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

Percentage Accurate: 88.0% → 99.8%

Time: 8.5s

Alternatives: 10

Speedup: 0.5×

• 21.5% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -45000000 \lor \neg \left(y \leq 6000000000\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -45000000.0) (not (<= y 6000000000.0)))
   (/ y (/ z (- z x)))
   (/ (fma y (- z x) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -45000000.0) || !(y <= 6000000000.0)) {
		tmp = y / (z / (z - x));
	} else {
		tmp = fma(y, (z - x), x) / z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -45000000.0) || !(y <= 6000000000.0))
		tmp = Float64(y / Float64(z / Float64(z - x)));
	else
		tmp = Float64(fma(y, Float64(z - x), x) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -45000000.0], N[Not[LessEqual[y, 6000000000.0]], $MachinePrecision]], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(z - x), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5e7 or 6e9 < y

   1. Initial program 72.8%

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

   3. Taylor expanded in y around inf 72.8%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -4.5e7 < y < 6e9

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -45000000 \lor \neg \left(y \leq 6000000000\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -45000000 \lor \neg \left(y \leq 3500000000\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -45000000.0) (not (<= y 3500000000.0)))
   (/ y (/ z (- z x)))
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -45000000.0) || !(y <= 3500000000.0)) {
		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 <= (-45000000.0d0)) .or. (.not. (y <= 3500000000.0d0))) 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 <= -45000000.0) || !(y <= 3500000000.0)) {
		tmp = y / (z / (z - x));
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -45000000.0) or not (y <= 3500000000.0):
		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 <= -45000000.0) || !(y <= 3500000000.0))
		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 <= -45000000.0) || ~((y <= 3500000000.0)))
		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, -45000000.0], N[Not[LessEqual[y, 3500000000.0]], $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 < -4.5e7 or 3.5e9 < y

   1. Initial program 72.8%

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

   3. Taylor expanded in y around inf 72.8%

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

      1. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -4.5e7 < y < 3.5e9

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -45000000 \lor \neg \left(y \leq 3500000000\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -950000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -950000000000.0) (not (<= y 1.0)))
   (/ y (/ z (- z x)))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -950000000000.0) || !(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 <= (-950000000000.0d0)) .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 <= -950000000000.0) || !(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 <= -950000000000.0) 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 <= -950000000000.0) || !(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 <= -950000000000.0) || ~((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, -950000000000.0], 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 < -9.5e11 or 1 < y

   1. Initial program 73.4%

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

   3. Taylor expanded in y around inf 73.1%

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

      1. associate-/l* 99.6%

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

   5. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.7%

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

   7. Applied egg-rr 99.7%

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

   if -9.5e11 < y < 1

   1. Initial program 99.2%

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

   3. Taylor expanded in z around inf 97.0%

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

   4. Taylor expanded in y around 0 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -950000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -950000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -950000000000.0) (not (<= y 1.0)))
   (* y (/ (- z x) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -950000000000.0) || !(y <= 1.0)) {
		tmp = y * ((z - 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 <= (-950000000000.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * ((z - 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 <= -950000000000.0) || !(y <= 1.0)) {
		tmp = y * ((z - x) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -950000000000.0) or not (y <= 1.0):
		tmp = y * ((z - x) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -950000000000.0) || !(y <= 1.0))
		tmp = Float64(y * Float64(Float64(z - 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 <= -950000000000.0) || ~((y <= 1.0)))
		tmp = y * ((z - x) / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -950000000000.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.5e11 or 1 < y

   1. Initial program 73.4%

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

   3. Taylor expanded in y around inf 73.1%

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

      1. associate-/l* 99.6%

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

   5. Simplified 99.6%

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

   if -9.5e11 < y < 1

   1. Initial program 99.2%

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

   3. Taylor expanded in z around inf 97.0%

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

   4. Taylor expanded in y around 0 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -950000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-85} \lor \neg \left(z \leq 1.28 \cdot 10^{-101}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.5e-85) (not (<= z 1.28e-101)))
   (+ y (/ x z))
   (* x (/ (- 1.0 y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.5e-85) || !(z <= 1.28e-101)) {
		tmp = y + (x / z);
	} else {
		tmp = x * ((1.0 - y) / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.5d-85)) .or. (.not. (z <= 1.28d-101))) then
        tmp = y + (x / z)
    else
        tmp = x * ((1.0d0 - y) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.5e-85) || !(z <= 1.28e-101)) {
		tmp = y + (x / z);
	} else {
		tmp = x * ((1.0 - y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.5e-85) or not (z <= 1.28e-101):
		tmp = y + (x / z)
	else:
		tmp = x * ((1.0 - y) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.5e-85) || !(z <= 1.28e-101))
		tmp = Float64(y + Float64(x / z));
	else
		tmp = Float64(x * Float64(Float64(1.0 - y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.5e-85) || ~((z <= 1.28e-101)))
		tmp = y + (x / z);
	else
		tmp = x * ((1.0 - y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.5e-85], N[Not[LessEqual[z, 1.28e-101]], $MachinePrecision]], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.50000000000000011e-85 or 1.27999999999999995e-101 < z

   1. Initial program 78.8%

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

   3. Taylor expanded in z around inf 93.5%

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

   4. Taylor expanded in y around 0 87.2%

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

   if -1.50000000000000011e-85 < z < 1.27999999999999995e-101

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 92.9%

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

      1. associate-/l* 86.1%

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

      2. mul-1-neg 86.1%

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

      3. unsub-neg 86.1%

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

   5. Simplified 86.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-85} \lor \neg \left(z \leq 1.28 \cdot 10^{-101}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+179}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+253}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e+179)
   (+ y (/ x z))
   (if (<= y 4.3e+253) (* x (/ y (- z))) (* z (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e+179) {
		tmp = y + (x / z);
	} else if (y <= 4.3e+253) {
		tmp = x * (y / -z);
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 6.5d+179) then
        tmp = y + (x / z)
    else if (y <= 4.3d+253) then
        tmp = x * (y / -z)
    else
        tmp = z * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e+179) {
		tmp = y + (x / z);
	} else if (y <= 4.3e+253) {
		tmp = x * (y / -z);
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.5e+179:
		tmp = y + (x / z)
	elif y <= 4.3e+253:
		tmp = x * (y / -z)
	else:
		tmp = z * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.5e+179)
		tmp = Float64(y + Float64(x / z));
	elseif (y <= 4.3e+253)
		tmp = Float64(x * Float64(y / Float64(-z)));
	else
		tmp = Float64(z * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.5e+179)
		tmp = y + (x / z);
	elseif (y <= 4.3e+253)
		tmp = x * (y / -z);
	else
		tmp = z * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.5e+179], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.3e+253], N[(x * N[(y / (-z)), $MachinePrecision]), $MachinePrecision], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.50000000000000052e179

   1. Initial program 88.8%

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

   3. Taylor expanded in z around inf 94.3%

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

   4. Taylor expanded in y around 0 84.1%

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

   if 6.50000000000000052e179 < y < 4.2999999999999999e253

   1. Initial program 87.7%

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

   3. Taylor expanded in x around inf 80.8%

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

      1. associate-/l* 86.8%

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

      2. mul-1-neg 86.8%

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

      3. unsub-neg 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in y around inf 86.8%

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

      1. neg-mul-1 86.8%

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

   8. Simplified 86.8%

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

   if 4.2999999999999999e253 < y

   1. Initial program 59.9%

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

   3. Taylor expanded in y around inf 59.9%

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

   4. Taylor expanded in z around inf 22.8%

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

      1. *-commutative 22.8%

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

      2. associate-/l* 73.3%

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

   6. Applied egg-rr 73.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+179}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+253}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-86}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.1e-86) y (if (<= y 1.1e-11) (/ x z) (* z (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.1e-86) {
		tmp = y;
	} else if (y <= 1.1e-11) {
		tmp = x / z;
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.1d-86)) then
        tmp = y
    else if (y <= 1.1d-11) then
        tmp = x / z
    else
        tmp = z * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.1e-86) {
		tmp = y;
	} else if (y <= 1.1e-11) {
		tmp = x / z;
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.1e-86:
		tmp = y
	elif y <= 1.1e-11:
		tmp = x / z
	else:
		tmp = z * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.1e-86)
		tmp = y;
	elseif (y <= 1.1e-11)
		tmp = Float64(x / z);
	else
		tmp = Float64(z * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.1e-86)
		tmp = y;
	elseif (y <= 1.1e-11)
		tmp = x / z;
	else
		tmp = z * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.1e-86], y, If[LessEqual[y, 1.1e-11], N[(x / z), $MachinePrecision], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1000000000000001e-86

   1. Initial program 78.8%

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

   3. Taylor expanded in x around 0 61.8%

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

   if -1.1000000000000001e-86 < y < 1.1000000000000001e-11

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 77.0%

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

   if 1.1000000000000001e-11 < y

   1. Initial program 75.1%

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

   3. Taylor expanded in y around inf 75.1%

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

   4. Taylor expanded in z around inf 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/l* 54.4%

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

   6. Applied egg-rr 54.4%

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

Alternative 8: 60.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-86}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.6e-86) y (if (<= y 5.5e-10) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.6e-86) {
		tmp = y;
	} else if (y <= 5.5e-10) {
		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 <= (-1.6d-86)) then
        tmp = y
    else if (y <= 5.5d-10) 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 <= -1.6e-86) {
		tmp = y;
	} else if (y <= 5.5e-10) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.6e-86:
		tmp = y
	elif y <= 5.5e-10:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.6e-86)
		tmp = y;
	elseif (y <= 5.5e-10)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.6e-86)
		tmp = y;
	elseif (y <= 5.5e-10)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.6e-86], y, If[LessEqual[y, 5.5e-10], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -1.60000000000000003e-86 or 5.4999999999999996e-10 < y

   1. Initial program 77.0%

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

   3. Taylor expanded in x around 0 55.8%

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

   if -1.60000000000000003e-86 < y < 5.4999999999999996e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 77.0%

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

Alternative 9: 79.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.0d0) then
        tmp = y + (x / z)
    else
        tmp = z * (y / 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 + (x / z);
	} else {
		tmp = z * (y / z);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 91.1%

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

   3. Taylor expanded in z around inf 95.2%

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

   4. Taylor expanded in y around 0 89.4%

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

   if 1 < y

   1. Initial program 75.1%

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

   3. Taylor expanded in y around inf 75.1%

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

   4. Taylor expanded in z around inf 28.2%

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

      1. *-commutative 28.2%

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

      2. associate-/l* 54.4%

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

   6. Applied egg-rr 54.4%

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

Alternative 10: 41.1% 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 86.7%

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

3. Taylor expanded in x around 0 42.6%

   \[\leadsto \color{blue}{y} \]
4. Add Preprocessing

Developer Target 1: 93.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(y + \frac{x}{z}\right) - \frac{y}{\frac{z}{x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024116 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- (+ y (/ x z)) (/ y (/ z x))))

  (/ (+ x (* y (- z x))) z))