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

Percentage Accurate: 87.9% → 99.9%

Time: 7.5s

Alternatives: 9

Speedup: 1.1×

• 21.1% 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: 87.9% 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 91.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 91.2

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

4. Applied rewrites 91.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{y + x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right)} \]
6. 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. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. sub-neg N/A

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

   9. mul-1-neg N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. mul-1-neg N/A

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

   15. sub-neg N/A

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

   16. lower--.f64 99.9

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

7. Applied rewrites 99.9%

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

Alternative 2: 77.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 91.2

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

4. Applied rewrites 91.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{y + x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right)} \]
6. 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. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. sub-neg N/A

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

   9. mul-1-neg N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. mul-1-neg N/A

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

   15. sub-neg N/A

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

   16. lower--.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{y + x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right)} \]
9. 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. *-commutative N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. sub-neg N/A

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

   6. div-sub N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. mul-1-neg N/A

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

   10. lower-/.f64 N/A

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

   11. mul-1-neg N/A

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

   12. sub-neg N/A

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

   13. lower--.f64 95.5

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

10. Applied rewrites 95.5%

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

11. Taylor expanded in y around 0

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

13. Applied rewrites 72.3%

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

14. Final simplification 72.3%

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

Alternative 3: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 5.4 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{z - x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 5.4e-11)))
   (* (/ (- 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 <= 5.4e-11)) {
		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 <= 5.4e-11))
		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, 5.4e-11]], $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 5.40000000000000009e-11 < y

   1. Initial program 84.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 84.6

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - x\right)}{z}} \]
   6. 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 99.2

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

   7. Applied rewrites 99.2%

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

   if -1 < y < 5.40000000000000009e-11

   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

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right)} \]
   6. 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. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 5.4 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{z - x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (/ (* (- z x) y) z)
   (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) * y) / z;
	} 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) * y) / z);
	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] * y), $MachinePrecision] / z), $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 84.5%

      \[\frac{x + y \cdot \left(z - x\right)}{z} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 83.8

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

   5. Applied rewrites 83.8%

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

   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

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right)} \]
   6. 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. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 98.9%

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

4. Final simplification 90.4%

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

Alternative 5: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-5} \lor \neg \left(x \leq 1.62 \cdot 10^{-84}\right):\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.5e-5) (not (<= x 1.62e-84)))
   (* (- 1.0 y) (/ x z))
   (fma (/ x z) 1.0 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.5e-5) || !(x <= 1.62e-84)) {
		tmp = (1.0 - y) * (x / z);
	} else {
		tmp = fma((x / z), 1.0, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6.5e-5) || !(x <= 1.62e-84))
		tmp = Float64(Float64(1.0 - y) * Float64(x / z));
	else
		tmp = fma(Float64(x / z), 1.0, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6.5e-5], N[Not[LessEqual[x, 1.62e-84]], $MachinePrecision]], N[(N[(1.0 - y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.49999999999999943e-5 or 1.62000000000000008e-84 < x

   1. Initial program 92.3%

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. neg-mul-1 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if -6.49999999999999943e-5 < x < 1.62000000000000008e-84

   1. Initial program 89.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 89.9

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

   4. Applied rewrites 89.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right)} \]
   6. 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. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 89.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-5} \lor \neg \left(x \leq 1.62 \cdot 10^{-84}\right):\\ \;\;\;\;\left(1 - y\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.9e+15) (fma (/ x z) 1.0 y) (* (- y) (/ x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.9e+15) {
		tmp = fma((x / z), 1.0, y);
	} else {
		tmp = -y * (x / z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.9e+15)
		tmp = fma(Float64(x / z), 1.0, y);
	else
		tmp = Float64(Float64(-y) * Float64(x / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.9e+15], N[(N[(x / z), $MachinePrecision] * 1.0 + y), $MachinePrecision], N[((-y) * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9e15

   1. Initial program 91.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 91.8

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

   4. Applied rewrites 91.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y + x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right)} \]
   6. 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. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. lower--.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 86.7%

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

   if 1.9e15 < y

   1. Initial program 89.4%

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. neg-mul-1 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 69.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 1, y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+17} \lor \neg \left(y \leq 1.4 \cdot 10^{-11}\right):\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.2e+17) (not (<= y 1.4e-11))) (* 1.0 y) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.2e+17) || !(y <= 1.4e-11)) {
		tmp = 1.0 * y;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.2d+17)) .or. (.not. (y <= 1.4d-11))) then
        tmp = 1.0d0 * y
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.2e+17) || !(y <= 1.4e-11)) {
		tmp = 1.0 * y;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.2e+17) or not (y <= 1.4e-11):
		tmp = 1.0 * y
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.2e+17) || !(y <= 1.4e-11))
		tmp = Float64(1.0 * y);
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.2e+17) || ~((y <= 1.4e-11)))
		tmp = 1.0 * y;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.2e+17], N[Not[LessEqual[y, 1.4e-11]], $MachinePrecision]], N[(1.0 * y), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2e17 or 1.4e-11 < y

   1. Initial program 84.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 84.2

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

   4. Applied rewrites 84.2%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{y \cdot \left(z - x\right)}{z}} \]
   6. 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 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto 1 \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 47.0%

       \[\leadsto 1 \cdot y \]

   if -1.2e17 < y < 1.4e-11

   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 72.2

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

   5. Applied rewrites 72.2%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+17} \lor \neg \left(y \leq 1.4 \cdot 10^{-11}\right):\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.5% 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 91.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 91.2

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

4. Applied rewrites 91.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{y + x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right)} \]
6. 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. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. sub-neg N/A

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

   9. mul-1-neg N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. mul-1-neg N/A

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

   15. sub-neg N/A

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

   16. lower--.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 72.3%

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

Alternative 9: 40.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 1.0 y))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 91.2

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

4. Applied rewrites 91.2%

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

5. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{y \cdot \left(z - x\right)}{z}} \]
6. 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 71.8

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

7. Applied rewrites 71.8%

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

8. Taylor expanded in x around 0

   \[\leadsto 1 \cdot y \]
9. Step-by-step derivation

10. Applied rewrites 38.6%

    \[\leadsto 1 \cdot y \]
11. 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 2024324 
(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))