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

Percentage Accurate: 88.5% → 99.7%

Time: 7.4s

Alternatives: 10

Speedup: 0.8×

• 20.7% 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: 88.5% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+38} \lor \neg \left(y \leq 195\right):\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.8e+38) (not (<= y 195.0)))
   (- y (/ y (/ z x)))
   (/ (+ x (* y (- z x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.8e+38) || !(y <= 195.0)) {
		tmp = y - (y / (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 <= (-4.8d+38)) .or. (.not. (y <= 195.0d0))) then
        tmp = y - (y / (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 <= -4.8e+38) || !(y <= 195.0)) {
		tmp = y - (y / (z / x));
	} else {
		tmp = (x + (y * (z - x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.8e+38) or not (y <= 195.0):
		tmp = y - (y / (z / x))
	else:
		tmp = (x + (y * (z - x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.8e+38) || !(y <= 195.0))
		tmp = Float64(y - Float64(y / 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 <= -4.8e+38) || ~((y <= 195.0)))
		tmp = y - (y / (z / x));
	else
		tmp = (x + (y * (z - x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.8e+38], N[Not[LessEqual[y, 195.0]], $MachinePrecision]], N[(y - N[(y / 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 < -4.80000000000000035e38 or 195 < y

   1. Initial program 77.2%

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

   3. Taylor expanded in y around inf 77.2%

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

      1. associate-/l* 99.9%

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

      2. div-sub 99.9%

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

      3. sub-neg 99.9%

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

      4. *-inverses 99.9%

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

      5. sub-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 89.5%

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

      1. associate-*r/ 89.5%

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

      2. mul-1-neg 89.5%

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

      3. distribute-rgt-neg-out 89.5%

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

   8. Simplified 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-*l/ 87.7%

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

      3. add-sqr-sqrt 41.2%

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

      4. sqrt-unprod 46.7%

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

      5. sqr-neg 46.7%

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

      6. sqrt-unprod 26.3%

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

      7. add-sqr-sqrt 43.4%

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

      8. cancel-sign-sub 43.4%

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

      9. distribute-frac-neg 43.4%

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

      10. *-commutative 43.4%

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

      11. associate-*r/ 48.1%

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

      12. div-inv 48.1%

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

      13. *-commutative 48.1%

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

      14. associate-*l* 50.5%

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

      15. add-sqr-sqrt 19.3%

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

      16. sqrt-unprod 46.8%

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

      17. sqr-neg 46.8%

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

      18. sqrt-unprod 54.8%

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

      19. add-sqr-sqrt 99.9%

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

      20. div-inv 99.9%

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

      21. clear-num 99.9%

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

   10. Applied egg-rr 99.9%

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

   if -4.80000000000000035e38 < y < 195

   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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+38} \lor \neg \left(y \leq 195\right):\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -550000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -550000000000.0) (not (<= y 1.0)))
   (- y (/ y (/ z x)))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -550000000000.0) || !(y <= 1.0)) {
		tmp = y - (y / (z / x));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-550000000000.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y - (y / (z / x))
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -550000000000.0) || !(y <= 1.0)) {
		tmp = y - (y / (z / x));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -550000000000.0) or not (y <= 1.0):
		tmp = y - (y / (z / x))
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -550000000000.0) || !(y <= 1.0))
		tmp = Float64(y - Float64(y / 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 <= -550000000000.0) || ~((y <= 1.0)))
		tmp = y - (y / (z / x));
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -550000000000.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y - N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5e11 or 1 < y

   1. Initial program 78.0%

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

   3. Taylor expanded in y around inf 78.0%

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

      1. associate-/l* 99.9%

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

      2. div-sub 99.9%

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

      3. sub-neg 99.9%

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

      4. *-inverses 99.9%

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

      5. sub-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 89.8%

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

      1. associate-*r/ 89.8%

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

      2. mul-1-neg 89.8%

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

      3. distribute-rgt-neg-out 89.8%

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

   8. Simplified 89.8%

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

      1. *-commutative 89.8%

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

      2. associate-*l/ 88.0%

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

      3. add-sqr-sqrt 43.0%

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

      4. sqrt-unprod 48.3%

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

      5. sqr-neg 48.3%

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

      6. sqrt-unprod 25.5%

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

      7. add-sqr-sqrt 42.7%

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

      8. cancel-sign-sub 42.7%

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

      9. distribute-frac-neg 42.7%

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

      10. *-commutative 42.7%

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

      11. associate-*r/ 47.2%

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

      12. div-inv 47.2%

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

      13. *-commutative 47.2%

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

      14. associate-*l* 49.6%

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

      15. add-sqr-sqrt 19.3%

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

      16. sqrt-unprod 46.0%

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

      17. sqr-neg 46.0%

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

      18. sqrt-unprod 53.0%

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

      19. add-sqr-sqrt 99.9%

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

      20. div-inv 99.9%

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

      21. clear-num 99.9%

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

   10. Applied egg-rr 99.9%

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

   if -5.5e11 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 98.4%

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

   4. Taylor expanded in x around 0 97.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -550000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -550000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -550000000000.0) (not (<= y 1.0)))
   (* y (- 1.0 (/ x z)))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -550000000000.0) || !(y <= 1.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-550000000000.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = y * (1.0d0 - (x / z))
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -550000000000.0) || !(y <= 1.0)) {
		tmp = y * (1.0 - (x / z));
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -550000000000.0) or not (y <= 1.0):
		tmp = y * (1.0 - (x / z))
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -550000000000.0) || !(y <= 1.0))
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -550000000000.0) || ~((y <= 1.0)))
		tmp = y * (1.0 - (x / z));
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -550000000000.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5e11 or 1 < y

   1. Initial program 78.0%

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

   3. Taylor expanded in y around inf 78.0%

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

      1. associate-/l* 99.9%

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

      2. div-sub 99.9%

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

      3. sub-neg 99.9%

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

      4. *-inverses 99.9%

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

      5. sub-neg 99.9%

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

   5. Simplified 99.9%

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

   if -5.5e11 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 98.4%

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

   4. Taylor expanded in x around 0 97.6%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -550000000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+156} \lor \neg \left(x \leq 4.9 \cdot 10^{+26}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.5e+156) (not (<= x 4.9e+26)))
   (* x (/ (- 1.0 y) z))
   (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.5e+156) || !(x <= 4.9e+26)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-8.5d+156)) .or. (.not. (x <= 4.9d+26))) then
        tmp = x * ((1.0d0 - y) / z)
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.5e+156) || !(x <= 4.9e+26)) {
		tmp = x * ((1.0 - y) / z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.5e+156) or not (x <= 4.9e+26):
		tmp = x * ((1.0 - y) / z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.5e+156) || !(x <= 4.9e+26))
		tmp = Float64(x * Float64(Float64(1.0 - y) / z));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.5e+156) || ~((x <= 4.9e+26)))
		tmp = x * ((1.0 - y) / z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.5e+156], N[Not[LessEqual[x, 4.9e+26]], $MachinePrecision]], N[(x * N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.49999999999999948e156 or 4.89999999999999974e26 < x

   1. Initial program 83.8%

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

   3. Taylor expanded in x around inf 78.8%

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

      1. associate-/l* 92.0%

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

      2. mul-1-neg 92.0%

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

      3. unsub-neg 92.0%

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

   5. Simplified 92.0%

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

   if -8.49999999999999948e156 < x < 4.89999999999999974e26

   1. Initial program 90.8%

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

   3. Taylor expanded in y around 0 99.4%

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

   4. Taylor expanded in x around 0 83.0%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+156} \lor \neg \left(x \leq 4.9 \cdot 10^{+26}\right):\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.065:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -0.065) y (if (<= y 6.4e-71) (/ x z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -0.065) {
		tmp = y;
	} else if (y <= 6.4e-71) {
		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 <= (-0.065d0)) then
        tmp = y
    else if (y <= 6.4d-71) 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 <= -0.065) {
		tmp = y;
	} else if (y <= 6.4e-71) {
		tmp = x / z;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -0.065:
		tmp = y
	elif y <= 6.4e-71:
		tmp = x / z
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -0.065)
		tmp = y;
	elseif (y <= 6.4e-71)
		tmp = Float64(x / z);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -0.065)
		tmp = y;
	elseif (y <= 6.4e-71)
		tmp = x / z;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -0.065], y, If[LessEqual[y, 6.4e-71], N[(x / z), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if y < -0.065000000000000002 or 6.3999999999999998e-71 < y

   1. Initial program 80.7%

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

   3. Taylor expanded in x around 0 52.8%

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

   if -0.065000000000000002 < y < 6.3999999999999998e-71

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 71.8%

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

Alternative 6: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+115}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.8e+115) (* y (/ x (- z))) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.8e+115) {
		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 <= (-1.8d+115)) 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 <= -1.8e+115) {
		tmp = y * (x / -z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.8e+115:
		tmp = y * (x / -z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.8e+115)
		tmp = Float64(y * Float64(x / Float64(-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 <= -1.8e+115)
		tmp = y * (x / -z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.8e+115], N[(y * N[(x / (-z)), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e115

   1. Initial program 68.3%

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

   3. Taylor expanded in y around inf 68.3%

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

      1. associate-/l* 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. *-inverses 100.0%

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

      5. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 62.6%

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

      1. mul-1-neg 62.6%

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

      2. distribute-frac-neg2 62.6%

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

   8. Simplified 62.6%

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

   if -1.8e115 < y

   1. Initial program 92.9%

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

   3. Taylor expanded in y around 0 98.1%

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

   4. Taylor expanded in x around 0 81.6%

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

Alternative 7: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+213}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.5e+213) (* x (/ y (- z))) (+ y (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.5e+213) {
		tmp = x * (y / -z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3.5d+213)) then
        tmp = x * (y / -z)
    else
        tmp = y + (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.5e+213) {
		tmp = x * (y / -z);
	} else {
		tmp = y + (x / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.5e+213:
		tmp = x * (y / -z)
	else:
		tmp = y + (x / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.5e+213)
		tmp = Float64(x * Float64(y / Float64(-z)));
	else
		tmp = Float64(y + Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.5e+213)
		tmp = x * (y / -z);
	else
		tmp = y + (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.5e+213], N[(x * N[(y / (-z)), $MachinePrecision]), $MachinePrecision], N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4999999999999997e213

   1. Initial program 63.8%

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

   3. Taylor expanded in x around inf 63.8%

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

      1. associate-/l* 95.4%

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

      2. mul-1-neg 95.4%

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

      3. unsub-neg 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in y around inf 68.3%

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

      1. neg-mul-1 68.3%

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

      2. distribute-neg-frac 68.3%

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

   8. Simplified 68.3%

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

   if -3.4999999999999997e213 < x

   1. Initial program 90.8%

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

   3. Taylor expanded in y around 0 99.5%

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

   4. Taylor expanded in x around 0 79.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+213}:\\ \;\;\;\;x \cdot \frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (x / z) + (y * (1.0 - (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 / z) + (y * (1.0d0 - (x / z)))
end function

Java

public static double code(double x, double y, double z) {
	return (x / z) + (y * (1.0 - (x / z)));
}

Python

def code(x, y, z):
	return (x / z) + (y * (1.0 - (x / z)))

Julia

function code(x, y, z)
	return Float64(Float64(x / z) + Float64(y * Float64(1.0 - Float64(x / z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x / z) + (y * (1.0 - (x / z)));
end

Wolfram

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

Derivation

1. Initial program 88.9%

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

3. Taylor expanded in y around 0 98.4%

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

4. Final simplification 98.4%

   \[\leadsto \frac{x}{z} + y \cdot \left(1 - \frac{x}{z}\right) \]
5. Add Preprocessing

Alternative 9: 78.5% 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 88.9%

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

3. Taylor expanded in y around 0 98.4%

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

4. Taylor expanded in x around 0 76.0%

   \[\leadsto \color{blue}{y} + \frac{x}{z} \]
5. Add Preprocessing

Alternative 10: 40.5% 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 88.9%

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

3. Taylor expanded in x around 0 42.4%

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

Developer Target 1: 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 2024165 
(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))