ab-angle->ABCF D

Percentage Accurate: 82.3% → 99.7%

Time: 7.2s

Alternatives: 7

Speedup: 1.0×

• 27.5% 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: 82.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. neg-sub0 N/A

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

   9. flip-- N/A

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

   10. +-lft-identity N/A

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

   11. associate-*r/ N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. pow2 N/A

       \[\leadsto \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(0 - \color{blue}{{a}^{2}}\right)}{a} \]

   16. neg-sub0 N/A

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

   17. pow2 N/A

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

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

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

   19. lift-neg.f64 N/A

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

   20. lower-*.f64 79.8

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

6. Applied rewrites 79.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. *-inverses N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. associate-/r/ N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r* N/A

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

   12. lift-neg.f64 N/A

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

   13. distribute-rgt-neg-out N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/r* N/A

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

   16. unpow-1 N/A

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

   17. lift-pow.f64 N/A

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

   18. clear-num N/A

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

   19. distribute-frac-neg N/A

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

8. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{b \cdot a}{\frac{-1}{b \cdot a}}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. *-commutative N/A

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

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

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

   6. rem-square-sqrt N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. associate-*r* N/A

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

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

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

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

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

   14. lower-*.f64 N/A

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

   15. lower-neg.f64 50.1

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

6. Applied rewrites 50.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-neg.f64 N/A

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

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

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. distribute-rgt-neg-out N/A

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

   11. lift-*.f64 N/A

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

   12. /-rgt-identity N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/l* N/A

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

   15. clear-num N/A

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

   16. div-inv N/A

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

   17. distribute-neg-frac2 N/A

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

   18. lower-/.f64 N/A

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

   19. distribute-neg-frac N/A

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

   20. metadata-eval N/A

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

   21. lower-/.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

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

Alternative 4: 82.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

3. Final simplification 82.3%

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

Alternative 5: 29.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 30.7%

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

5. Final simplification 30.7%

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

Alternative 6: 29.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 30.6%

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

5. Final simplification 30.6%

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

Alternative 7: 29.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 30.6%

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

5. Taylor expanded in b around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. remove-double-neg N/A

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

   4. mul-1-neg N/A

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

   5. neg-mul-1 N/A

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

   6. rem-square-sqrt N/A

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

   7. associate-*r* N/A

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

   8. mul-1-neg N/A

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

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

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

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

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

   14. mul-1-neg N/A

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

   15. associate-*r* N/A

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

   16. rem-square-sqrt N/A

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

   17. neg-mul-1 N/A

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

   18. mul-1-neg N/A

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

   19. remove-double-neg N/A

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

   20. remove-double-neg N/A

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

   21. mul-1-neg N/A

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

   22. neg-mul-1 N/A

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

7. Applied rewrites 30.6%

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

8. Final simplification 30.6%

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

Reproduce

herbie shell --seed 2024242 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))