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

Percentage Accurate: 88.1% → 98.6%

Time: 4.6s

Alternatives: 8

Speedup: 0.8×

• 20.4% 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: 88.1% 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: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+27}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.7e+27)
   (- 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 <= -2.7e+27) {
		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 <= (-2.7d+27)) 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 <= -2.7e+27) {
		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 <= -2.7e+27:
		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 <= -2.7e+27)
		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 <= -2.7e+27)
		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, -2.7e+27], 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 < -2.6999999999999997e27

   1. Initial program 70.0%

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

   2. Taylor expanded in x around inf 88.9%

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

   3. Taylor expanded in y around inf 88.9%

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

      1. associate-*r/ 88.9%

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

      2. mul-1-neg 88.9%

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

      3. *-commutative 88.9%

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

      4. distribute-lft-neg-in 88.9%

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

      5. associate-*l/ 99.8%

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

      6. distribute-neg-frac 99.8%

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

      7. mul-1-neg 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*l* 99.8%

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

      10. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   if -2.6999999999999997e27 < y < 1

   1. Initial program 99.2%

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

   2. Taylor expanded in z around inf 97.2%

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

   3. Taylor expanded in x around 0 98.0%

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

      1. +-commutative 98.0%

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

   5. Simplified 98.0%

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

   if 1 < y

   1. Initial program 80.4%

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

   2. Taylor expanded in x around inf 94.2%

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

   3. Taylor expanded in y around 0 94.2%

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

      1. +-commutative 94.2%

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

      2. mul-1-neg 94.2%

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

      3. unsub-neg 94.2%

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

      4. *-commutative 94.2%

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

   5. Simplified 94.2%

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

   6. Taylor expanded in y around inf 99.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+27}:\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \]

Alternative 2: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+27} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.7e+27) (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 <= -2.7e+27) || !(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 <= (-2.7d+27)) .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 <= -2.7e+27) || !(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 <= -2.7e+27) 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 <= -2.7e+27) || !(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 <= -2.7e+27) || ~((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, -2.7e+27], 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 < -2.6999999999999997e27 or 1 < y

   1. Initial program 75.5%

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

   2. Taylor expanded in x around inf 91.7%

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

   3. Taylor expanded in y around 0 91.7%

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

      1. +-commutative 91.7%

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

      2. mul-1-neg 91.7%

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

      3. unsub-neg 91.7%

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

      4. *-commutative 91.7%

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

   5. Simplified 91.7%

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

   6. Taylor expanded in y around inf 99.7%

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

   if -2.6999999999999997e27 < y < 1

   1. Initial program 99.2%

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

   2. Taylor expanded in z around inf 97.2%

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

   3. Taylor expanded in x around 0 98.0%

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

      1. +-commutative 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+27} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 3: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+183}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x - y \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.4e+183) (* y (- 1.0 (/ x z))) (+ y (/ (- x (* y x)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+183) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + ((x - (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 <= (-2.4d+183)) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = y + ((x - (y * x)) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e+183) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + ((x - (y * x)) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.4e+183:
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + ((x - (y * x)) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.4e+183)
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(y + Float64(Float64(x - Float64(y * x)) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.4e+183)
		tmp = y * (1.0 - (x / z));
	else
		tmp = y + ((x - (y * x)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.4e+183], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(N[(x - N[(y * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4000000000000002e183

   1. Initial program 46.9%

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

   2. Taylor expanded in x around inf 75.0%

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

   3. Taylor expanded in y around 0 75.0%

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

      1. +-commutative 75.0%

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

      2. mul-1-neg 75.0%

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

      3. unsub-neg 75.0%

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

      4. *-commutative 75.0%

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

   5. Simplified 75.0%

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

   6. Taylor expanded in y around inf 99.9%

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

   if -2.4000000000000002e183 < y

   1. Initial program 92.1%

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

   2. Taylor expanded in x around inf 98.3%

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

   3. Taylor expanded in y around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. mul-1-neg 98.3%

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

      3. unsub-neg 98.3%

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

      4. *-commutative 98.3%

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

   5. Simplified 98.3%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+183}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x - y \cdot x}{z}\\ \end{array} \]

Alternative 4: 61.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-50} \lor \neg \left(y \leq 8 \cdot 10^{-5}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.3e-50) (not (<= y 8e-5))) (* z (/ y z)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.3e-50) || !(y <= 8e-5)) {
		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 <= (-1.3d-50)) .or. (.not. (y <= 8d-5))) 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 <= -1.3e-50) || !(y <= 8e-5)) {
		tmp = z * (y / z);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.3e-50) or not (y <= 8e-5):
		tmp = z * (y / z)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.3e-50) || !(y <= 8e-5))
		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 <= -1.3e-50) || ~((y <= 8e-5)))
		tmp = z * (y / z);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.3e-50], N[Not[LessEqual[y, 8e-5]], $MachinePrecision]], N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3000000000000001e-50 or 8.00000000000000065e-5 < y

   1. Initial program 77.6%

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

   2. Taylor expanded in y around inf 75.4%

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

   3. Taylor expanded in z around inf 36.2%

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

      1. associate-/l* 54.5%

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

      2. associate-/r/ 55.5%

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

   5. Applied egg-rr 55.5%

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

   if -1.3000000000000001e-50 < y < 8.00000000000000065e-5

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 72.8%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-50} \lor \neg \left(y \leq 8 \cdot 10^{-5}\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 5: 60.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-49}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-49) y (if (<= y 8e-5) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-49) {
		tmp = y;
	} else if (y <= 8e-5) {
		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 <= (-1d-49)) then
        tmp = y
    else if (y <= 8d-5) 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 <= -1e-49) {
		tmp = y;
	} else if (y <= 8e-5) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1e-49:
		tmp = y
	elif y <= 8e-5:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-49)
		tmp = y;
	elseif (y <= 8e-5)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e-49)
		tmp = y;
	elseif (y <= 8e-5)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-49], y, If[LessEqual[y, 8e-5], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999936e-50 or 8.00000000000000065e-5 < y

   1. Initial program 77.6%

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

   2. Taylor expanded in x around 0 54.5%

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

   if -9.99999999999999936e-50 < y < 8.00000000000000065e-5

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 72.8%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-49}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6: 81.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.0:
		tmp = y + (x / z)
	else:
		tmp = y - (x / 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(y - Float64(x / 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 = y - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 90.1%

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

   2. Taylor expanded in z around inf 79.7%

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

   3. Taylor expanded in x around 0 87.5%

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

      1. +-commutative 87.5%

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

   5. Simplified 87.5%

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

   if 1 < y

   1. Initial program 80.4%

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

   2. Taylor expanded in z around inf 32.4%

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

   3. Taylor expanded in x around 0 49.1%

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

      1. +-commutative 49.1%

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

   5. Simplified 49.1%

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

      1. +-commutative 49.1%

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

      2. add-cube-cbrt 48.1%

         \[\leadsto \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}} + \frac{x}{z} \]

      3. fma-def 48.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \frac{x}{z}\right)} \]

      4. add-sqr-sqrt 31.5%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \color{blue}{\sqrt{\frac{x}{z}} \cdot \sqrt{\frac{x}{z}}}\right) \]

      5. sqrt-unprod 54.7%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \color{blue}{\sqrt{\frac{x}{z} \cdot \frac{x}{z}}}\right) \]

      6. clear-num 54.7%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \sqrt{\color{blue}{\frac{1}{\frac{z}{x}}} \cdot \frac{x}{z}}\right) \]

      7. clear-num 54.7%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \sqrt{\frac{1}{\frac{z}{x}} \cdot \color{blue}{\frac{1}{\frac{z}{x}}}}\right) \]

      8. frac-times 54.7%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \sqrt{\color{blue}{\frac{1 \cdot 1}{\frac{z}{x} \cdot \frac{z}{x}}}}\right) \]

      9. metadata-eval 54.7%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \sqrt{\frac{\color{blue}{1}}{\frac{z}{x} \cdot \frac{z}{x}}}\right) \]

      10. metadata-eval 54.7%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \sqrt{\frac{\color{blue}{-1 \cdot -1}}{\frac{z}{x} \cdot \frac{z}{x}}}\right) \]

      11. frac-times 54.7%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \sqrt{\color{blue}{\frac{-1}{\frac{z}{x}} \cdot \frac{-1}{\frac{z}{x}}}}\right) \]

      12. sqrt-unprod 28.9%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \color{blue}{\sqrt{\frac{-1}{\frac{z}{x}}} \cdot \sqrt{\frac{-1}{\frac{z}{x}}}}\right) \]

      13. add-sqr-sqrt 64.6%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \color{blue}{\frac{-1}{\frac{z}{x}}}\right) \]

      14. div-inv 64.6%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \color{blue}{-1 \cdot \frac{1}{\frac{z}{x}}}\right) \]

      15. clear-num 64.6%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, -1 \cdot \color{blue}{\frac{x}{z}}\right) \]

      16. mul-1-neg 64.6%

          \[\leadsto \mathsf{fma}\left(\sqrt[3]{y} \cdot \sqrt[3]{y}, \sqrt[3]{y}, \color{blue}{-\frac{x}{z}}\right) \]

      17. fma-neg 64.6%

          \[\leadsto \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y} - \frac{x}{z}} \]

      18. add-cube-cbrt 65.7%

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

   7. Applied egg-rr 65.7%

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

4. Final simplification 81.8%

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

Alternative 7: 78.3% 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 87.5%

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

2. Taylor expanded in z around inf 67.3%

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

3. Taylor expanded in x around 0 77.5%

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

   1. +-commutative 77.5%

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

5. Simplified 77.5%

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

6. Final simplification 77.5%

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

Alternative 8: 42.2% 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 87.5%

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

2. Taylor expanded in x around 0 43.0%

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

3. Final simplification 43.0%

   \[\leadsto y \]

Developer target: 94.3% 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 2023196 
(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))