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

Percentage Accurate: 87.8% → 98.5%

Time: 4.9s

Alternatives: 9

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: 87.8% 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: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-165} \lor \neg \left(z \leq 10^{-136}\right):\\ \;\;\;\;\frac{x}{z} + \left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.6e-165) (not (<= z 1e-136)))
   (+ (/ 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 <= -1.6e-165) || !(z <= 1e-136)) {
		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 <= (-1.6d-165)) .or. (.not. (z <= 1d-136))) 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 <= -1.6e-165) || !(z <= 1e-136)) {
		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 <= -1.6e-165) or not (z <= 1e-136):
		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 <= -1.6e-165) || !(z <= 1e-136))
		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 <= -1.6e-165) || ~((z <= 1e-136)))
		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, -1.6e-165], N[Not[LessEqual[z, 1e-136]], $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 < -1.60000000000000006e-165 or 1e-136 < z

   1. Initial program 85.3%

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

   2. Taylor expanded in y around 0 99.9%

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

   if -1.60000000000000006e-165 < z < 1e-136

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-165} \lor \neg \left(z \leq 10^{-136}\right):\\ \;\;\;\;\frac{x}{z} + \left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7000000000:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{elif}\;y \leq 27000:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7000000000.0)
   (/ y (/ z (- z x)))
   (if (<= y 27000.0) (/ (+ x (* y (- z x))) z) (* (- 1.0 (/ x z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7000000000.0) {
		tmp = y / (z / (z - x));
	} else if (y <= 27000.0) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = (1.0 - (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 <= (-7000000000.0d0)) then
        tmp = y / (z / (z - x))
    else if (y <= 27000.0d0) then
        tmp = (x + (y * (z - x))) / z
    else
        tmp = (1.0d0 - (x / z)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -7000000000.0) {
		tmp = y / (z / (z - x));
	} else if (y <= 27000.0) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = (1.0 - (x / z)) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7000000000.0:
		tmp = y / (z / (z - x))
	elif y <= 27000.0:
		tmp = (x + (y * (z - x))) / z
	else:
		tmp = (1.0 - (x / z)) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7000000000.0)
		tmp = Float64(y / Float64(z / Float64(z - x)));
	elseif (y <= 27000.0)
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	else
		tmp = Float64(Float64(1.0 - Float64(x / z)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -7000000000.0)
		tmp = y / (z / (z - x));
	elseif (y <= 27000.0)
		tmp = (x + (y * (z - x))) / z;
	else
		tmp = (1.0 - (x / z)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7000000000.0], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 27000.0], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7e9

   1. Initial program 79.8%

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

   2. Taylor expanded in y around inf 79.8%

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

      1. associate-/l* 99.9%

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

   4. Simplified 99.9%

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

   if -7e9 < y < 27000

   1. Initial program 100.0%

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

   if 27000 < y

   1. Initial program 78.1%

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

   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 y around inf 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7000000000:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{elif}\;y \leq 27000:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \end{array} \]

Alternative 3: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -440000000000 \lor \neg \left(y \leq 1.55 \cdot 10^{-14}\right):\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -440000000000.0) (not (<= y 1.55e-14)))
   (* (- 1.0 (/ x z)) y)
   (+ (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -440000000000.0) || !(y <= 1.55e-14)) {
		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 <= (-440000000000.0d0)) .or. (.not. (y <= 1.55d-14))) 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 <= -440000000000.0) || !(y <= 1.55e-14)) {
		tmp = (1.0 - (x / z)) * y;
	} else {
		tmp = (x / z) + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -440000000000.0) or not (y <= 1.55e-14):
		tmp = (1.0 - (x / z)) * y
	else:
		tmp = (x / z) + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -440000000000.0) || !(y <= 1.55e-14))
		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 <= -440000000000.0) || ~((y <= 1.55e-14)))
		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, -440000000000.0], N[Not[LessEqual[y, 1.55e-14]], $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 < -4.4e11 or 1.55000000000000002e-14 < y

   1. Initial program 79.7%

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

   2. Taylor expanded in y around 0 96.2%

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

   3. Taylor expanded in y around inf 99.5%

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

   if -4.4e11 < y < 1.55000000000000002e-14

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 91.8%

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

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -440000000000 \lor \neg \left(y \leq 1.55 \cdot 10^{-14}\right):\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y\\ \end{array} \]

Alternative 4: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -440000000000:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -440000000000.0)
   (/ y (/ z (- z x)))
   (if (<= y 1.55e-14) (+ (/ x z) y) (* (- 1.0 (/ x z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -440000000000.0) {
		tmp = y / (z / (z - x));
	} else if (y <= 1.55e-14) {
		tmp = (x / z) + y;
	} else {
		tmp = (1.0 - (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 <= (-440000000000.0d0)) then
        tmp = y / (z / (z - x))
    else if (y <= 1.55d-14) then
        tmp = (x / z) + y
    else
        tmp = (1.0d0 - (x / z)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -440000000000.0) {
		tmp = y / (z / (z - x));
	} else if (y <= 1.55e-14) {
		tmp = (x / z) + y;
	} else {
		tmp = (1.0 - (x / z)) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -440000000000.0:
		tmp = y / (z / (z - x))
	elif y <= 1.55e-14:
		tmp = (x / z) + y
	else:
		tmp = (1.0 - (x / z)) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -440000000000.0)
		tmp = Float64(y / Float64(z / Float64(z - x)));
	elseif (y <= 1.55e-14)
		tmp = Float64(Float64(x / z) + y);
	else
		tmp = Float64(Float64(1.0 - Float64(x / z)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -440000000000.0)
		tmp = y / (z / (z - x));
	elseif (y <= 1.55e-14)
		tmp = (x / z) + y;
	else
		tmp = (1.0 - (x / z)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -440000000000.0], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e-14], N[(N[(x / z), $MachinePrecision] + y), $MachinePrecision], N[(N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4e11

   1. Initial program 79.1%

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

   2. Taylor expanded in y around inf 79.1%

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

      1. associate-/l* 99.9%

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

   4. Simplified 99.9%

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

   if -4.4e11 < y < 1.55000000000000002e-14

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 91.8%

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

   3. Taylor expanded in x around 0 99.3%

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

   if 1.55000000000000002e-14 < y

   1. Initial program 80.1%

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

   2. Taylor expanded in y around 0 93.3%

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

   3. Taylor expanded in y around inf 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -440000000000:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \end{array} \]

Alternative 5: 60.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.0) y (if (<= y 1.7e-16) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.0) {
		tmp = y;
	} else if (y <= 1.7e-16) {
		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 <= (-3.0d0)) then
        tmp = y
    else if (y <= 1.7d-16) 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 <= -3.0) {
		tmp = y;
	} else if (y <= 1.7e-16) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.0:
		tmp = y
	elif y <= 1.7e-16:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.0)
		tmp = y;
	elseif (y <= 1.7e-16)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.0)
		tmp = y;
	elseif (y <= 1.7e-16)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.0], y, If[LessEqual[y, 1.7e-16], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -3 or 1.7e-16 < y

   1. Initial program 80.0%

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

   2. Taylor expanded in x around 0 57.8%

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

   if -3 < y < 1.7e-16

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 80.9%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6: 61.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.0) (* z (/ y z)) (if (<= y 1.75e-17) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.0) {
		tmp = z * (y / z);
	} else if (y <= 1.75e-17) {
		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 <= (-3.0d0)) then
        tmp = z * (y / z)
    else if (y <= 1.75d-17) 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 <= -3.0) {
		tmp = z * (y / z);
	} else if (y <= 1.75e-17) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.0:
		tmp = z * (y / z)
	elif y <= 1.75e-17:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.0)
		tmp = Float64(z * Float64(y / z));
	elseif (y <= 1.75e-17)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.0)
		tmp = z * (y / z);
	elseif (y <= 1.75e-17)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.0], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-17], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 3 regimes

2. if y < -3

   1. Initial program 79.8%

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

   2. Taylor expanded in y around inf 79.8%

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

   3. Taylor expanded in z around inf 39.7%

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

      1. associate-/l* 55.3%

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

      2. associate-/r/ 58.2%

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

   5. Applied egg-rr 58.2%

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

   if -3 < y < 1.7500000000000001e-17

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 80.9%

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

   if 1.7500000000000001e-17 < y

   1. Initial program 80.1%

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

   2. Taylor expanded in x around 0 59.7%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7: 81.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.55e-14) (+ (/ x z) y) (- y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.55e-14) {
		tmp = (x / z) + y;
	} 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 <= 1.55d-14) then
        tmp = (x / z) + y
    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 <= 1.55e-14) {
		tmp = (x / z) + y;
	} else {
		tmp = y - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.55e-14:
		tmp = (x / z) + y
	else:
		tmp = y - (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.55e-14)
		tmp = Float64(Float64(x / z) + y);
	else
		tmp = Float64(y - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.55e-14)
		tmp = (x / z) + y;
	else
		tmp = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.55e-14], N[(N[(x / z), $MachinePrecision] + y), $MachinePrecision], N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55000000000000002e-14

   1. Initial program 93.2%

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

   2. Taylor expanded in y around 0 94.4%

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

   3. Taylor expanded in x around 0 88.6%

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

   if 1.55000000000000002e-14 < y

   1. Initial program 80.1%

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

   2. Taylor expanded in y around 0 93.3%

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

   3. Taylor expanded in x around 0 59.2%

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

      1. div-inv 59.2%

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

      2. add-sqr-sqrt 26.8%

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

      3. sqrt-unprod 62.2%

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

      4. sqr-neg 62.2%

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

      5. sqrt-unprod 37.0%

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

      6. add-sqr-sqrt 67.1%

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

      7. cancel-sign-sub-inv 67.1%

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

      8. div-inv 67.1%

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

   5. Applied egg-rr 67.1%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{else}:\\ \;\;\;\;y - \frac{x}{z}\\ \end{array} \]

Alternative 8: 78.4% 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 89.3%

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

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 79.9%

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

4. Final simplification 79.9%

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

Alternative 9: 41.3% 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 89.3%

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

2. Taylor expanded in x around 0 40.6%

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

3. Final simplification 40.6%

   \[\leadsto y \]

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 2023268 
(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))