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

Percentage Accurate: 88.6% → 100.0%

Time: 7.3s

Alternatives: 8

Speedup: 1.1×

• 20.6% 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: 100.0% 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 84.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. div-add-rev N/A

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

   4. +-commutative N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. fp-cancel-sign-sub-inv N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

   14. lower--.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 71.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - x\right)}{z}} \]
   4. 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 98.1

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

   5. Applied rewrites 98.1%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 94.1

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.9%

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

4. Final simplification 98.5%

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

Alternative 3: 81.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-34} \lor \neg \left(z \leq 2.35 \cdot 10^{+71}\right):\\ \;\;\;\;\mathsf{fma}\left(1, y, \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.8e-34) (not (<= z 2.35e+71)))
   (fma 1.0 y (/ x z))
   (* (/ (- 1.0 y) z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.8e-34) || !(z <= 2.35e+71)) {
		tmp = fma(1.0, y, (x / z));
	} else {
		tmp = ((1.0 - y) / z) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.8e-34) || !(z <= 2.35e+71))
		tmp = fma(1.0, y, Float64(x / z));
	else
		tmp = Float64(Float64(Float64(1.0 - y) / z) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.8e-34], N[Not[LessEqual[z, 2.35e+71]], $MachinePrecision]], N[(1.0 * y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8000000000000001e-34 or 2.3499999999999998e71 < z

   1. Initial program 67.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 85.9%

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

   if -3.8000000000000001e-34 < z < 2.3499999999999998e71

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-add-rev N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 85.6

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

   5. Applied rewrites 85.6%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-34} \lor \neg \left(z \leq 2.35 \cdot 10^{+71}\right):\\ \;\;\;\;\mathsf{fma}\left(1, y, \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 72.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - x\right)}{z}} \]
   4. 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 98.7

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

   5. Applied rewrites 98.7%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 94.1

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.9%

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

   if 1 < y

   1. Initial program 70.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 97.6%

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

Alternative 5: 51.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+65} \lor \neg \left(y \leq 2.4 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.5e+65) (not (<= y 2.4e-10))) (/ (* z y) z) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.5e+65) || !(y <= 2.4e-10)) {
		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 <= (-6.5d+65)) .or. (.not. (y <= 2.4d-10))) 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 <= -6.5e+65) || !(y <= 2.4e-10)) {
		tmp = (z * y) / z;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6.5e+65) or not (y <= 2.4e-10):
		tmp = (z * y) / z
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.5e+65) || !(y <= 2.4e-10))
		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 ((y <= -6.5e+65) || ~((y <= 2.4e-10)))
		tmp = (z * y) / z;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.5e+65], N[Not[LessEqual[y, 2.4e-10]], $MachinePrecision]], N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5000000000000003e65 or 2.4e-10 < y

   1. Initial program 70.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 26.3

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

   5. Applied rewrites 26.3%

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

   if -6.5000000000000003e65 < y < 2.4e-10

   1. Initial program 98.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 74.9

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

   5. Applied rewrites 74.9%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+65} \lor \neg \left(y \leq 2.4 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(1, y, \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.2e+122) (fma 1.0 y (/ x z)) (* (/ x z) (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.2e+122) {
		tmp = fma(1.0, y, (x / z));
	} else {
		tmp = (x / z) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.2e+122)
		tmp = fma(1.0, y, Float64(x / z));
	else
		tmp = Float64(Float64(x / z) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.2e+122], N[(1.0 * y + N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.20000000000000032e122

   1. Initial program 87.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 81.0%

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

   if 4.20000000000000032e122 < y

   1. Initial program 69.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. div-add-rev N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 64.3%

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

   10. Applied rewrites 67.3%

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

Alternative 7: 78.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 96.4

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

4. Applied rewrites 96.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 72.9%

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

Alternative 8: 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 84.9%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 40.8

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

5. Applied rewrites 40.8%

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

Developer Target 1: 94.3% 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 2024337 
(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))