Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C

Percentage Accurate: 99.9% → 100.0%

Time: 4.8s

Alternatives: 8

Speedup: 1.1×

• 20.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(y + z\right) + z \cdot 5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (+ y z)) (* z 5.0)))

C

double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

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)) + (z * 5.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

Python

def code(x, y, z):
	return (x * (y + z)) + (z * 5.0)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) + Float64(z * 5.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y + z)) + (z * 5.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(y + z\right) + z \cdot 5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (+ y z)) (* z 5.0)))

C

double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

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)) + (z * 5.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

Python

def code(x, y, z):
	return (x * (y + z)) + (z * 5.0)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) + Float64(z * 5.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y + z)) + (z * 5.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, 5, \left(z + y\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma z 5.0 (* (+ z y) x)))

C

double code(double x, double y, double z) {
	return fma(z, 5.0, ((z + y) * x));
}

Julia

function code(x, y, z)
	return fma(z, 5.0, Float64(Float64(z + y) * x))
end

Wolfram

code[x_, y_, z_] := N[(z * 5.0 + N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(y + z\right) + z \cdot 5} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{z \cdot 5 + x \cdot \left(y + z\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{z \cdot 5} + x \cdot \left(y + z\right) \]

   4. lower-fma.f64 100.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, x \cdot \left(y + z\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{x \cdot \left(y + z\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right) \cdot x}\right) \]

   7. lower-*.f64 100.0

      \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right) \cdot x}\right) \]

   8. lift-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right)} \cdot x\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(z + y\right)} \cdot x\right) \]

   10. lower-+.f64 100.0

       \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(z + y\right)} \cdot x\right) \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, \left(z + y\right) \cdot x\right)} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -72000 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, z, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -72000.0) (not (<= x 5.0))) (* (+ z y) x) (fma 5.0 z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -72000.0) || !(x <= 5.0)) {
		tmp = (z + y) * x;
	} else {
		tmp = fma(5.0, z, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -72000.0) || !(x <= 5.0))
		tmp = Float64(Float64(z + y) * x);
	else
		tmp = fma(5.0, z, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -72000.0], N[Not[LessEqual[x, 5.0]], $MachinePrecision]], N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision], N[(5.0 * z + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -72000 or 5 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right) + z \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot 5 + x \cdot \left(y + z\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5} + x \cdot \left(y + z\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, x \cdot \left(y + z\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{x \cdot \left(y + z\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right)} \cdot x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(z + y\right)} \cdot x\right) \]

      10. lower-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(z + y\right)} \cdot x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, \left(z + y\right) \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5 + \left(z + y\right) \cdot x} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5} + \left(z + y\right) \cdot x \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + y\right) \cdot x + z \cdot 5} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(z + y\right) \cdot x} + z \cdot 5 \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(z + y\right)} + z \cdot 5 \]

      6. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(z + y\right)} + z \cdot 5 \]

      7. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y + z\right)} + z \cdot 5 \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(y \cdot x + z \cdot x\right)} + z \cdot 5 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot x + \color{blue}{z \cdot x}\right) + z \cdot 5 \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{y \cdot x} + z \cdot x\right) + z \cdot 5 \]

      11. associate-+l+ N/A

          \[\leadsto \color{blue}{y \cdot x + \left(z \cdot x + z \cdot 5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot x} + \left(z \cdot x + z \cdot 5\right) \]

      13. lift-*.f64 N/A

          \[\leadsto y \cdot x + \left(\color{blue}{z \cdot x} + z \cdot 5\right) \]

      14. lift-*.f64 N/A

          \[\leadsto y \cdot x + \left(z \cdot x + \color{blue}{z \cdot 5}\right) \]

      15. distribute-lft-in N/A

          \[\leadsto y \cdot x + \color{blue}{z \cdot \left(x + 5\right)} \]

      16. metadata-eval N/A

          \[\leadsto y \cdot x + z \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(-5\right)\right)}\right) \]

      17. sub-neg N/A

          \[\leadsto y \cdot x + z \cdot \color{blue}{\left(x - -5\right)} \]

      18. lift--.f64 N/A

          \[\leadsto y \cdot x + z \cdot \color{blue}{\left(x - -5\right)} \]

      19. *-commutative N/A

          \[\leadsto y \cdot x + \color{blue}{\left(x - -5\right) \cdot z} \]

      20. lift-*.f64 N/A

          \[\leadsto y \cdot x + \color{blue}{\left(x - -5\right) \cdot z} \]

      21. lower-fma.f64 95.5

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(x - -5\right) \cdot z\right)} \]

   6. Applied rewrites 95.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(x - -5\right) \cdot z\right)} \]

   7. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-lft-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 99.5

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

   9. Applied rewrites 99.5%

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

   if -72000 < x < 5

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right) + z \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot 5 + x \cdot \left(y + z\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5} + x \cdot \left(y + z\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, x \cdot \left(y + z\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{x \cdot \left(y + z\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right)} \cdot x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(z + y\right)} \cdot x\right) \]

      10. lower-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(z + y\right)} \cdot x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, \left(z + y\right) \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5 + \left(z + y\right) \cdot x} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5} + \left(z + y\right) \cdot x \]

      3. lift-*.f64 N/A

         \[\leadsto z \cdot 5 + \color{blue}{\left(z + y\right) \cdot x} \]

      4. *-commutative N/A

         \[\leadsto z \cdot 5 + \color{blue}{x \cdot \left(z + y\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto z \cdot 5 + x \cdot \color{blue}{\left(z + y\right)} \]

      6. +-commutative N/A

         \[\leadsto z \cdot 5 + x \cdot \color{blue}{\left(y + z\right)} \]

      7. +-commutative N/A

         \[\leadsto z \cdot 5 + x \cdot \color{blue}{\left(z + y\right)} \]

      8. distribute-rgt-in N/A

         \[\leadsto z \cdot 5 + \color{blue}{\left(z \cdot x + y \cdot x\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto z \cdot 5 + \left(\color{blue}{z \cdot x} + y \cdot x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto z \cdot 5 + \left(z \cdot x + \color{blue}{y \cdot x}\right) \]

      11. associate-+r+ N/A

          \[\leadsto \color{blue}{\left(z \cdot 5 + z \cdot x\right) + y \cdot x} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{z \cdot 5} + z \cdot x\right) + y \cdot x \]

      13. lift-*.f64 N/A

          \[\leadsto \left(z \cdot 5 + \color{blue}{z \cdot x}\right) + y \cdot x \]

      14. distribute-lft-in N/A

          \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} + y \cdot x \]

      15. +-commutative N/A

          \[\leadsto z \cdot \color{blue}{\left(x + 5\right)} + y \cdot x \]

      16. metadata-eval N/A

          \[\leadsto z \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(-5\right)\right)}\right) + y \cdot x \]

      17. sub-neg N/A

          \[\leadsto z \cdot \color{blue}{\left(x - -5\right)} + y \cdot x \]

      18. lift--.f64 N/A

          \[\leadsto z \cdot \color{blue}{\left(x - -5\right)} + y \cdot x \]

      19. *-commutative N/A

          \[\leadsto \color{blue}{\left(x - -5\right) \cdot z} + y \cdot x \]

      20. lower-fma.f64 99.9

          \[\leadsto \color{blue}{\mathsf{fma}\left(x - -5, z, y \cdot x\right)} \]

      21. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - -5, z, \color{blue}{y \cdot x}\right) \]

      22. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x - -5, z, \color{blue}{x \cdot y}\right) \]

      23. lower-*.f64 99.9

          \[\leadsto \mathsf{fma}\left(x - -5, z, \color{blue}{x \cdot y}\right) \]

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - -5, z, x \cdot y\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{5}, z, x \cdot y\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 97.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{5}, z, x \cdot y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -72000 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5, z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -44 \lor \neg \left(x \leq 6.5 \cdot 10^{-9}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -44.0) (not (<= x 6.5e-9))) (* (+ z y) x) (fma z 5.0 (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -44.0) || !(x <= 6.5e-9)) {
		tmp = (z + y) * x;
	} else {
		tmp = fma(z, 5.0, (x * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -44.0) || !(x <= 6.5e-9))
		tmp = Float64(Float64(z + y) * x);
	else
		tmp = fma(z, 5.0, Float64(x * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -44.0], N[Not[LessEqual[x, 6.5e-9]], $MachinePrecision]], N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision], N[(z * 5.0 + N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -44 or 6.5000000000000003e-9 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right) + z \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot 5 + x \cdot \left(y + z\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5} + x \cdot \left(y + z\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, x \cdot \left(y + z\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{x \cdot \left(y + z\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right)} \cdot x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(z + y\right)} \cdot x\right) \]

      10. lower-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(z + y\right)} \cdot x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, \left(z + y\right) \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5 + \left(z + y\right) \cdot x} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5} + \left(z + y\right) \cdot x \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + y\right) \cdot x + z \cdot 5} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(z + y\right) \cdot x} + z \cdot 5 \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(z + y\right)} + z \cdot 5 \]

      6. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(z + y\right)} + z \cdot 5 \]

      7. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y + z\right)} + z \cdot 5 \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(y \cdot x + z \cdot x\right)} + z \cdot 5 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot x + \color{blue}{z \cdot x}\right) + z \cdot 5 \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{y \cdot x} + z \cdot x\right) + z \cdot 5 \]

      11. associate-+l+ N/A

          \[\leadsto \color{blue}{y \cdot x + \left(z \cdot x + z \cdot 5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot x} + \left(z \cdot x + z \cdot 5\right) \]

      13. lift-*.f64 N/A

          \[\leadsto y \cdot x + \left(\color{blue}{z \cdot x} + z \cdot 5\right) \]

      14. lift-*.f64 N/A

          \[\leadsto y \cdot x + \left(z \cdot x + \color{blue}{z \cdot 5}\right) \]

      15. distribute-lft-in N/A

          \[\leadsto y \cdot x + \color{blue}{z \cdot \left(x + 5\right)} \]

      16. metadata-eval N/A

          \[\leadsto y \cdot x + z \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(-5\right)\right)}\right) \]

      17. sub-neg N/A

          \[\leadsto y \cdot x + z \cdot \color{blue}{\left(x - -5\right)} \]

      18. lift--.f64 N/A

          \[\leadsto y \cdot x + z \cdot \color{blue}{\left(x - -5\right)} \]

      19. *-commutative N/A

          \[\leadsto y \cdot x + \color{blue}{\left(x - -5\right) \cdot z} \]

      20. lift-*.f64 N/A

          \[\leadsto y \cdot x + \color{blue}{\left(x - -5\right) \cdot z} \]

      21. lower-fma.f64 95.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(x - -5\right) \cdot z\right)} \]

   6. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(x - -5\right) \cdot z\right)} \]

   7. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-lft-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 98.9

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

   9. Applied rewrites 98.9%

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

   if -44 < x < 6.5000000000000003e-9

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot z} + z \cdot 5 \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot x} + z \cdot 5 \]

      2. lower-*.f64 75.6

         \[\leadsto \color{blue}{z \cdot x} + z \cdot 5 \]

   5. Applied rewrites 75.6%

      \[\leadsto \color{blue}{z \cdot x} + z \cdot 5 \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{z \cdot x + z \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot 5 + z \cdot x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5} + z \cdot x \]

      4. lower-fma.f64 75.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, z \cdot x\right)} \]

   7. Applied rewrites 75.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, x \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -44 \lor \neg \left(x \leq 6.5 \cdot 10^{-9}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -44 \lor \neg \left(x \leq 6.5 \cdot 10^{-9}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x - -5\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -44.0) (not (<= x 6.5e-9))) (* (+ z y) x) (* (- x -5.0) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -44.0) || !(x <= 6.5e-9)) {
		tmp = (z + y) * x;
	} else {
		tmp = (x - -5.0) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-44.0d0)) .or. (.not. (x <= 6.5d-9))) then
        tmp = (z + y) * x
    else
        tmp = (x - (-5.0d0)) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -44.0) || !(x <= 6.5e-9)) {
		tmp = (z + y) * x;
	} else {
		tmp = (x - -5.0) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -44.0) or not (x <= 6.5e-9):
		tmp = (z + y) * x
	else:
		tmp = (x - -5.0) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -44.0) || !(x <= 6.5e-9))
		tmp = Float64(Float64(z + y) * x);
	else
		tmp = Float64(Float64(x - -5.0) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -44.0) || ~((x <= 6.5e-9)))
		tmp = (z + y) * x;
	else
		tmp = (x - -5.0) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -44.0], N[Not[LessEqual[x, 6.5e-9]], $MachinePrecision]], N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x - -5.0), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -44 or 6.5000000000000003e-9 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right) + z \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot 5 + x \cdot \left(y + z\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5} + x \cdot \left(y + z\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, x \cdot \left(y + z\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{x \cdot \left(y + z\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right)} \cdot x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(z + y\right)} \cdot x\right) \]

      10. lower-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(z + y\right)} \cdot x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, \left(z + y\right) \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5 + \left(z + y\right) \cdot x} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5} + \left(z + y\right) \cdot x \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + y\right) \cdot x + z \cdot 5} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(z + y\right) \cdot x} + z \cdot 5 \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(z + y\right)} + z \cdot 5 \]

      6. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(z + y\right)} + z \cdot 5 \]

      7. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y + z\right)} + z \cdot 5 \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(y \cdot x + z \cdot x\right)} + z \cdot 5 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot x + \color{blue}{z \cdot x}\right) + z \cdot 5 \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{y \cdot x} + z \cdot x\right) + z \cdot 5 \]

      11. associate-+l+ N/A

          \[\leadsto \color{blue}{y \cdot x + \left(z \cdot x + z \cdot 5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot x} + \left(z \cdot x + z \cdot 5\right) \]

      13. lift-*.f64 N/A

          \[\leadsto y \cdot x + \left(\color{blue}{z \cdot x} + z \cdot 5\right) \]

      14. lift-*.f64 N/A

          \[\leadsto y \cdot x + \left(z \cdot x + \color{blue}{z \cdot 5}\right) \]

      15. distribute-lft-in N/A

          \[\leadsto y \cdot x + \color{blue}{z \cdot \left(x + 5\right)} \]

      16. metadata-eval N/A

          \[\leadsto y \cdot x + z \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(-5\right)\right)}\right) \]

      17. sub-neg N/A

          \[\leadsto y \cdot x + z \cdot \color{blue}{\left(x - -5\right)} \]

      18. lift--.f64 N/A

          \[\leadsto y \cdot x + z \cdot \color{blue}{\left(x - -5\right)} \]

      19. *-commutative N/A

          \[\leadsto y \cdot x + \color{blue}{\left(x - -5\right) \cdot z} \]

      20. lift-*.f64 N/A

          \[\leadsto y \cdot x + \color{blue}{\left(x - -5\right) \cdot z} \]

      21. lower-fma.f64 95.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(x - -5\right) \cdot z\right)} \]

   6. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(x - -5\right) \cdot z\right)} \]

   7. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-lft-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 98.9

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

   9. Applied rewrites 98.9%

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

   if -44 < x < 6.5000000000000003e-9

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{5 \cdot z + x \cdot z} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x + 5\right)} \cdot z \]

      5. metadata-eval N/A

         \[\leadsto \left(x + \color{blue}{\left(\mathsf{neg}\left(-5\right)\right)}\right) \cdot z \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(x - -5\right)} \cdot z \]

      7. lower--.f64 75.5

         \[\leadsto \color{blue}{\left(x - -5\right)} \cdot z \]

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\left(x - -5\right) \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -44 \lor \neg \left(x \leq 6.5 \cdot 10^{-9}\right):\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x - -5\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+126} \lor \neg \left(y \leq 0.38\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x - -5\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.05e+126) (not (<= y 0.38))) (* y x) (* (- x -5.0) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.05e+126) || !(y <= 0.38)) {
		tmp = y * x;
	} else {
		tmp = (x - -5.0) * 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 <= (-1.05d+126)) .or. (.not. (y <= 0.38d0))) then
        tmp = y * x
    else
        tmp = (x - (-5.0d0)) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.05e+126) || !(y <= 0.38)) {
		tmp = y * x;
	} else {
		tmp = (x - -5.0) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.05e+126) or not (y <= 0.38):
		tmp = y * x
	else:
		tmp = (x - -5.0) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.05e+126) || !(y <= 0.38))
		tmp = Float64(y * x);
	else
		tmp = Float64(Float64(x - -5.0) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.05e+126) || ~((y <= 0.38)))
		tmp = y * x;
	else
		tmp = (x - -5.0) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.05e+126], N[Not[LessEqual[y, 0.38]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(N[(x - -5.0), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e126 or 0.38 < y

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if -1.05e126 < y < 0.38

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{5 \cdot z + x \cdot z} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x + 5\right)} \cdot z \]

      5. metadata-eval N/A

         \[\leadsto \left(x + \color{blue}{\left(\mathsf{neg}\left(-5\right)\right)}\right) \cdot z \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(x - -5\right)} \cdot z \]

      7. lower--.f64 80.0

         \[\leadsto \color{blue}{\left(x - -5\right)} \cdot z \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\left(x - -5\right) \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+126} \lor \neg \left(y \leq 0.38\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x - -5\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot \left(x + 5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1e+18) (fma y x (* z (+ x 5.0))) (* (+ z y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e+18) {
		tmp = fma(y, x, (z * (x + 5.0)));
	} else {
		tmp = (z + y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1e+18)
		tmp = fma(y, x, Float64(z * Float64(x + 5.0)));
	else
		tmp = Float64(Float64(z + y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1e+18], N[(y * x + N[(z * N[(x + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e18

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right) + z \cdot 5} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right)} + z \cdot 5 \]

      3. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(y + z\right)} + z \cdot 5 \]

      4. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(y \cdot x + z \cdot x\right)} + z \cdot 5 \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{y \cdot x + \left(z \cdot x + z \cdot 5\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, z \cdot x + z \cdot 5\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, z \cdot x + \color{blue}{z \cdot 5}\right) \]

      8. distribute-lft-out N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{z \cdot \left(x + 5\right)}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{z \cdot \left(x + 5\right)}\right) \]

      10. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(y, x, z \cdot \color{blue}{\left(x + 5\right)}\right) \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, z \cdot \left(x + 5\right)\right)} \]

   if 1e18 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(y + z\right) + z \cdot 5} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot 5 + x \cdot \left(y + z\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5} + x \cdot \left(y + z\right) \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, x \cdot \left(y + z\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{x \cdot \left(y + z\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right) \cdot x}\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right) \cdot x}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(y + z\right)} \cdot x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(z + y\right)} \cdot x\right) \]

      10. lower-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(z, 5, \color{blue}{\left(z + y\right)} \cdot x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5, \left(z + y\right) \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5 + \left(z + y\right) \cdot x} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot 5} + \left(z + y\right) \cdot x \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + y\right) \cdot x + z \cdot 5} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(z + y\right) \cdot x} + z \cdot 5 \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(z + y\right)} + z \cdot 5 \]

      6. lift-+.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(z + y\right)} + z \cdot 5 \]

      7. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(y + z\right)} + z \cdot 5 \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(y \cdot x + z \cdot x\right)} + z \cdot 5 \]

      9. lift-*.f64 N/A

         \[\leadsto \left(y \cdot x + \color{blue}{z \cdot x}\right) + z \cdot 5 \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\color{blue}{y \cdot x} + z \cdot x\right) + z \cdot 5 \]

      11. associate-+l+ N/A

          \[\leadsto \color{blue}{y \cdot x + \left(z \cdot x + z \cdot 5\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \color{blue}{y \cdot x} + \left(z \cdot x + z \cdot 5\right) \]

      13. lift-*.f64 N/A

          \[\leadsto y \cdot x + \left(\color{blue}{z \cdot x} + z \cdot 5\right) \]

      14. lift-*.f64 N/A

          \[\leadsto y \cdot x + \left(z \cdot x + \color{blue}{z \cdot 5}\right) \]

      15. distribute-lft-in N/A

          \[\leadsto y \cdot x + \color{blue}{z \cdot \left(x + 5\right)} \]

      16. metadata-eval N/A

          \[\leadsto y \cdot x + z \cdot \left(x + \color{blue}{\left(\mathsf{neg}\left(-5\right)\right)}\right) \]

      17. sub-neg N/A

          \[\leadsto y \cdot x + z \cdot \color{blue}{\left(x - -5\right)} \]

      18. lift--.f64 N/A

          \[\leadsto y \cdot x + z \cdot \color{blue}{\left(x - -5\right)} \]

      19. *-commutative N/A

          \[\leadsto y \cdot x + \color{blue}{\left(x - -5\right) \cdot z} \]

      20. lift-*.f64 N/A

          \[\leadsto y \cdot x + \color{blue}{\left(x - -5\right) \cdot z} \]

      21. lower-fma.f64 91.7

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(x - -5\right) \cdot z\right)} \]

   6. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, \left(x - -5\right) \cdot z\right)} \]

   7. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-lft-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 100.0

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

   9. Applied rewrites 100.0%

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

Alternative 7: 61.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.46 \lor \neg \left(x \leq 6.5 \cdot 10^{-12}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.46) (not (<= x 6.5e-12))) (* y x) (* 5.0 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.46) || !(x <= 6.5e-12)) {
		tmp = y * x;
	} else {
		tmp = 5.0 * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-0.46d0)) .or. (.not. (x <= 6.5d-12))) then
        tmp = y * x
    else
        tmp = 5.0d0 * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.46) || !(x <= 6.5e-12)) {
		tmp = y * x;
	} else {
		tmp = 5.0 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -0.46) or not (x <= 6.5e-12):
		tmp = y * x
	else:
		tmp = 5.0 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.46) || !(x <= 6.5e-12))
		tmp = Float64(y * x);
	else
		tmp = Float64(5.0 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -0.46) || ~((x <= 6.5e-12)))
		tmp = y * x;
	else
		tmp = 5.0 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.46], N[Not[LessEqual[x, 6.5e-12]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(5.0 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.46000000000000002 or 6.5000000000000002e-12 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

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

   if -0.46000000000000002 < x < 6.5000000000000002e-12

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 74.0

         \[\leadsto \color{blue}{5 \cdot z} \]

   5. Applied rewrites 74.0%

      \[\leadsto \color{blue}{5 \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.46 \lor \neg \left(x \leq 6.5 \cdot 10^{-12}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ 5 \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 5.0 z))

C

double code(double x, double y, double z) {
	return 5.0 * 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 = 5.0d0 * z
end function

Java

public static double code(double x, double y, double z) {
	return 5.0 * z;
}

Python

def code(x, y, z):
	return 5.0 * z

Julia

function code(x, y, z)
	return Float64(5.0 * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 5.0 * z;
end

Wolfram

code[x_, y_, z_] := N[(5.0 * z), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{5 \cdot z} \]
4. Step-by-step derivation

   1. lower-*.f64 35.3

      \[\leadsto \color{blue}{5 \cdot z} \]

5. Applied rewrites 35.3%

   \[\leadsto \color{blue}{5 \cdot z} \]
6. Add Preprocessing

Developer Target 1: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + 5\right) \cdot z + x \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* (+ x 5.0) z) (* x y)))

C

double code(double x, double y, double z) {
	return ((x + 5.0) * z) + (x * 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 = ((x + 5.0d0) * z) + (x * y)
end function

Java

public static double code(double x, double y, double z) {
	return ((x + 5.0) * z) + (x * y);
}

Python

def code(x, y, z):
	return ((x + 5.0) * z) + (x * y)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x + 5.0) * z) + Float64(x * y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x + 5.0) * z) + (x * y);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x + 5.0), $MachinePrecision] * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024313 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (+ x 5) z) (* x y)))

  (+ (* x (+ y z)) (* z 5.0)))