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

Percentage Accurate: 88.0% → 99.8%

Time: 8.8s

Alternatives: 9

Speedup: 0.8×

• 20.8% 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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -115500000 \lor \neg \left(y \leq 5.3 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -115500000.0) (not (<= y 5.3e+14)))
   (/ y (/ z (- z x)))
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -115500000.0) || !(y <= 5.3e+14)) {
		tmp = y / (z / (z - x));
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-115500000.0d0)) .or. (.not. (y <= 5.3d+14))) then
        tmp = y / (z / (z - x))
    else
        tmp = (x + (y * (z - x))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -115500000.0) || !(y <= 5.3e+14)) {
		tmp = y / (z / (z - x));
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -115500000.0) or not (y <= 5.3e+14):
		tmp = y / (z / (z - x))
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -115500000.0) || !(y <= 5.3e+14))
		tmp = Float64(y / Float64(z / Float64(z - x)));
	else
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -115500000.0) || ~((y <= 5.3e+14)))
		tmp = y / (z / (z - x));
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -115500000.0], N[Not[LessEqual[y, 5.3e+14]], $MachinePrecision]], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.155e8 or 5.3e14 < y

   1. Initial program 66.8%

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

   2. Taylor expanded in y around inf 66.6%

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

      1. associate-/l* 99.4%

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

   4. Simplified 99.4%

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

   if -1.155e8 < y < 5.3e14

   1. Initial program 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -115500000 \lor \neg \left(y \leq 5.3 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]

Alternative 2: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e+16) (+ (/ x z) (* y (- 1.0 (/ x z)))) (/ y (/ z (- z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e+16) {
		tmp = (x / z) + (y * (1.0 - (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 <= 5d+16) then
        tmp = (x / z) + (y * (1.0d0 - (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 <= 5e+16) {
		tmp = (x / z) + (y * (1.0 - (x / z)));
	} else {
		tmp = y / (z / (z - x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5e+16:
		tmp = (x / z) + (y * (1.0 - (x / z)))
	else:
		tmp = y / (z / (z - x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e+16)
		tmp = Float64(Float64(x / z) + Float64(y * Float64(1.0 - 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 <= 5e+16)
		tmp = (x / z) + (y * (1.0 - (x / z)));
	else
		tmp = y / (z / (z - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e+16], N[(N[(x / z), $MachinePrecision] + N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5e16

   1. Initial program 91.8%

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

   2. Taylor expanded in y around 0 98.3%

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

   if 5e16 < y

   1. Initial program 60.9%

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

   2. Taylor expanded in y around inf 60.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \]

Alternative 3: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-61} \lor \neg \left(z \leq 3.2 \cdot 10^{-13}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.75e-61) (not (<= z 3.2e-13)))
   (+ y (/ x z))
   (* (/ x z) (- 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.75e-61) || !(z <= 3.2e-13)) {
		tmp = y + (x / z);
	} else {
		tmp = (x / z) * (1.0 - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.75d-61)) .or. (.not. (z <= 3.2d-13))) then
        tmp = y + (x / z)
    else
        tmp = (x / z) * (1.0d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.75e-61) || !(z <= 3.2e-13)) {
		tmp = y + (x / z);
	} else {
		tmp = (x / z) * (1.0 - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.75e-61) or not (z <= 3.2e-13):
		tmp = y + (x / z)
	else:
		tmp = (x / z) * (1.0 - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.75e-61) || !(z <= 3.2e-13))
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(Float64(x / z) * Float64(1.0 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.75e-61) || ~((z <= 3.2e-13)))
		tmp = y + (x / z);
	else
		tmp = (x / z) * (1.0 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.75e-61], N[Not[LessEqual[z, 3.2e-13]], $MachinePrecision]], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7500000000000002e-61 or 3.2e-13 < z

   1. Initial program 71.1%

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

   2. Taylor expanded in y around 0 99.9%

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

      1. flip-- 94.6%

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

      2. associate-*l/ 93.4%

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

      3. metadata-eval 93.4%

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

      4. pow2 93.4%

         \[\leadsto \frac{\left(1 - \color{blue}{{\left(\frac{x}{z}\right)}^{2}}\right) \cdot y}{1 + \frac{x}{z}} + \frac{x}{z} \]

   4. Applied egg-rr 93.4%

      \[\leadsto \color{blue}{\frac{\left(1 - {\left(\frac{x}{z}\right)}^{2}\right) \cdot y}{1 + \frac{x}{z}}} + \frac{x}{z} \]

   5. Taylor expanded in x around 0 86.8%

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

   if -1.7500000000000002e-61 < z < 3.2e-13

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-/l* 87.1%

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

      3. mul-1-neg 87.1%

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

      4. unsub-neg 87.1%

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

   4. Simplified 87.1%

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

      1. associate-/r/ 93.9%

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

   6. Applied egg-rr 93.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-61} \lor \neg \left(z \leq 3.2 \cdot 10^{-13}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 - y\right)\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = y / (z / (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 ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y / (z / (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 ((y <= -1.0) || !(y <= 1.0)) {
		tmp = y / (z / (z - x));
	} 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 / (z / (z - x))
	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(z / 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 ((y <= -1.0) || ~((y <= 1.0)))
		tmp = y / (z / (z - x));
	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[(z / N[(z - x), $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 68.0%

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

   2. Taylor expanded in y around inf 67.1%

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

      1. associate-/l* 98.6%

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

   4. Simplified 98.6%

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

   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 97.5%

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

      1. flip-- 75.2%

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

      2. associate-*l/ 75.2%

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

      3. metadata-eval 75.2%

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

      4. pow2 75.2%

         \[\leadsto \frac{\left(1 - \color{blue}{{\left(\frac{x}{z}\right)}^{2}}\right) \cdot y}{1 + \frac{x}{z}} + \frac{x}{z} \]

   4. Applied egg-rr 75.2%

      \[\leadsto \color{blue}{\frac{\left(1 - {\left(\frac{x}{z}\right)}^{2}\right) \cdot y}{1 + \frac{x}{z}}} + \frac{x}{z} \]

   5. Taylor expanded in x around 0 99.2%

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

4. Final simplification 98.9%

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

Alternative 5: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-115} \lor \neg \left(z \leq 1.82 \cdot 10^{-183}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.75e-115) (not (<= z 1.82e-183)))
   (+ y (/ x z))
   (* (/ x z) (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.75e-115) || !(z <= 1.82e-183)) {
		tmp = y + (x / z);
	} 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 ((z <= (-1.75d-115)) .or. (.not. (z <= 1.82d-183))) then
        tmp = y + (x / z)
    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 ((z <= -1.75e-115) || !(z <= 1.82e-183)) {
		tmp = y + (x / z);
	} else {
		tmp = (x / z) * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.75e-115) or not (z <= 1.82e-183):
		tmp = y + (x / z)
	else:
		tmp = (x / z) * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.75e-115) || !(z <= 1.82e-183))
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(Float64(x / z) * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.75e-115) || ~((z <= 1.82e-183)))
		tmp = y + (x / z);
	else
		tmp = (x / z) * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.75e-115], N[Not[LessEqual[z, 1.82e-183]], $MachinePrecision]], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7500000000000001e-115 or 1.82e-183 < z

   1. Initial program 77.3%

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

   2. Taylor expanded in y around 0 99.4%

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

      1. flip-- 87.9%

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

      2. associate-*l/ 85.9%

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

      3. metadata-eval 85.9%

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

      4. pow2 85.9%

         \[\leadsto \frac{\left(1 - \color{blue}{{\left(\frac{x}{z}\right)}^{2}}\right) \cdot y}{1 + \frac{x}{z}} + \frac{x}{z} \]

   4. Applied egg-rr 85.9%

      \[\leadsto \color{blue}{\frac{\left(1 - {\left(\frac{x}{z}\right)}^{2}\right) \cdot y}{1 + \frac{x}{z}}} + \frac{x}{z} \]

   5. Taylor expanded in x around 0 84.9%

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

   if -1.7500000000000001e-115 < z < 1.82e-183

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 95.8%

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

      1. +-commutative 95.8%

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

      2. distribute-rgt1-in 95.8%

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

      3. associate-*r* 95.8%

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

      4. mul-1-neg 95.8%

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

      5. unsub-neg 95.8%

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

   4. Simplified 95.8%

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

   5. Taylor expanded in y around inf 64.3%

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

      1. mul-1-neg 64.3%

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

      2. distribute-rgt-neg-out 64.3%

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

   7. Simplified 64.3%

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

      1. div-inv 64.2%

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

      2. distribute-rgt-neg-out 64.2%

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

      3. remove-double-neg 64.2%

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

      4. distribute-rgt-neg-out 64.2%

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

      5. distribute-lft-neg-out 64.2%

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

      6. distribute-rgt-neg-out 64.2%

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

      7. remove-double-neg 64.2%

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

      8. associate-*r* 65.5%

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

      9. div-inv 65.6%

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

   9. Applied egg-rr 65.6%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-115} \lor \neg \left(z \leq 1.82 \cdot 10^{-183}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \end{array} \]

Alternative 6: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15500000 \lor \neg \left(y \leq 5 \cdot 10^{-28}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -15500000.0) (not (<= y 5e-28))) (* z (/ y z)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -15500000.0) || !(y <= 5e-28)) {
		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 <= (-15500000.0d0)) .or. (.not. (y <= 5d-28))) 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 <= -15500000.0) || !(y <= 5e-28)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -15500000.0) or not (y <= 5e-28):
		tmp = z * (y / z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -15500000.0) || !(y <= 5e-28))
		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 <= -15500000.0) || ~((y <= 5e-28)))
		tmp = z * (y / z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -15500000.0], N[Not[LessEqual[y, 5e-28]], $MachinePrecision]], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55e7 or 5.0000000000000002e-28 < y

   1. Initial program 68.5%

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

   2. Taylor expanded in x around 0 25.3%

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

      1. div-inv 25.3%

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

      2. *-commutative 25.3%

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

      3. associate-*l* 56.6%

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

   4. Applied egg-rr 56.6%

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

      1. associate-*r/ 56.8%

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

      2. *-rgt-identity 56.8%

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

   6. Simplified 56.8%

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

   if -1.55e7 < y < 5.0000000000000002e-28

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 76.7%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15500000 \lor \neg \left(y \leq 5 \cdot 10^{-28}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 7: 61.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15500000:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -15500000.0) y (if (<= y 1.85e-28) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -15500000.0) {
		tmp = y;
	} else if (y <= 1.85e-28) {
		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 <= (-15500000.0d0)) then
        tmp = y
    else if (y <= 1.85d-28) 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 <= -15500000.0) {
		tmp = y;
	} else if (y <= 1.85e-28) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -15500000.0:
		tmp = y
	elif y <= 1.85e-28:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -15500000.0)
		tmp = y;
	elseif (y <= 1.85e-28)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -15500000.0)
		tmp = y;
	elseif (y <= 1.85e-28)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -15500000.0], y, If[LessEqual[y, 1.85e-28], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55e7 or 1.8500000000000001e-28 < y

   1. Initial program 68.5%

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

   2. Taylor expanded in x around 0 50.4%

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

   if -1.55e7 < y < 1.8500000000000001e-28

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 76.7%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15500000:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 8: 78.8% 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 83.4%

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

2. Taylor expanded in y around 0 96.4%

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

   1. flip-- 81.5%

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

   2. associate-*l/ 79.0%

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

   3. metadata-eval 79.0%

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

   4. pow2 79.0%

      \[\leadsto \frac{\left(1 - \color{blue}{{\left(\frac{x}{z}\right)}^{2}}\right) \cdot y}{1 + \frac{x}{z}} + \frac{x}{z} \]

4. Applied egg-rr 79.0%

   \[\leadsto \color{blue}{\frac{\left(1 - {\left(\frac{x}{z}\right)}^{2}\right) \cdot y}{1 + \frac{x}{z}}} + \frac{x}{z} \]

5. Taylor expanded in x around 0 75.4%

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

6. Final simplification 75.4%

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

Alternative 9: 41.0% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

Derivation

1. Initial program 83.4%

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

2. Taylor expanded in x around 0 37.6%

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

3. Final simplification 37.6%

   \[\leadsto y \]

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 2023274 
(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))