ab-angle->ABCF D

Percentage Accurate: 83.1% → 99.7%

Time: 5.8s

Alternatives: 7

Speedup: 1.0×

• 27.4% 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: 83.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

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 77.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. unswap-sqr N/A

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

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

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

   8. lower-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. 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

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. unpow-prod-down N/A

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

   8. associate-/r* N/A

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

   9. pow-flip N/A

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

   10. metadata-eval N/A

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

   11. unpow1 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-pow.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. frac-2neg N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 N/A

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

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

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

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

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

   10. lift-neg.f64 N/A

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

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

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

   12. neg-mul-1 N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. neg-mul-1 N/A

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

   16. lift-neg.f64 N/A

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

   17. remove-double-neg N/A

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

   18. *-commutative N/A

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

   19. lift-*.f64 N/A

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

   20. lower-/.f64 N/A

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

   21. distribute-neg-frac N/A

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

   22. lower-/.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

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 (/ -1.0 b_m)) (* a b_m)))

C

b_m = fabs(b);
assert(a < b_m);
double code(double a, double b_m) {
	return (a / (-1.0 / 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 / ((-1.0d0) / 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 / (-1.0 / 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 / (-1.0 / 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 / Float64(-1.0 / 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 / (-1.0 / 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 / N[(-1.0 / b$95$m), $MachinePrecision]), $MachinePrecision] * N[(a * b$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. unswap-sqr N/A

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

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

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

   8. lower-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. 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

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. unpow-prod-down N/A

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

   8. associate-/r* N/A

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

   9. pow-flip N/A

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

   10. metadata-eval N/A

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

   11. unpow1 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-pow.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow-1 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 0.5× speedup

Math

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

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 b_m) (/ -1.0 (* a b_m))))

C

b_m = fabs(b);
assert(a < b_m);
double code(double a, double b_m) {
	return (a * b_m) / (-1.0 / (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) / ((-1.0d0) / (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) / (-1.0 / (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) / (-1.0 / (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(-1.0 / 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) / (-1.0 / (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[(-1.0 / N[(a * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.9%

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

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{-1}{{\left(b \cdot a\right)}^{-2}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. lift-pow.f64 N/A

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

   5. sqr-pow N/A

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

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

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. unpow-1 N/A

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

   10. remove-double-div N/A

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

   11. lower-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. unpow-1 N/A

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

   17. distribute-neg-frac N/A

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

   18. metadata-eval N/A

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

   19. lower-/.f64 99.6

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{a \cdot b}{\frac{-1}{a \cdot b}}} \]
7. Add Preprocessing

Alternative 4: 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 77.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. unswap-sqr N/A

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

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

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

   8. lower-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 5: 83.1% 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 77.9%

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

3. Final simplification 77.9%

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

Alternative 6: 29.4% 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 77.9%

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

   1. lift-neg.f64 N/A

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

   2. +-lft-identity N/A

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

   3. flip3-+ N/A

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

   4. distribute-neg-frac N/A

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

4. Applied rewrites 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: 29.3% accurate, 1.1× speedup

Math

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

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

Derivation

1. Initial program 77.9%

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

   1. lift-neg.f64 N/A

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

   2. +-lft-identity N/A

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

   3. flip3-+ N/A

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

   4. distribute-neg-frac N/A

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

4. Applied rewrites 29.6%

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

5. Taylor expanded in b around 0

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 29.6

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

7. Applied rewrites 29.6%

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

8. Final simplification 29.6%

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

Reproduce

herbie shell --seed 2024249 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))