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

Percentage Accurate: 88.2% → 99.9%

Time: 6.8s

Alternatives: 8

Speedup: 1.1×

• 21.3% 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: 88.2% 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 87.2%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + y \cdot \left(z - x\right)}{z} \leq 10^{+308}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (+ x (* y (- z x))) z) 1e+308)
   (fma (/ x z) 1.0 y)
   (* (/ x z) (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x + (y * (z - x))) / z) <= 1e+308) {
		tmp = fma((x / z), 1.0, y);
	} else {
		tmp = (x / z) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x + Float64(y * Float64(z - x))) / z) <= 1e+308)
		tmp = fma(Float64(x / z), 1.0, y);
	else
		tmp = Float64(Float64(x / z) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e+308], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (*.f64 y (-.f64 z x))) z) < 1e308

   1. Initial program 91.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. Taylor expanded in y around 0

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

   8. Applied rewrites 80.3%

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

   if 1e308 < (/.f64 (+.f64 x (*.f64 y (-.f64 z x))) z)

   1. Initial program 69.4%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 62.0%

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

   10. Applied rewrites 68.1%

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

Alternative 3: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 77.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. 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 98.0%

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

   if -1 < y < 1

   1. Initial program 99.9%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 98.9%

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

Alternative 4: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

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((x / z), 1.0, y);
	}
	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(Float64(x / z), 1.0, y);
	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[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 77.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   3. 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 98.9%

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

Alternative 5: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+148} \lor \neg \left(x \leq 7.2 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{1 - y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.75e+148) (not (<= x 7.2e+55)))
   (* (/ (- 1.0 y) z) x)
   (fma (/ x z) 1.0 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.75e+148) || !(x <= 7.2e+55)) {
		tmp = ((1.0 - y) / z) * x;
	} else {
		tmp = fma((x / z), 1.0, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.75e+148) || !(x <= 7.2e+55))
		tmp = Float64(Float64(Float64(1.0 - y) / z) * x);
	else
		tmp = fma(Float64(x / z), 1.0, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.75e+148], N[Not[LessEqual[x, 7.2e+55]], $MachinePrecision]], N[(N[(N[(1.0 - y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7499999999999999e148 or 7.19999999999999975e55 < x

   1. Initial program 89.1%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 90.7

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

   5. Applied rewrites 90.7%

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

   if -1.7499999999999999e148 < x < 7.19999999999999975e55

   1. Initial program 86.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. Taylor expanded in y around 0

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

   8. Applied rewrites 85.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+148} \lor \neg \left(x \leq 7.2 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{1 - y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-54} \lor \neg \left(y \leq 7.2 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.6e-54) (not (<= y 7.2e-25))) (/ (* z y) z) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.6e-54) || !(y <= 7.2e-25)) {
		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 <= (-3.6d-54)) .or. (.not. (y <= 7.2d-25))) 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 <= -3.6e-54) || !(y <= 7.2e-25)) {
		tmp = (z * y) / z;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.6e-54) or not (y <= 7.2e-25):
		tmp = (z * y) / z
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.6e-54) || !(y <= 7.2e-25))
		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 <= -3.6e-54) || ~((y <= 7.2e-25)))
		tmp = (z * y) / z;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.6e-54], N[Not[LessEqual[y, 7.2e-25]], $MachinePrecision]], N[(N[(z * y), $MachinePrecision] / z), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.59999999999999976e-54 or 7.1999999999999998e-25 < y

   1. Initial program 79.0%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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 33.7

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

   5. Applied rewrites 33.7%

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

   if -3.59999999999999976e-54 < y < 7.1999999999999998e-25

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 77.7

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

   5. Applied rewrites 77.7%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-54} \lor \neg \left(y \leq 7.2 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{z \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.2%

   \[\frac{x + y \cdot \left(z - x\right)}{z} \]
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 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 75.1%

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

Alternative 8: 38.4% 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 87.2%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 36.3

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

5. Applied rewrites 36.3%

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

Developer Target 1: 94.1% 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 2024339 
(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))