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

Percentage Accurate: 88.6% → 99.9%

Time: 8.5s

Alternatives: 9

Speedup: 1.1×

• 21.0% 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: 88.6% 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: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x}{z}, 1 - y, y\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return fma((x / z), (1.0 - y), y);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 97.3%

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

7. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(\frac{x}{z}, \color{blue}{1 - y}, y\right) \]
8. Add Preprocessing

Alternative 2: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, -y, y\right)\\ \mathbf{if}\;y \leq -0.0115:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{z \cdot y + x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) (- y) y)))
   (if (<= y -0.0115) t_0 (if (<= y 3.2e-11) (/ (+ (* z y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / z), -y, y);
	double tmp;
	if (y <= -0.0115) {
		tmp = t_0;
	} else if (y <= 3.2e-11) {
		tmp = ((z * y) + x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / z), Float64(-y), y)
	tmp = 0.0
	if (y <= -0.0115)
		tmp = t_0;
	elseif (y <= 3.2e-11)
		tmp = Float64(Float64(Float64(z * y) + x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * (-y) + y), $MachinePrecision]}, If[LessEqual[y, -0.0115], t$95$0, If[LessEqual[y, 3.2e-11], N[(N[(N[(z * y), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0115 or 3.19999999999999994e-11 < y

   1. Initial program 75.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.1%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 98.9%

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

   if -0.0115 < y < 3.19999999999999994e-11

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0115:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, -y, y\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{z \cdot y + x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, -y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, -y, y\right)\\ \mathbf{if}\;y \leq -17:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) (- y) y)))
   (if (<= y -17.0) t_0 (if (<= y 3.2e-11) (fma (/ 1.0 z) x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / z), -y, y);
	double tmp;
	if (y <= -17.0) {
		tmp = t_0;
	} else if (y <= 3.2e-11) {
		tmp = fma((1.0 / z), x, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / z), Float64(-y), y)
	tmp = 0.0
	if (y <= -17.0)
		tmp = t_0;
	elseif (y <= 3.2e-11)
		tmp = fma(Float64(1.0 / z), x, y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * (-y) + y), $MachinePrecision]}, If[LessEqual[y, -17.0], t$95$0, If[LessEqual[y, 3.2e-11], N[(N[(1.0 / z), $MachinePrecision] * x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -17 or 3.19999999999999994e-11 < y

   1. Initial program 75.2%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 98.9%

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

   if -17 < y < 3.19999999999999994e-11

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z - x}{z} \cdot y\\ \mathbf{if}\;y \leq -17:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (- z x) z) y)))
   (if (<= y -17.0) t_0 (if (<= y 3.2e-11) (fma (/ 1.0 z) x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((z - x) / z) * y;
	double tmp;
	if (y <= -17.0) {
		tmp = t_0;
	} else if (y <= 3.2e-11) {
		tmp = fma((1.0 / z), x, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(z - x) / z) * y)
	tmp = 0.0
	if (y <= -17.0)
		tmp = t_0;
	elseif (y <= 3.2e-11)
		tmp = fma(Float64(1.0 / z), x, y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -17.0], t$95$0, If[LessEqual[y, 3.2e-11], N[(N[(1.0 / z), $MachinePrecision] * x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -17 or 3.19999999999999994e-11 < y

   1. Initial program 75.2%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if -17 < y < 3.19999999999999994e-11

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - y}{z} \cdot x\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (- 1.0 y) z) x)))
   (if (<= x -2.1e+17) t_0 (if (<= x 3.4) (fma (/ 1.0 z) x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 - y) / z) * x;
	double tmp;
	if (x <= -2.1e+17) {
		tmp = t_0;
	} else if (x <= 3.4) {
		tmp = fma((1.0 / z), x, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 - y) / z) * x)
	tmp = 0.0
	if (x <= -2.1e+17)
		tmp = t_0;
	elseif (x <= 3.4)
		tmp = fma(Float64(1.0 / z), x, y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.1e+17], t$95$0, If[LessEqual[x, 3.4], N[(N[(1.0 / z), $MachinePrecision] * x + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e17 or 3.39999999999999991 < x

   1. Initial program 91.2%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-out-- N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. div-sub N/A

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

      17. unsub-neg N/A

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

      18. mul-1-neg N/A

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

      19. lower-/.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if -2.1e17 < x < 3.39999999999999991

   1. Initial program 83.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 91.0%

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 52.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;x \leq 1.08 \cdot 10^{-54}:\\ \;\;\;\;\frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.2e+19) (/ x z) (if (<= x 1.08e-54) (/ (* z y) z) (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.2e+19) {
		tmp = x / z;
	} else if (x <= 1.08e-54) {
		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 (x <= (-3.2d+19)) then
        tmp = x / z
    else if (x <= 1.08d-54) 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 (x <= -3.2e+19) {
		tmp = x / z;
	} else if (x <= 1.08e-54) {
		tmp = (z * y) / z;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.2e+19:
		tmp = x / z
	elif x <= 1.08e-54:
		tmp = (z * y) / z
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.2e+19)
		tmp = Float64(x / z);
	elseif (x <= 1.08e-54)
		tmp = Float64(Float64(z * y) / z);
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.2e+19)
		tmp = x / z;
	elseif (x <= 1.08e-54)
		tmp = (z * y) / z;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.2e+19], N[(x / z), $MachinePrecision], If[LessEqual[x, 1.08e-54], N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision], N[(x / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2e19 or 1.08000000000000002e-54 < x

   1. Initial program 91.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 53.1

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

   5. Applied rewrites 53.1%

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

   if -3.2e19 < x < 1.08000000000000002e-54

   1. Initial program 82.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

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

Alternative 7: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.5e+196) (fma (/ 1.0 z) x y) (* (- y) (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e+196) {
		tmp = fma((1.0 / z), x, y);
	} else {
		tmp = -y * (x / z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.5e+196)
		tmp = fma(Float64(1.0 / z), x, y);
	else
		tmp = Float64(Float64(-y) * Float64(x / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.5e+196], N[(N[(1.0 / z), $MachinePrecision] * x + y), $MachinePrecision], N[((-y) * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.50000000000000041e196

   1. Initial program 88.8%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 83.3%

      \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]

   if 8.50000000000000041e196 < y

   1. Initial program 69.2%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. div-sub N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. associate-/l* N/A

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

      7. distribute-lft-out-- N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. div-sub N/A

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

      17. unsub-neg N/A

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

      18. mul-1-neg N/A

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

      19. lower-/.f64 N/A

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. lower--.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 61.9%

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

   10. Applied rewrites 61.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+196}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{1}{z}, x, y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (/ 1.0 z) x y))

C

double code(double x, double y, double z) {
	return fma((1.0 / z), x, y);
}

Julia

function code(x, y, z)
	return fma(Float64(1.0 / z), x, y)
end

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Taylor expanded in z around 0

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

5. Applied rewrites 97.3%

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

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
7. Step-by-step derivation

8. Applied rewrites 78.6%

   \[\leadsto \mathsf{fma}\left(\frac{1}{z}, x, y\right) \]
9. Add Preprocessing

Alternative 9: 40.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{x}{z} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 36.0

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

5. Applied rewrites 36.0%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
6. Add Preprocessing

Developer Target 1: 93.7% accurate, 0.6× 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 2024268 
(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))