ab-angle->ABCF D

Percentage Accurate: 82.1% → 99.7%

Time: 5.0s

Alternatives: 7

Speedup: 1.0×

• 28.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ -\left(\left(a \cdot a\right) \cdot b\right) \cdot b \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* (* (* a a) b) b)))

C

double code(double a, double b) {
	return -(((a * a) * b) * b);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -(((a * a) * b) * b)
end function

Java

public static double code(double a, double b) {
	return -(((a * a) * b) * b);
}

Python

def code(a, b):
	return -(((a * a) * b) * b)

Julia

function code(a, b)
	return Float64(-Float64(Float64(Float64(a * a) * b) * b))
end

MATLAB

function tmp = code(a, b)
	tmp = -(((a * a) * b) * b);
end

Wolfram

code[a_, b_] := (-N[(N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision])

Initial Program: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -\left(\left(a \cdot a\right) \cdot b\right) \cdot b \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* (* (* a a) b) b)))

C

double code(double a, double b) {
	return -(((a * a) * b) * b);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -(((a * a) * b) * b)
end function

Java

public static double code(double a, double b) {
	return -(((a * a) * b) * b);
}

Python

def code(a, b):
	return -(((a * a) * b) * b)

Julia

function code(a, b)
	return Float64(-Float64(Float64(Float64(a * a) * b) * b))
end

MATLAB

function tmp = code(a, b)
	tmp = -(((a * a) * b) * b);
end

Wolfram

code[a_, b_] := (-N[(N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision])

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ [a_m, b_m] = \mathsf{sort}([a_m, b_m])\\ \\ \frac{a\_m \cdot b\_m}{\frac{\frac{-1}{b\_m}}{a\_m}} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
(FPCore (a_m b_m) :precision binary64 (/ (* a_m b_m) (/ (/ -1.0 b_m) a_m)))

C

b_m = fabs(b);
a_m = fabs(a);
assert(a_m < b_m);
double code(double a_m, double b_m) {
	return (a_m * b_m) / ((-1.0 / b_m) / a_m);
}

Fortran

b_m = abs(b)
a_m = abs(a)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
real(8) function code(a_m, b_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    code = (a_m * b_m) / (((-1.0d0) / b_m) / a_m)
end function

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
assert a_m < b_m;
public static double code(double a_m, double b_m) {
	return (a_m * b_m) / ((-1.0 / b_m) / a_m);
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
[a_m, b_m] = sort([a_m, b_m])
def code(a_m, b_m):
	return (a_m * b_m) / ((-1.0 / b_m) / a_m)

Julia

b_m = abs(b)
a_m = abs(a)
a_m, b_m = sort([a_m, b_m])
function code(a_m, b_m)
	return Float64(Float64(a_m * b_m) / Float64(Float64(-1.0 / b_m) / a_m))
end

MATLAB

b_m = abs(b);
a_m = abs(a);
a_m, b_m = num2cell(sort([a_m, b_m])){:}
function tmp = code(a_m, b_m)
	tmp = (a_m * b_m) / ((-1.0 / b_m) / a_m);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
code[a$95$m_, b$95$m_] := N[(N[(a$95$m * b$95$m), $MachinePrecision] / N[(N[(-1.0 / b$95$m), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.9%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -\color{blue}{\left(\left(a \cdot a\right) \cdot b\right) \cdot b} \]

   2. lift-*.f64 N/A

      \[\leadsto -\color{blue}{\left(\left(a \cdot a\right) \cdot b\right)} \cdot b \]

   3. associate-*l* N/A

      \[\leadsto -\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto -\color{blue}{\left(a \cdot a\right)} \cdot \left(b \cdot b\right) \]

   5. associate-*l* N/A

      \[\leadsto -\color{blue}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto -\color{blue}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a} \]

   7. unpow1 N/A

      \[\leadsto -\left(a \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{{a}^{1}} \]

   8. metadata-eval N/A

      \[\leadsto -\left(a \cdot \left(b \cdot b\right)\right) \cdot {a}^{\color{blue}{\left(\frac{2}{2}\right)}} \]

   9. sqr-pow N/A

      \[\leadsto -\left(a \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left({a}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)} \]

   10. associate-*r* N/A

       \[\leadsto -\color{blue}{\left(\left(a \cdot \left(b \cdot b\right)\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}} \]

   11. lower-*.f64 N/A

       \[\leadsto -\color{blue}{\left(\left(a \cdot \left(b \cdot b\right)\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}} \]

   12. lower-*.f64 N/A

       \[\leadsto -\color{blue}{\left(\left(a \cdot \left(b \cdot b\right)\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)} \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

   13. associate-*r* N/A

       \[\leadsto -\left(\color{blue}{\left(\left(a \cdot b\right) \cdot b\right)} \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto -\left(\color{blue}{\left(\left(a \cdot b\right) \cdot b\right)} \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

   15. *-commutative N/A

       \[\leadsto -\left(\left(\color{blue}{\left(b \cdot a\right)} \cdot b\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

   16. lower-*.f64 N/A

       \[\leadsto -\left(\left(\color{blue}{\left(b \cdot a\right)} \cdot b\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

   17. metadata-eval N/A

       \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot {a}^{\left(\frac{\color{blue}{1}}{2}\right)}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

   18. metadata-eval N/A

       \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot {a}^{\color{blue}{\frac{1}{2}}}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

   19. unpow1/2 N/A

       \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \color{blue}{\sqrt{a}}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

   20. lower-sqrt.f64 N/A

       \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \color{blue}{\sqrt{a}}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

   21. metadata-eval N/A

       \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right) \cdot {a}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

   22. metadata-eval N/A

       \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right) \cdot {a}^{\color{blue}{\frac{1}{2}}} \]

   23. unpow1/2 N/A

       \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right) \cdot \color{blue}{\sqrt{a}} \]

   24. lower-sqrt.f64 43.6

       \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right) \cdot \color{blue}{\sqrt{a}} \]

4. Applied rewrites 43.6%

   \[\leadsto -\color{blue}{\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right) \cdot \sqrt{a}} \]
5. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right) \cdot \sqrt{a}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right) \cdot \sqrt{a}}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right)} \cdot \sqrt{a}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\left(b \cdot a\right) \cdot b\right)} \cdot \sqrt{a}\right) \cdot \sqrt{a}\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(b \cdot a\right) \cdot \left(b \cdot \sqrt{a}\right)\right)} \cdot \sqrt{a}\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(b \cdot a\right) \cdot \left(\left(b \cdot \sqrt{a}\right) \cdot \sqrt{a}\right)}\right) \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \left(\left(b \cdot \sqrt{a}\right) \cdot \sqrt{a}\right)} \]

   8. associate-*r* N/A

      \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \color{blue}{\left(b \cdot \left(\sqrt{a} \cdot \sqrt{a}\right)\right)} \]

   9. lift-sqrt.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \left(b \cdot \left(\color{blue}{\sqrt{a}} \cdot \sqrt{a}\right)\right) \]

   10. lift-sqrt.f64 N/A

       \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \left(b \cdot \left(\sqrt{a} \cdot \color{blue}{\sqrt{a}}\right)\right) \]

   11. rem-square-sqrt N/A

       \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \left(b \cdot \color{blue}{a}\right) \]

   12. *-commutative N/A

       \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot b\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot b\right)} \]

   14. rem-square-sqrt N/A

       \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \color{blue}{\left(\sqrt{a \cdot b} \cdot \sqrt{a \cdot b}\right)} \]

   15. sqrt-unprod N/A

       \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \color{blue}{\sqrt{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}} \]

   16. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \sqrt{\color{blue}{\left(a \cdot b\right)} \cdot \left(a \cdot b\right)} \]

   17. *-commutative N/A

       \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \sqrt{\color{blue}{\left(b \cdot a\right)} \cdot \left(a \cdot b\right)} \]

   18. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \sqrt{\color{blue}{\left(b \cdot a\right)} \cdot \left(a \cdot b\right)} \]

   19. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \sqrt{\left(b \cdot a\right) \cdot \color{blue}{\left(a \cdot b\right)}} \]

   20. associate-*l* N/A

       \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \sqrt{\color{blue}{\left(\left(b \cdot a\right) \cdot a\right) \cdot b}} \]

   21. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \sqrt{\color{blue}{\left(\left(b \cdot a\right) \cdot a\right)} \cdot b} \]

   22. *-commutative N/A

       \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \sqrt{\color{blue}{b \cdot \left(\left(b \cdot a\right) \cdot a\right)}} \]

6. Applied rewrites 47.5%

   \[\leadsto \color{blue}{\left(\left(\left(-a\right) \cdot b\right) \cdot \sqrt{b}\right) \cdot \left(\sqrt{b} \cdot a\right)} \]

8. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{b \cdot a}{\frac{\frac{-1}{b}}{a}}} \]

9. Final simplification 99.8%

   \[\leadsto \frac{a \cdot b}{\frac{\frac{-1}{b}}{a}} \]
10. Add Preprocessing

Alternative 2: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ [a_m, b_m] = \mathsf{sort}([a_m, b_m])\\ \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 6 \cdot 10^{-199}:\\ \;\;\;\;\left(\left(\left(-a\_m\right) \cdot b\_m\right) \cdot b\_m\right) \cdot a\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a\_m \cdot b\_m\right) \cdot a\_m}{\frac{-1}{b\_m}}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
(FPCore (a_m b_m)
 :precision binary64
 (if (<= a_m 6e-199)
   (* (* (* (- a_m) b_m) b_m) a_m)
   (/ (* (* a_m b_m) a_m) (/ -1.0 b_m))))

C

b_m = fabs(b);
a_m = fabs(a);
assert(a_m < b_m);
double code(double a_m, double b_m) {
	double tmp;
	if (a_m <= 6e-199) {
		tmp = ((-a_m * b_m) * b_m) * a_m;
	} else {
		tmp = ((a_m * b_m) * a_m) / (-1.0 / b_m);
	}
	return tmp;
}

Fortran

b_m = abs(b)
a_m = abs(a)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
real(8) function code(a_m, b_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8) :: tmp
    if (a_m <= 6d-199) then
        tmp = ((-a_m * b_m) * b_m) * a_m
    else
        tmp = ((a_m * b_m) * a_m) / ((-1.0d0) / b_m)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
assert a_m < b_m;
public static double code(double a_m, double b_m) {
	double tmp;
	if (a_m <= 6e-199) {
		tmp = ((-a_m * b_m) * b_m) * a_m;
	} else {
		tmp = ((a_m * b_m) * a_m) / (-1.0 / b_m);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
[a_m, b_m] = sort([a_m, b_m])
def code(a_m, b_m):
	tmp = 0
	if a_m <= 6e-199:
		tmp = ((-a_m * b_m) * b_m) * a_m
	else:
		tmp = ((a_m * b_m) * a_m) / (-1.0 / b_m)
	return tmp

Julia

b_m = abs(b)
a_m = abs(a)
a_m, b_m = sort([a_m, b_m])
function code(a_m, b_m)
	tmp = 0.0
	if (a_m <= 6e-199)
		tmp = Float64(Float64(Float64(Float64(-a_m) * b_m) * b_m) * a_m);
	else
		tmp = Float64(Float64(Float64(a_m * b_m) * a_m) / Float64(-1.0 / b_m));
	end
	return tmp
end

MATLAB

b_m = abs(b);
a_m = abs(a);
a_m, b_m = num2cell(sort([a_m, b_m])){:}
function tmp_2 = code(a_m, b_m)
	tmp = 0.0;
	if (a_m <= 6e-199)
		tmp = ((-a_m * b_m) * b_m) * a_m;
	else
		tmp = ((a_m * b_m) * a_m) / (-1.0 / b_m);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
code[a$95$m_, b$95$m_] := If[LessEqual[a$95$m, 6e-199], N[(N[(N[((-a$95$m) * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * a$95$m), $MachinePrecision], N[(N[(N[(a$95$m * b$95$m), $MachinePrecision] * a$95$m), $MachinePrecision] / N[(-1.0 / b$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.99999999999999966e-199

   1. Initial program 82.7%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(a \cdot a\right) \cdot b\right) \cdot b}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(a \cdot a\right) \cdot b\right)} \cdot b\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(a \cdot a\right)} \cdot \left(b \cdot b\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot \left(a \cdot \left(b \cdot b\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(b \cdot b\right)\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(b \cdot b\right)\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(a \cdot b\right) \cdot b\right)} \cdot \left(\mathsf{neg}\left(a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(a \cdot b\right) \cdot b\right)} \cdot \left(\mathsf{neg}\left(a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(b \cdot a\right)} \cdot b\right) \cdot \left(\mathsf{neg}\left(a\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(b \cdot a\right)} \cdot b\right) \cdot \left(\mathsf{neg}\left(a\right)\right) \]

      14. lower-neg.f64 94.7

          \[\leadsto \left(\left(b \cdot a\right) \cdot b\right) \cdot \color{blue}{\left(-a\right)} \]

   4. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)} \]

   if 5.99999999999999966e-199 < a

   1. Initial program 88.7%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -\color{blue}{\left(\left(a \cdot a\right) \cdot b\right) \cdot b} \]

      2. lift-*.f64 N/A

         \[\leadsto -\color{blue}{\left(\left(a \cdot a\right) \cdot b\right)} \cdot b \]

      3. associate-*l* N/A

         \[\leadsto -\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto -\color{blue}{\left(a \cdot a\right)} \cdot \left(b \cdot b\right) \]

      5. associate-*l* N/A

         \[\leadsto -\color{blue}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto -\color{blue}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a} \]

      7. unpow1 N/A

         \[\leadsto -\left(a \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{{a}^{1}} \]

      8. metadata-eval N/A

         \[\leadsto -\left(a \cdot \left(b \cdot b\right)\right) \cdot {a}^{\color{blue}{\left(\frac{2}{2}\right)}} \]

      9. sqr-pow N/A

         \[\leadsto -\left(a \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left({a}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)} \]

      10. associate-*r* N/A

          \[\leadsto -\color{blue}{\left(\left(a \cdot \left(b \cdot b\right)\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto -\color{blue}{\left(\left(a \cdot \left(b \cdot b\right)\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}} \]

      12. lower-*.f64 N/A

          \[\leadsto -\color{blue}{\left(\left(a \cdot \left(b \cdot b\right)\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)} \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

      13. associate-*r* N/A

          \[\leadsto -\left(\color{blue}{\left(\left(a \cdot b\right) \cdot b\right)} \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto -\left(\color{blue}{\left(\left(a \cdot b\right) \cdot b\right)} \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

      15. *-commutative N/A

          \[\leadsto -\left(\left(\color{blue}{\left(b \cdot a\right)} \cdot b\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto -\left(\left(\color{blue}{\left(b \cdot a\right)} \cdot b\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

      17. metadata-eval N/A

          \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot {a}^{\left(\frac{\color{blue}{1}}{2}\right)}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

      18. metadata-eval N/A

          \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot {a}^{\color{blue}{\frac{1}{2}}}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

      19. unpow1/2 N/A

          \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \color{blue}{\sqrt{a}}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

      20. lower-sqrt.f64 N/A

          \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \color{blue}{\sqrt{a}}\right) \cdot {a}^{\left(\frac{\frac{2}{2}}{2}\right)} \]

      21. metadata-eval N/A

          \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right) \cdot {a}^{\left(\frac{\color{blue}{1}}{2}\right)} \]

      22. metadata-eval N/A

          \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right) \cdot {a}^{\color{blue}{\frac{1}{2}}} \]

      23. unpow1/2 N/A

          \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right) \cdot \color{blue}{\sqrt{a}} \]

      24. lower-sqrt.f64 93.2

          \[\leadsto -\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right) \cdot \color{blue}{\sqrt{a}} \]

   4. Applied rewrites 93.2%

      \[\leadsto -\color{blue}{\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right) \cdot \sqrt{a}} \]
   5. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right) \cdot \sqrt{a}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right) \cdot \sqrt{a}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \sqrt{a}\right)} \cdot \sqrt{a}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(\color{blue}{\left(\left(b \cdot a\right) \cdot b\right)} \cdot \sqrt{a}\right) \cdot \sqrt{a}\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(b \cdot a\right) \cdot \left(b \cdot \sqrt{a}\right)\right)} \cdot \sqrt{a}\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(b \cdot a\right) \cdot \left(\left(b \cdot \sqrt{a}\right) \cdot \sqrt{a}\right)}\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \left(\left(b \cdot \sqrt{a}\right) \cdot \sqrt{a}\right)} \]

      8. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \color{blue}{\left(b \cdot \left(\sqrt{a} \cdot \sqrt{a}\right)\right)} \]

      9. lift-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \left(b \cdot \left(\color{blue}{\sqrt{a}} \cdot \sqrt{a}\right)\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \left(b \cdot \left(\sqrt{a} \cdot \color{blue}{\sqrt{a}}\right)\right) \]

      11. rem-square-sqrt N/A

          \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \left(b \cdot \color{blue}{a}\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot b\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot b\right)} \]

      14. rem-square-sqrt N/A

          \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \color{blue}{\left(\sqrt{a \cdot b} \cdot \sqrt{a \cdot b}\right)} \]

      15. sqrt-unprod N/A

          \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \color{blue}{\sqrt{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}} \]

      16. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \sqrt{\color{blue}{\left(a \cdot b\right)} \cdot \left(a \cdot b\right)} \]

      17. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \sqrt{\color{blue}{\left(b \cdot a\right)} \cdot \left(a \cdot b\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \sqrt{\color{blue}{\left(b \cdot a\right)} \cdot \left(a \cdot b\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \sqrt{\left(b \cdot a\right) \cdot \color{blue}{\left(a \cdot b\right)}} \]

      20. associate-*l* N/A

          \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \sqrt{\color{blue}{\left(\left(b \cdot a\right) \cdot a\right) \cdot b}} \]

      21. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \sqrt{\color{blue}{\left(\left(b \cdot a\right) \cdot a\right)} \cdot b} \]

      22. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(b \cdot a\right)\right) \cdot \sqrt{\color{blue}{b \cdot \left(\left(b \cdot a\right) \cdot a\right)}} \]

   6. Applied rewrites 47.8%

      \[\leadsto \color{blue}{\left(\left(\left(-a\right) \cdot b\right) \cdot \sqrt{b}\right) \cdot \left(\sqrt{b} \cdot a\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(-a\right) \cdot b\right) \cdot \sqrt{b}\right) \cdot \left(\sqrt{b} \cdot a\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(-a\right) \cdot b\right) \cdot \sqrt{b}\right)} \cdot \left(\sqrt{b} \cdot a\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(-a\right) \cdot b\right) \cdot \left(\sqrt{b} \cdot \left(\sqrt{b} \cdot a\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-a\right) \cdot b\right)} \cdot \left(\sqrt{b} \cdot \left(\sqrt{b} \cdot a\right)\right) \]

      5. lift-neg.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot b\right) \cdot \left(\sqrt{b} \cdot \left(\sqrt{b} \cdot a\right)\right) \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a \cdot b\right)\right)} \cdot \left(\sqrt{b} \cdot \left(\sqrt{b} \cdot a\right)\right) \]

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

         \[\leadsto \color{blue}{\left(a \cdot \left(\mathsf{neg}\left(b\right)\right)\right)} \cdot \left(\sqrt{b} \cdot \left(\sqrt{b} \cdot a\right)\right) \]

      8. lift-neg.f64 N/A

         \[\leadsto \left(a \cdot \color{blue}{\left(-b\right)}\right) \cdot \left(\sqrt{b} \cdot \left(\sqrt{b} \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(-b\right) \cdot a\right)} \cdot \left(\sqrt{b} \cdot \left(\sqrt{b} \cdot a\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(-b\right) \cdot a\right)} \cdot \left(\sqrt{b} \cdot \left(\sqrt{b} \cdot a\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\left(-b\right) \cdot a\right) \cdot \color{blue}{\left(\left(\sqrt{b} \cdot a\right) \cdot \sqrt{b}\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(-b\right) \cdot a\right) \cdot \left(\color{blue}{\left(\sqrt{b} \cdot a\right)} \cdot \sqrt{b}\right) \]

      13. *-commutative N/A

          \[\leadsto \left(\left(-b\right) \cdot a\right) \cdot \left(\color{blue}{\left(a \cdot \sqrt{b}\right)} \cdot \sqrt{b}\right) \]

      14. associate-*l* N/A

          \[\leadsto \left(\left(-b\right) \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(\sqrt{b} \cdot \sqrt{b}\right)\right)} \]

      15. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(-b\right) \cdot a\right) \cdot \left(a \cdot \left(\color{blue}{\sqrt{b}} \cdot \sqrt{b}\right)\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(-b\right) \cdot a\right) \cdot \left(a \cdot \left(\sqrt{b} \cdot \color{blue}{\sqrt{b}}\right)\right) \]

      17. rem-square-sqrt N/A

          \[\leadsto \left(\left(-b\right) \cdot a\right) \cdot \left(a \cdot \color{blue}{b}\right) \]

      18. remove-double-div N/A

          \[\leadsto \left(\left(-b\right) \cdot a\right) \cdot \left(a \cdot \color{blue}{\frac{1}{\frac{1}{b}}}\right) \]

      19. metadata-eval N/A

          \[\leadsto \left(\left(-b\right) \cdot a\right) \cdot \left(a \cdot \frac{1}{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{b}}\right) \]

      20. distribute-neg-frac N/A

          \[\leadsto \left(\left(-b\right) \cdot a\right) \cdot \left(a \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{-1}{b}\right)}}\right) \]

      21. lift-/.f64 N/A

          \[\leadsto \left(\left(-b\right) \cdot a\right) \cdot \left(a \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{\frac{-1}{b}}\right)}\right) \]

      22. lift-neg.f64 N/A

          \[\leadsto \left(\left(-b\right) \cdot a\right) \cdot \left(a \cdot \frac{1}{\color{blue}{-\frac{-1}{b}}}\right) \]

   8. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\frac{\left(b \cdot a\right) \cdot a}{\frac{-1}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6 \cdot 10^{-199}:\\ \;\;\;\;\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot a}{\frac{-1}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ [a_m, b_m] = \mathsf{sort}([a_m, b_m])\\ \\ \begin{array}{l} t_0 := \left(-a\_m\right) \cdot b\_m\\ \mathbf{if}\;a\_m \leq 1.6 \cdot 10^{-199}:\\ \;\;\;\;\left(t\_0 \cdot b\_m\right) \cdot a\_m\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot a\_m\right) \cdot b\_m\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
(FPCore (a_m b_m)
 :precision binary64
 (let* ((t_0 (* (- a_m) b_m)))
   (if (<= a_m 1.6e-199) (* (* t_0 b_m) a_m) (* (* t_0 a_m) b_m))))

C

b_m = fabs(b);
a_m = fabs(a);
assert(a_m < b_m);
double code(double a_m, double b_m) {
	double t_0 = -a_m * b_m;
	double tmp;
	if (a_m <= 1.6e-199) {
		tmp = (t_0 * b_m) * a_m;
	} else {
		tmp = (t_0 * a_m) * b_m;
	}
	return tmp;
}

Fortran

b_m = abs(b)
a_m = abs(a)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
real(8) function code(a_m, b_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -a_m * b_m
    if (a_m <= 1.6d-199) then
        tmp = (t_0 * b_m) * a_m
    else
        tmp = (t_0 * a_m) * b_m
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
assert a_m < b_m;
public static double code(double a_m, double b_m) {
	double t_0 = -a_m * b_m;
	double tmp;
	if (a_m <= 1.6e-199) {
		tmp = (t_0 * b_m) * a_m;
	} else {
		tmp = (t_0 * a_m) * b_m;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
[a_m, b_m] = sort([a_m, b_m])
def code(a_m, b_m):
	t_0 = -a_m * b_m
	tmp = 0
	if a_m <= 1.6e-199:
		tmp = (t_0 * b_m) * a_m
	else:
		tmp = (t_0 * a_m) * b_m
	return tmp

Julia

b_m = abs(b)
a_m = abs(a)
a_m, b_m = sort([a_m, b_m])
function code(a_m, b_m)
	t_0 = Float64(Float64(-a_m) * b_m)
	tmp = 0.0
	if (a_m <= 1.6e-199)
		tmp = Float64(Float64(t_0 * b_m) * a_m);
	else
		tmp = Float64(Float64(t_0 * a_m) * b_m);
	end
	return tmp
end

MATLAB

b_m = abs(b);
a_m = abs(a);
a_m, b_m = num2cell(sort([a_m, b_m])){:}
function tmp_2 = code(a_m, b_m)
	t_0 = -a_m * b_m;
	tmp = 0.0;
	if (a_m <= 1.6e-199)
		tmp = (t_0 * b_m) * a_m;
	else
		tmp = (t_0 * a_m) * b_m;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
code[a$95$m_, b$95$m_] := Block[{t$95$0 = N[((-a$95$m) * b$95$m), $MachinePrecision]}, If[LessEqual[a$95$m, 1.6e-199], N[(N[(t$95$0 * b$95$m), $MachinePrecision] * a$95$m), $MachinePrecision], N[(N[(t$95$0 * a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.6e-199

   1. Initial program 82.7%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(a \cdot a\right) \cdot b\right) \cdot b}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(a \cdot a\right) \cdot b\right)} \cdot b\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(a \cdot a\right)} \cdot \left(b \cdot b\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot \left(a \cdot \left(b \cdot b\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(b \cdot b\right)\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(b \cdot b\right)\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(a \cdot b\right) \cdot b\right)} \cdot \left(\mathsf{neg}\left(a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(a \cdot b\right) \cdot b\right)} \cdot \left(\mathsf{neg}\left(a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(b \cdot a\right)} \cdot b\right) \cdot \left(\mathsf{neg}\left(a\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(b \cdot a\right)} \cdot b\right) \cdot \left(\mathsf{neg}\left(a\right)\right) \]

      14. lower-neg.f64 94.7

          \[\leadsto \left(\left(b \cdot a\right) \cdot b\right) \cdot \color{blue}{\left(-a\right)} \]

   4. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)} \]

   if 1.6e-199 < a

   1. Initial program 88.7%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(a \cdot a\right) \cdot b\right) \cdot b}\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(a \cdot a\right) \cdot b\right)\right) \cdot b} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(a \cdot a\right) \cdot b\right)\right) \cdot b} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot a\right) \cdot b}\right)\right) \cdot b \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot a\right)} \cdot b\right)\right) \cdot b \]

      7. associate-*l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(a \cdot b\right)}\right)\right) \cdot b \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(a \cdot b\right) \cdot a}\right)\right) \cdot b \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot b\right)\right) \cdot a\right)} \cdot b \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot b\right)\right) \cdot a\right)} \cdot b \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \left(\color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) \cdot b\right)} \cdot a\right) \cdot b \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) \cdot b\right)} \cdot a\right) \cdot b \]

      13. lower-neg.f64 97.8

          \[\leadsto \left(\left(\color{blue}{\left(-a\right)} \cdot b\right) \cdot a\right) \cdot b \]

   4. Applied rewrites 97.8%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-199}:\\ \;\;\;\;\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-a\right) \cdot b\right) \cdot a\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ [a_m, b_m] = \mathsf{sort}([a_m, b_m])\\ \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 7.4 \cdot 10^{-155}:\\ \;\;\;\;\left(\left(\left(-a\_m\right) \cdot b\_m\right) \cdot b\_m\right) \cdot a\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-b\_m\right) \cdot \left(\left(a\_m \cdot a\_m\right) \cdot b\_m\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
(FPCore (a_m b_m)
 :precision binary64
 (if (<= a_m 7.4e-155)
   (* (* (* (- a_m) b_m) b_m) a_m)
   (* (- b_m) (* (* a_m a_m) b_m))))

C

b_m = fabs(b);
a_m = fabs(a);
assert(a_m < b_m);
double code(double a_m, double b_m) {
	double tmp;
	if (a_m <= 7.4e-155) {
		tmp = ((-a_m * b_m) * b_m) * a_m;
	} else {
		tmp = -b_m * ((a_m * a_m) * b_m);
	}
	return tmp;
}

Fortran

b_m = abs(b)
a_m = abs(a)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
real(8) function code(a_m, b_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8) :: tmp
    if (a_m <= 7.4d-155) then
        tmp = ((-a_m * b_m) * b_m) * a_m
    else
        tmp = -b_m * ((a_m * a_m) * b_m)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
assert a_m < b_m;
public static double code(double a_m, double b_m) {
	double tmp;
	if (a_m <= 7.4e-155) {
		tmp = ((-a_m * b_m) * b_m) * a_m;
	} else {
		tmp = -b_m * ((a_m * a_m) * b_m);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
[a_m, b_m] = sort([a_m, b_m])
def code(a_m, b_m):
	tmp = 0
	if a_m <= 7.4e-155:
		tmp = ((-a_m * b_m) * b_m) * a_m
	else:
		tmp = -b_m * ((a_m * a_m) * b_m)
	return tmp

Julia

b_m = abs(b)
a_m = abs(a)
a_m, b_m = sort([a_m, b_m])
function code(a_m, b_m)
	tmp = 0.0
	if (a_m <= 7.4e-155)
		tmp = Float64(Float64(Float64(Float64(-a_m) * b_m) * b_m) * a_m);
	else
		tmp = Float64(Float64(-b_m) * Float64(Float64(a_m * a_m) * b_m));
	end
	return tmp
end

MATLAB

b_m = abs(b);
a_m = abs(a);
a_m, b_m = num2cell(sort([a_m, b_m])){:}
function tmp_2 = code(a_m, b_m)
	tmp = 0.0;
	if (a_m <= 7.4e-155)
		tmp = ((-a_m * b_m) * b_m) * a_m;
	else
		tmp = -b_m * ((a_m * a_m) * b_m);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
code[a$95$m_, b$95$m_] := If[LessEqual[a$95$m, 7.4e-155], N[(N[(N[((-a$95$m) * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * a$95$m), $MachinePrecision], N[((-b$95$m) * N[(N[(a$95$m * a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.4000000000000001e-155

   1. Initial program 82.0%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(a \cdot a\right) \cdot b\right) \cdot b}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(a \cdot a\right) \cdot b\right)} \cdot b\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(a \cdot a\right)} \cdot \left(b \cdot b\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{a \cdot \left(a \cdot \left(b \cdot b\right)\right)}\right) \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot \left(a \cdot \left(b \cdot b\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(b \cdot b\right)\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(b \cdot b\right)\right) \cdot \left(\mathsf{neg}\left(a\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(a \cdot b\right) \cdot b\right)} \cdot \left(\mathsf{neg}\left(a\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(a \cdot b\right) \cdot b\right)} \cdot \left(\mathsf{neg}\left(a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(b \cdot a\right)} \cdot b\right) \cdot \left(\mathsf{neg}\left(a\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(b \cdot a\right)} \cdot b\right) \cdot \left(\mathsf{neg}\left(a\right)\right) \]

      14. lower-neg.f64 93.9

          \[\leadsto \left(\left(b \cdot a\right) \cdot b\right) \cdot \color{blue}{\left(-a\right)} \]

   4. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)} \]

   if 7.4000000000000001e-155 < a

   1. Initial program 90.8%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.4 \cdot 10^{-155}:\\ \;\;\;\;\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(\left(a \cdot a\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ [a_m, b_m] = \mathsf{sort}([a_m, b_m])\\ \\ \left(-b\_m\right) \cdot \left(\left(a\_m \cdot a\_m\right) \cdot b\_m\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
(FPCore (a_m b_m) :precision binary64 (* (- b_m) (* (* a_m a_m) b_m)))

C

b_m = fabs(b);
a_m = fabs(a);
assert(a_m < b_m);
double code(double a_m, double b_m) {
	return -b_m * ((a_m * a_m) * b_m);
}

Fortran

b_m = abs(b)
a_m = abs(a)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
real(8) function code(a_m, b_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    code = -b_m * ((a_m * a_m) * b_m)
end function

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
assert a_m < b_m;
public static double code(double a_m, double b_m) {
	return -b_m * ((a_m * a_m) * b_m);
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
[a_m, b_m] = sort([a_m, b_m])
def code(a_m, b_m):
	return -b_m * ((a_m * a_m) * b_m)

Julia

b_m = abs(b)
a_m = abs(a)
a_m, b_m = sort([a_m, b_m])
function code(a_m, b_m)
	return Float64(Float64(-b_m) * Float64(Float64(a_m * a_m) * b_m))
end

MATLAB

b_m = abs(b);
a_m = abs(a);
a_m, b_m = num2cell(sort([a_m, b_m])){:}
function tmp = code(a_m, b_m)
	tmp = -b_m * ((a_m * a_m) * b_m);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
code[a$95$m_, b$95$m_] := N[((-b$95$m) * N[(N[(a$95$m * a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.9%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
2. Add Preprocessing

3. Final simplification 84.9%

   \[\leadsto \left(-b\right) \cdot \left(\left(a \cdot a\right) \cdot b\right) \]
4. Add Preprocessing

Alternative 6: 28.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ [a_m, b_m] = \mathsf{sort}([a_m, b_m])\\ \\ \left(\left(a\_m \cdot a\_m\right) \cdot b\_m\right) \cdot b\_m \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
(FPCore (a_m b_m) :precision binary64 (* (* (* a_m a_m) b_m) b_m))

C

b_m = fabs(b);
a_m = fabs(a);
assert(a_m < b_m);
double code(double a_m, double b_m) {
	return ((a_m * a_m) * b_m) * b_m;
}

Fortran

b_m = abs(b)
a_m = abs(a)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
real(8) function code(a_m, b_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    code = ((a_m * a_m) * b_m) * b_m
end function

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
assert a_m < b_m;
public static double code(double a_m, double b_m) {
	return ((a_m * a_m) * b_m) * b_m;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
[a_m, b_m] = sort([a_m, b_m])
def code(a_m, b_m):
	return ((a_m * a_m) * b_m) * b_m

Julia

b_m = abs(b)
a_m = abs(a)
a_m, b_m = sort([a_m, b_m])
function code(a_m, b_m)
	return Float64(Float64(Float64(a_m * a_m) * b_m) * b_m)
end

MATLAB

b_m = abs(b);
a_m = abs(a);
a_m, b_m = num2cell(sort([a_m, b_m])){:}
function tmp = code(a_m, b_m)
	tmp = ((a_m * a_m) * b_m) * b_m;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
code[a$95$m_, b$95$m_] := N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]

Derivation

1. Initial program 84.9%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)} \]

   2. +-lft-identity N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(0 + \left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}\right) \]

   3. flip3-+ N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}}\right) \]

   4. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}\right)\right)}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}} \]

4. Applied rewrites 34.2%

   \[\leadsto \color{blue}{\left(b \cdot \left(a \cdot a\right)\right) \cdot b} \]

5. Final simplification 34.2%

   \[\leadsto \left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
6. Add Preprocessing

Alternative 7: 28.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ [a_m, b_m] = \mathsf{sort}([a_m, b_m])\\ \\ \left(\left(a\_m \cdot b\_m\right) \cdot a\_m\right) \cdot b\_m \end{array} \]

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
(FPCore (a_m b_m) :precision binary64 (* (* (* a_m b_m) a_m) b_m))

C

b_m = fabs(b);
a_m = fabs(a);
assert(a_m < b_m);
double code(double a_m, double b_m) {
	return ((a_m * b_m) * a_m) * b_m;
}

Fortran

b_m = abs(b)
a_m = abs(a)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
real(8) function code(a_m, b_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    code = ((a_m * b_m) * a_m) * b_m
end function

Java

b_m = Math.abs(b);
a_m = Math.abs(a);
assert a_m < b_m;
public static double code(double a_m, double b_m) {
	return ((a_m * b_m) * a_m) * b_m;
}

Python

b_m = math.fabs(b)
a_m = math.fabs(a)
[a_m, b_m] = sort([a_m, b_m])
def code(a_m, b_m):
	return ((a_m * b_m) * a_m) * b_m

Julia

b_m = abs(b)
a_m = abs(a)
a_m, b_m = sort([a_m, b_m])
function code(a_m, b_m)
	return Float64(Float64(Float64(a_m * b_m) * a_m) * b_m)
end

MATLAB

b_m = abs(b);
a_m = abs(a);
a_m, b_m = num2cell(sort([a_m, b_m])){:}
function tmp = code(a_m, b_m)
	tmp = ((a_m * b_m) * a_m) * b_m;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
code[a$95$m_, b$95$m_] := N[(N[(N[(a$95$m * b$95$m), $MachinePrecision] * a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]

Derivation

1. Initial program 84.9%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)} \]

   2. +-lft-identity N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(0 + \left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}\right) \]

   3. flip3-+ N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}}\right) \]

   4. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}\right)\right)}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}} \]

4. Applied rewrites 34.1%

   \[\leadsto \color{blue}{\left(\left(b \cdot a\right) \cdot b\right) \cdot a} \]

5. Taylor expanded in b around 0

   \[\leadsto \color{blue}{{a}^{2} \cdot {b}^{2}} \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto {a}^{2} \cdot \color{blue}{\left(b \cdot b\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left({a}^{2} \cdot b\right) \cdot b} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left({a}^{2} \cdot b\right) \cdot b} \]

   4. unpow2 N/A

      \[\leadsto \left(\color{blue}{\left(a \cdot a\right)} \cdot b\right) \cdot b \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\left(a \cdot \left(a \cdot b\right)\right)} \cdot b \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(a \cdot b\right) \cdot a\right)} \cdot b \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(a \cdot b\right) \cdot a\right)} \cdot b \]

   8. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(b \cdot a\right)} \cdot a\right) \cdot b \]

   9. lower-*.f64 34.2

      \[\leadsto \left(\color{blue}{\left(b \cdot a\right)} \cdot a\right) \cdot b \]

7. Applied rewrites 34.2%

   \[\leadsto \color{blue}{\left(\left(b \cdot a\right) \cdot a\right) \cdot b} \]

8. Final simplification 34.2%

   \[\leadsto \left(\left(a \cdot b\right) \cdot a\right) \cdot b \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024248 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))