ab-angle->ABCF D

Percentage Accurate: 82.9% → 99.7%

Time: 5.5s

Alternatives: 8

Speedup: 1.0×

• 28.7% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. remove-double-div N/A

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

   2. unpow-1 N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. unpow-prod-down N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. pow-flip N/A

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

   12. metadata-eval N/A

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

   13. unpow1 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-pow.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lower-/.f64 99.7

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

8. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. lift-/.f64 N/A

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

   9. remove-double-div N/A

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

   10. lift-*.f64 N/A

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

   11. remove-double-div N/A

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

   12. metadata-eval N/A

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

   13. lift-neg.f64 N/A

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

   14. frac-2neg N/A

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

   15. lift-/.f64 N/A

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

   16. div-inv N/A

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

   17. clear-num N/A

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

   18. un-div-inv N/A

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

   19. lower-/.f64 N/A

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

   20. lift-/.f64 N/A

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

   21. associate-/l/ N/A

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

   22. associate-/r* N/A

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

   23. metadata-eval N/A

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

   24. distribute-neg-frac N/A

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

   25. lift-/.f64 N/A

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

   26. lower-/.f64 N/A

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. remove-double-div N/A

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

   2. unpow-1 N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. unpow-prod-down N/A

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. pow-flip N/A

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

   12. metadata-eval N/A

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

   13. unpow1 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-pow.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lower-/.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. remove-double-div N/A

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

   3. unpow-1 N/A

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

   4. metadata-eval N/A

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

   5. div-inv N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 N/A

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

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

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-neg-frac N/A

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

   12. distribute-neg-frac2 N/A

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

   13. lower-/.f64 N/A

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

   14. metadata-eval N/A

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

   15. unpow-1 N/A

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

   16. metadata-eval N/A

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

   17. distribute-neg-frac N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 5: 82.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

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

3. Final simplification 81.4%

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

Alternative 6: 29.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 29.6%

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

5. Final simplification 29.6%

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

Alternative 7: 29.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 29.5%

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

5. Final simplification 29.5%

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

Alternative 8: 29.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 29.4%

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

5. Taylor expanded in b around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{{b}^{2} \cdot {a}^{2}} \]

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 29.5

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

7. Applied rewrites 29.5%

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

Reproduce

herbie shell --seed 2024271 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))