ab-angle->ABCF D

Percentage Accurate: 82.2% → 99.7%

Time: 6.1s

Alternatives: 6

Speedup: 1.0×

• 28.0% 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: 82.2% 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

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

FPCore

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

C

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

Fortran

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

Java

assert a < b;
public static double code(double a, double b) {
	return (a * b) * (b / (-1.0 / a));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. pow2 N/A

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

   9. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left({\left(b \cdot a\right)}^{\color{blue}{\left(3 + -1\right)}}\right) \]

   10. pow-prod-up N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(b \cdot a\right)}^{3} \cdot {\left(b \cdot a\right)}^{-1}}\right) \]

   11. lift-pow.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(b \cdot a\right)}^{3}} \cdot {\left(b \cdot a\right)}^{-1}\right) \]

   12. inv-pow N/A

       \[\leadsto \mathsf{neg}\left({\left(b \cdot a\right)}^{3} \cdot \color{blue}{\frac{1}{b \cdot a}}\right) \]

   13. div-inv N/A

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

   14. distribute-neg-frac2 N/A

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

   15. lift-pow.f64 N/A

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

   16. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}}{\mathsf{neg}\left(b \cdot a\right)} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}{\mathsf{neg}\left(\color{blue}{b \cdot a}\right)} \]

   18. *-commutative N/A

       \[\leadsto \frac{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}{\mathsf{neg}\left(\color{blue}{a \cdot b}\right)} \]

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

       \[\leadsto \frac{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot b}} \]

   20. lift-neg.f64 N/A

       \[\leadsto \frac{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{\left(-a\right)} \cdot b} \]

   21. lift-*.f64 N/A

       \[\leadsto \frac{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{\left(-a\right) \cdot b}} \]

   22. associate-/l* N/A

       \[\leadsto \color{blue}{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot \frac{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}{\left(-a\right) \cdot b}} \]

   23. lower-*.f64 N/A

       \[\leadsto \color{blue}{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot \frac{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}{\left(-a\right) \cdot b}} \]

6. Applied rewrites 57.6%

   \[\leadsto \color{blue}{{\left(b \cdot a\right)}^{1.5} \cdot \left(-{\left(b \cdot a\right)}^{0.5}\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{{\left(b \cdot a\right)}^{\frac{3}{2}} \cdot \left(-{\left(b \cdot a\right)}^{\frac{1}{2}}\right)} \]

   2. lift-neg.f64 N/A

      \[\leadsto {\left(b \cdot a\right)}^{\frac{3}{2}} \cdot \color{blue}{\left(\mathsf{neg}\left({\left(b \cdot a\right)}^{\frac{1}{2}}\right)\right)} \]

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

      \[\leadsto \color{blue}{\mathsf{neg}\left({\left(b \cdot a\right)}^{\frac{3}{2}} \cdot {\left(b \cdot a\right)}^{\frac{1}{2}}\right)} \]

   4. lift-pow.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(b \cdot a\right)}^{\frac{3}{2}}} \cdot {\left(b \cdot a\right)}^{\frac{1}{2}}\right) \]

   5. lift-pow.f64 N/A

      \[\leadsto \mathsf{neg}\left({\left(b \cdot a\right)}^{\frac{3}{2}} \cdot \color{blue}{{\left(b \cdot a\right)}^{\frac{1}{2}}}\right) \]

   6. pow-prod-up N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(b \cdot a\right)}^{\left(\frac{3}{2} + \frac{1}{2}\right)}}\right) \]

   7. metadata-eval N/A

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

   8. pow2 N/A

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

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

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

   10. lift-*.f64 N/A

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

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

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

   12. lift-neg.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. /-rgt-identity N/A

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

   20. associate-*r/ N/A

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

   21. associate-*l/ N/A

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

   22. associate-/r/ N/A

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

   23. lift-*.f64 N/A

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

   24. lift-*.f64 N/A

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

   25. associate-*l* N/A

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow-1 N/A

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

   7. remove-double-div N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. lift-neg.f64 N/A

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

   11. frac-2neg N/A

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

   12. lower-/.f64 99.7

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

assert a < b;
public static double code(double a, double b) {
	return (a / (-1.0 / b)) * (a * b);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. pow2 N/A

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

   9. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left({\left(b \cdot a\right)}^{\color{blue}{\left(3 + -1\right)}}\right) \]

   10. pow-prod-up N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(b \cdot a\right)}^{3} \cdot {\left(b \cdot a\right)}^{-1}}\right) \]

   11. lift-pow.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(b \cdot a\right)}^{3}} \cdot {\left(b \cdot a\right)}^{-1}\right) \]

   12. inv-pow N/A

       \[\leadsto \mathsf{neg}\left({\left(b \cdot a\right)}^{3} \cdot \color{blue}{\frac{1}{b \cdot a}}\right) \]

   13. div-inv N/A

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

   14. distribute-neg-frac2 N/A

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

   15. lift-pow.f64 N/A

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

   16. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}}{\mathsf{neg}\left(b \cdot a\right)} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}{\mathsf{neg}\left(\color{blue}{b \cdot a}\right)} \]

   18. *-commutative N/A

       \[\leadsto \frac{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}{\mathsf{neg}\left(\color{blue}{a \cdot b}\right)} \]

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

       \[\leadsto \frac{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot b}} \]

   20. lift-neg.f64 N/A

       \[\leadsto \frac{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{\left(-a\right)} \cdot b} \]

   21. lift-*.f64 N/A

       \[\leadsto \frac{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{\left(-a\right) \cdot b}} \]

   22. associate-/l* N/A

       \[\leadsto \color{blue}{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot \frac{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}{\left(-a\right) \cdot b}} \]

   23. lower-*.f64 N/A

       \[\leadsto \color{blue}{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)} \cdot \frac{{\left(b \cdot a\right)}^{\left(\frac{3}{2}\right)}}{\left(-a\right) \cdot b}} \]

6. Applied rewrites 57.6%

   \[\leadsto \color{blue}{{\left(b \cdot a\right)}^{1.5} \cdot \left(-{\left(b \cdot a\right)}^{0.5}\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{{\left(b \cdot a\right)}^{\frac{3}{2}} \cdot \left(-{\left(b \cdot a\right)}^{\frac{1}{2}}\right)} \]

   2. lift-neg.f64 N/A

      \[\leadsto {\left(b \cdot a\right)}^{\frac{3}{2}} \cdot \color{blue}{\left(\mathsf{neg}\left({\left(b \cdot a\right)}^{\frac{1}{2}}\right)\right)} \]

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

      \[\leadsto \color{blue}{\mathsf{neg}\left({\left(b \cdot a\right)}^{\frac{3}{2}} \cdot {\left(b \cdot a\right)}^{\frac{1}{2}}\right)} \]

   4. lift-pow.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(b \cdot a\right)}^{\frac{3}{2}}} \cdot {\left(b \cdot a\right)}^{\frac{1}{2}}\right) \]

   5. lift-pow.f64 N/A

      \[\leadsto \mathsf{neg}\left({\left(b \cdot a\right)}^{\frac{3}{2}} \cdot \color{blue}{{\left(b \cdot a\right)}^{\frac{1}{2}}}\right) \]

   6. pow-prod-up N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(b \cdot a\right)}^{\left(\frac{3}{2} + \frac{1}{2}\right)}}\right) \]

   7. metadata-eval N/A

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

   8. pow2 N/A

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

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

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

   10. lift-*.f64 N/A

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

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

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

   12. lift-neg.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. /-rgt-identity N/A

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

   20. associate-*r/ N/A

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

   21. associate-*l/ N/A

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

   22. associate-/r/ N/A

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

   23. lift-*.f64 N/A

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

   24. lift-*.f64 N/A

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

   25. associate-*l* N/A

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f64 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow-1 N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return (-a * b) * (a * b)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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: 82.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return -b * ((a * a) * b)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Final simplification 82.1%

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

Alternative 5: 28.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return ((a * a) * b) * b

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 25.7%

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

5. Final simplification 25.7%

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

Alternative 6: 28.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return (a * b) * (a * b)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 25.6%

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

5. Final simplification 25.6%

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

Reproduce

herbie shell --seed 2024255 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))