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

Percentage Accurate: 99.9% → 100.0%

Time: 6.1s

Alternatives: 8

Speedup: 1.1×

• 20.3% 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: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -5 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. 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 98.2

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

      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 98.2

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

   6. Applied rewrites 98.2%

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

   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. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 98.0

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

   9. Applied rewrites 98.0%

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

   10. Taylor expanded in x around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-lft-in N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. mul-1-neg N/A

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

       5. remove-double-neg N/A

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

       6. *-commutative 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. lower-+.f64 98.0

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

   12. Applied rewrites 98.0%

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

   if -5 < x < 5

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   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 98.2%

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

4. Final simplification 98.1%

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

Alternative 3: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-9} \lor \neg \left(x \leq 3.6 \cdot 10^{-90}\right):\\ \;\;\;\;\left(y + z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.7e-9) (not (<= x 3.6e-90)))
   (* (+ y z) x)
   (fma z 5.0 (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.7e-9) || !(x <= 3.6e-90)) {
		tmp = (y + z) * x;
	} else {
		tmp = fma(z, 5.0, (x * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.7e-9) || !(x <= 3.6e-90))
		tmp = Float64(Float64(y + z) * x);
	else
		tmp = fma(z, 5.0, Float64(x * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.7e-9], N[Not[LessEqual[x, 3.6e-90]], $MachinePrecision]], N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision], N[(z * 5.0 + N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7e-9 or 3.59999999999999981e-90 < x

   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 98.5

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

      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 98.5

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

   6. Applied rewrites 98.5%

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

   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. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 90.3

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

   9. Applied rewrites 90.3%

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

   10. Taylor expanded in x around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-lft-in N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. mul-1-neg N/A

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

       5. remove-double-neg N/A

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

       6. *-commutative 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. lower-+.f64 90.3

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

   12. Applied rewrites 90.3%

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

   if -3.7e-9 < x < 3.59999999999999981e-90

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   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 74.8

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

   5. Applied rewrites 74.8%

      \[\leadsto \color{blue}{z \cdot x} + z \cdot 5 \]
   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 74.9

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

   7. Applied rewrites 74.9%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-9} \lor \neg \left(x \leq 3.6 \cdot 10^{-90}\right):\\ \;\;\;\;\left(y + z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, x \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.55 \cdot 10^{-10}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-90}:\\ \;\;\;\;5 \cdot z\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+82}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.55e-10)
   (* y x)
   (if (<= x 3.6e-90) (* 5.0 z) (if (<= x 3.1e+82) (* y x) (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.55e-10) {
		tmp = y * x;
	} else if (x <= 3.6e-90) {
		tmp = 5.0 * z;
	} else if (x <= 3.1e+82) {
		tmp = y * x;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3.55d-10)) then
        tmp = y * x
    else if (x <= 3.6d-90) then
        tmp = 5.0d0 * z
    else if (x <= 3.1d+82) then
        tmp = y * x
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.55e-10) {
		tmp = y * x;
	} else if (x <= 3.6e-90) {
		tmp = 5.0 * z;
	} else if (x <= 3.1e+82) {
		tmp = y * x;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.55e-10:
		tmp = y * x
	elif x <= 3.6e-90:
		tmp = 5.0 * z
	elif x <= 3.1e+82:
		tmp = y * x
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.55e-10)
		tmp = Float64(y * x);
	elseif (x <= 3.6e-90)
		tmp = Float64(5.0 * z);
	elseif (x <= 3.1e+82)
		tmp = Float64(y * x);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.55e-10)
		tmp = y * x;
	elseif (x <= 3.6e-90)
		tmp = 5.0 * z;
	elseif (x <= 3.1e+82)
		tmp = y * x;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.55e-10], N[(y * x), $MachinePrecision], If[LessEqual[x, 3.6e-90], N[(5.0 * z), $MachinePrecision], If[LessEqual[x, 3.1e+82], N[(y * x), $MachinePrecision], N[(x * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5500000000000001e-10 or 3.59999999999999981e-90 < x < 3.10000000000000032e82

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   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 60.8

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

   5. Applied rewrites 60.8%

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

   if -3.5500000000000001e-10 < x < 3.59999999999999981e-90

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 74.4

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

   5. Applied rewrites 74.4%

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

   if 3.10000000000000032e82 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   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 60.1

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

   5. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\left(x - -5\right) \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.0%

      \[\leadsto \frac{z}{\color{blue}{\frac{1}{x - -5}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(z + 5 \cdot \frac{z}{x}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.1%

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

   11. Taylor expanded in x around inf

       \[\leadsto x \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 60.1%

       \[\leadsto x \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-9} \lor \neg \left(x \leq 3.6 \cdot 10^{-90}\right):\\ \;\;\;\;\left(y + z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x - -5\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.7e-9) (not (<= x 3.6e-90))) (* (+ y z) x) (* (- x -5.0) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.7e-9) || !(x <= 3.6e-90)) {
		tmp = (y + z) * 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 <= (-3.7d-9)) .or. (.not. (x <= 3.6d-90))) then
        tmp = (y + z) * 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 <= -3.7e-9) || !(x <= 3.6e-90)) {
		tmp = (y + z) * x;
	} else {
		tmp = (x - -5.0) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.7e-9) or not (x <= 3.6e-90):
		tmp = (y + z) * x
	else:
		tmp = (x - -5.0) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.7e-9) || !(x <= 3.6e-90))
		tmp = Float64(Float64(y + z) * 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 <= -3.7e-9) || ~((x <= 3.6e-90)))
		tmp = (y + z) * x;
	else
		tmp = (x - -5.0) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.7e-9], N[Not[LessEqual[x, 3.6e-90]], $MachinePrecision]], N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision], N[(N[(x - -5.0), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7e-9 or 3.59999999999999981e-90 < x

   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 98.5

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

      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 98.5

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

   6. Applied rewrites 98.5%

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

   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. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-+.f64 90.3

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

   9. Applied rewrites 90.3%

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

   10. Taylor expanded in x around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-lft-in N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. mul-1-neg N/A

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

       5. remove-double-neg N/A

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

       6. *-commutative 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. lower-+.f64 90.3

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

   12. Applied rewrites 90.3%

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

   if -3.7e-9 < x < 3.59999999999999981e-90

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   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 74.8

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

   5. Applied rewrites 74.8%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-9} \lor \neg \left(x \leq 3.6 \cdot 10^{-90}\right):\\ \;\;\;\;\left(y + z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x - -5\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{-78} \lor \neg \left(z \leq 1.15 \cdot 10^{-99}\right):\\ \;\;\;\;\left(x - -5\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.55e-78) (not (<= z 1.15e-99))) (* (- x -5.0) z) (* y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.55e-78) || !(z <= 1.15e-99)) {
		tmp = (x - -5.0) * z;
	} else {
		tmp = y * x;
	}
	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 ((z <= (-2.55d-78)) .or. (.not. (z <= 1.15d-99))) then
        tmp = (x - (-5.0d0)) * z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.55e-78) || !(z <= 1.15e-99)) {
		tmp = (x - -5.0) * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.55e-78) or not (z <= 1.15e-99):
		tmp = (x - -5.0) * z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.55e-78) || !(z <= 1.15e-99))
		tmp = Float64(Float64(x - -5.0) * z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.55e-78) || ~((z <= 1.15e-99)))
		tmp = (x - -5.0) * z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.55e-78], N[Not[LessEqual[z, 1.15e-99]], $MachinePrecision]], N[(N[(x - -5.0), $MachinePrecision] * z), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.55e-78 or 1.1499999999999999e-99 < z

   1. Initial program 99.8%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   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 83.0

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

   5. Applied rewrites 83.0%

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

   if -2.55e-78 < z < 1.1499999999999999e-99

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   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 76.7

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

   5. Applied rewrites 76.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.55 \cdot 10^{-78} \lor \neg \left(z \leq 1.15 \cdot 10^{-99}\right):\\ \;\;\;\;\left(x - -5\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.245 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.245) (not (<= x 5.0))) (* x z) (* 5.0 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.245) || !(x <= 5.0)) {
		tmp = x * z;
	} 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 <= (-0.245d0)) .or. (.not. (x <= 5.0d0))) then
        tmp = x * z
    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 <= -0.245) || !(x <= 5.0)) {
		tmp = x * z;
	} else {
		tmp = 5.0 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -0.245) or not (x <= 5.0):
		tmp = x * z
	else:
		tmp = 5.0 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.245) || !(x <= 5.0))
		tmp = Float64(x * z);
	else
		tmp = Float64(5.0 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -0.245) || ~((x <= 5.0)))
		tmp = x * z;
	else
		tmp = 5.0 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.245], N[Not[LessEqual[x, 5.0]], $MachinePrecision]], N[(x * z), $MachinePrecision], N[(5.0 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.245 or 5 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   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 50.5

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

   5. Applied rewrites 50.5%

      \[\leadsto \color{blue}{\left(x - -5\right) \cdot z} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.4%

      \[\leadsto \frac{z}{\color{blue}{\frac{1}{x - -5}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(z + 5 \cdot \frac{z}{x}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.5%

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

   11. Taylor expanded in x around inf

       \[\leadsto x \cdot z \]
   12. Step-by-step derivation

   13. Applied rewrites 48.6%

       \[\leadsto x \cdot z \]

   if -0.245 < x < 5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.245 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;5 \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 27.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 61.7

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

5. Applied rewrites 61.7%

   \[\leadsto \color{blue}{\left(x - -5\right) \cdot z} \]
6. Step-by-step derivation

7. Applied rewrites 61.6%

   \[\leadsto \frac{z}{\color{blue}{\frac{1}{x - -5}}} \]

8. Taylor expanded in x around inf

   \[\leadsto x \cdot \color{blue}{\left(z + 5 \cdot \frac{z}{x}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 49.6%

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

11. Taylor expanded in x around inf

    \[\leadsto x \cdot z \]
12. Step-by-step derivation

13. Applied rewrites 23.0%

    \[\leadsto x \cdot z \]
14. Add Preprocessing

Developer Target 1: 98.0% 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 2024318 
(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)))