000000000002e-4

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 73.7%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-44} \lor \neg \left(y \leq 0.00042\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 7: 76.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+187}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.7e+187) (* y (/ (- x) z)) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.7e+187) {
		tmp = y * (-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 (x <= (-4.7d+187)) then
        tmp = y * (-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 (x <= -4.7e+187) {
		tmp = y * (-x / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.7e+187:
		tmp = y * (-x / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.7e+187)
		tmp = Float64(y * Float64(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 (x <= -4.7e+187)
		tmp = y * (-x / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.7e+187], N[(y * N[((-x) / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.69999999999999989e187

   1. Initial program 79.5%

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

   2. Taylor expanded in y around inf 51.3%

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

   3. Taylor expanded in z around 0 51.3%

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

      1. mul-1-neg 51.3%

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

      2. associate-*r/ 76.7%

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

      3. *-commutative 76.7%

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

      4. distribute-rgt-neg-in 76.7%

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

   5. Simplified 76.7%

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

   if -4.69999999999999989e187 < x

   1. Initial program 86.8%

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

   2. Taylor expanded in y around 0 94.1%

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

   3. Taylor expanded in x around 0 83.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+187}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 8: 76.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+187}:\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.4e+187) (/ y (/ (- z) x)) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.4e+187) {
		tmp = y / (-z / x);
	} 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 (x <= (-5.4d+187)) then
        tmp = y / (-z / x)
    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 (x <= -5.4e+187) {
		tmp = y / (-z / x);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.4e+187:
		tmp = y / (-z / x)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.4e+187)
		tmp = Float64(y / Float64(Float64(-z) / x));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.4e+187)
		tmp = y / (-z / x);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.4e+187], N[(y / N[((-z) / x), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.40000000000000016e187

   1. Initial program 79.5%

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

   2. Taylor expanded in y around inf 51.3%

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

      1. associate-/l* 76.9%

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

   4. Simplified 76.9%

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

   5. Taylor expanded in z around 0 76.9%

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

      1. associate-*r/ 76.9%

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

      2. neg-mul-1 76.9%

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

   7. Simplified 76.9%

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

   if -5.40000000000000016e187 < x

   1. Initial program 86.8%

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

   2. Taylor expanded in y around 0 94.1%

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

   3. Taylor expanded in x around 0 83.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{+187}:\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 9: 57.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+83}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.6e+19) (/ x z) (if (<= x 4.8e+83) y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.6e+19) {
		tmp = x / z;
	} else if (x <= 4.8e+83) {
		tmp = y;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3.6d+19)) then
        tmp = x / z
    else if (x <= 4.8d+83) then
        tmp = y
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.6e+19) {
		tmp = x / z;
	} else if (x <= 4.8e+83) {
		tmp = y;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.6e+19:
		tmp = x / z
	elif x <= 4.8e+83:
		tmp = y
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.6e+19)
		tmp = Float64(x / z);
	elseif (x <= 4.8e+83)
		tmp = y;
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.6e+19)
		tmp = x / z;
	elseif (x <= 4.8e+83)
		tmp = y;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.6e+19], N[(x / z), $MachinePrecision], If[LessEqual[x, 4.8e+83], y, N[(x / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6e19 or 4.79999999999999982e83 < x

   1. Initial program 91.2%

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

   2. Taylor expanded in y around 0 59.7%

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

   if -3.6e19 < x < 4.79999999999999982e83

   1. Initial program 82.8%

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

   2. Taylor expanded in x around 0 69.7%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+83}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 10: 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 86.3%

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

2. Taylor expanded in y around 0 92.5%

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

3. Taylor expanded in x around 0 80.7%

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

4. Final simplification 80.7%

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

Alternative 11: 41.6% 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 86.3%

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

2. Taylor expanded in x around 0 45.3%

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

3. Final simplification 45.3%

   \[\leadsto y \]

Developer target: 94.4% 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 2023187 
(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))