ab-angle->ABCF D

Percentage Accurate: 82.9% → 99.7%

Time: 5.0s

Alternatives: 6

Speedup: 1.0×

• 27.3% 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: 82.9% 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, 0.3× speedup

Math

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

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
 (* (/ (* (* (sqrt a_m) b_m) (- a_m)) (sqrt a_m)) (* 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 (((sqrt(a_m) * b_m) * -a_m) / sqrt(a_m)) * (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 = (((sqrt(a_m) * b_m) * -a_m) / sqrt(a_m)) * (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 (((Math.sqrt(a_m) * b_m) * -a_m) / Math.sqrt(a_m)) * (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 (((math.sqrt(a_m) * b_m) * -a_m) / math.sqrt(a_m)) * (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(Float64(Float64(sqrt(a_m) * b_m) * Float64(-a_m)) / sqrt(a_m)) * Float64(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 = (((sqrt(a_m) * b_m) * -a_m) / sqrt(a_m)) * (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[(N[(N[(N[Sqrt[a$95$m], $MachinePrecision] * b$95$m), $MachinePrecision] * (-a$95$m)), $MachinePrecision] / N[Sqrt[a$95$m], $MachinePrecision]), $MachinePrecision] * N[(b$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. 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. unswap-sqr N/A

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

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

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

   8. lower-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. neg-sub0 N/A

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

   5. flip-- N/A

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

   6. +-lft-identity N/A

      \[\leadsto \left(b \cdot \frac{0 \cdot 0 - a \cdot a}{\color{blue}{a}}\right) \cdot \left(b \cdot a\right) \]

   7. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{b \cdot \left(0 \cdot 0 - a \cdot a\right)}{a}} \cdot \left(b \cdot a\right) \]

   8. metadata-eval N/A

      \[\leadsto \frac{b \cdot \left(\color{blue}{0} - a \cdot a\right)}{a} \cdot \left(b \cdot a\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{b \cdot \left(0 - \color{blue}{a \cdot a}\right)}{a} \cdot \left(b \cdot a\right) \]

   10. sub0-neg N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. lift-neg.f64 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. rem-square-sqrt N/A

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

   19. lift-sqrt.f64 N/A

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

   20. lift-sqrt.f64 N/A

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

   21. associate-/r* N/A

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

   22. lower-/.f64 N/A

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

6. Applied rewrites 44.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. div-inv N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow2 N/A

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

   10. inv-pow N/A

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

   11. pow-prod-up N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. sqrt-pow1 N/A

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

   15. pow2 N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. rem-square-sqrt N/A

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

   19. lift-sqrt.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. lift-neg.f64 N/A

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

   22. neg-mul-1 N/A

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

   23. associate-*r* N/A

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

   24. metadata-eval N/A

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

   25. div-inv N/A

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

8. Applied rewrites 49.1%

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

9. Final simplification 49.1%

   \[\leadsto \frac{\left(\sqrt{a} \cdot b\right) \cdot \left(-a\right)}{\sqrt{a}} \cdot \left(b \cdot a\right) \]
10. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

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 (* (/ a_m (/ -1.0 b_m)) (* 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 / (-1.0 / b_m)) * (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 / ((-1.0d0) / b_m)) * (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 / (-1.0 / b_m)) * (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 / (-1.0 / b_m)) * (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 / Float64(-1.0 / b_m)) * Float64(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 / (-1.0 / b_m)) * (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 / N[(-1.0 / b$95$m), $MachinePrecision]), $MachinePrecision] * N[(b$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. 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. unswap-sqr N/A

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

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

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

   8. lower-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

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

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

   4. rem-square-sqrt N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-neg.f64 49.1

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

6. Applied rewrites 49.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. unpow1 N/A

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

   11. unpow1 N/A

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

   12. unpow-prod-down N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. unpow1 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. /-rgt-identity N/A

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

   19. clear-num N/A

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

   20. un-div-inv N/A

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

   21. distribute-neg-frac2 N/A

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

   22. neg-mul-1 N/A

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

   23. neg-mul-1 N/A

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

   24. distribute-frac-neg2 N/A

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

   25. lift-neg.f64 N/A

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

   26. lower-/.f64 N/A

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

8. Applied rewrites 99.7%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

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 (* (* a_m b_m) (* (- 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) * (-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) * (-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) * (-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) * (-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(-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) * (-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[((-b$95$m) * a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. 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. unswap-sqr N/A

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

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

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

   8. lower-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 82.9% accurate, 1.0× speedup

Math

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

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 (* (* (* 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) * Float64(-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 82.9%

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

3. Final simplification 82.9%

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

Alternative 5: 29.6% accurate, 1.1× speedup

Math

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

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 (* (* 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(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[(N[(b$95$m * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision]

Derivation

1. Initial program 82.9%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 27.5%

   \[\leadsto \color{blue}{\left(b \cdot \left(a \cdot a\right)\right) \cdot b} \]
5. Add Preprocessing

Alternative 6: 29.3% accurate, 1.1× speedup

Math

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

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) (* 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 (b_m * a_m) * (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 = (b_m * a_m) * (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 (b_m * a_m) * (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 (b_m * a_m) * (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(b_m * a_m) * Float64(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 = (b_m * a_m) * (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[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.9%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 27.4%

   \[\leadsto \color{blue}{\left(b \cdot a\right) \cdot \left(b \cdot a\right)} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024322 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))