ab-angle->ABCF D

Percentage Accurate: 82.0% → 99.7%

Time: 8.3s

Alternatives: 6

Speedup: 0.9×

• 27.9% 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: 82.0% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right) \]

   2. unswap-sqr N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(a \cdot b\right), \left(a \cdot b\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(a \cdot b\right)\right)\right) \]

   5. *-lowering-*.f64 99.6%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

4. Applied egg-rr 99.6%

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

   1. unpow1 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(a \cdot {b}^{1}\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(a \cdot {b}^{\left(\frac{2}{2}\right)}\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   3. sqr-pow N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(a \cdot \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\left(a \cdot {b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot {b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right), \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{\left(\frac{1}{2}\right)}\right)\right), \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{\frac{1}{2}}\right)\right), \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   9. unpow1/2 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\sqrt{b}\right)\right), \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sqrt.f64}\left(b\right)\right), \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sqrt.f64}\left(b\right)\right), \left({b}^{\left(\frac{1}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sqrt.f64}\left(b\right)\right), \left({b}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   13. unpow1/2 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sqrt.f64}\left(b\right)\right), \left(\sqrt{b}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   14. sqrt-lowering-sqrt.f64 45.8%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sqrt.f64}\left(b\right)\right), \mathsf{sqrt.f64}\left(b\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

6. Applied egg-rr 45.8%

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

8. Applied egg-rr 99.5%

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

   1. associate-/l/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\frac{\frac{-1}{b}}{\color{blue}{a}}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\frac{\frac{\mathsf{neg}\left(1\right)}{b}}{a}\right)\right) \]

   3. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\frac{\mathsf{neg}\left(\frac{1}{b}\right)}{a}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right), \color{blue}{a}\right)\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{b}\right), a\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{/.f64}\left(\left(\frac{-1}{b}\right), a\right)\right) \]

   7. /-lowering-/.f64 99.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, b\right), a\right)\right) \]

10. Applied egg-rr 99.6%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right) \]

   2. unswap-sqr N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(a \cdot b\right), \left(a \cdot b\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(a \cdot b\right)\right)\right) \]

   5. *-lowering-*.f64 99.6%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

4. Applied egg-rr 99.6%

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

   1. unpow1 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(a \cdot {b}^{1}\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(a \cdot {b}^{\left(\frac{2}{2}\right)}\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   3. sqr-pow N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(a \cdot \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\left(a \cdot {b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot {b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot {b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right), \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{\left(\frac{1}{2}\right)}\right)\right), \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{\frac{1}{2}}\right)\right), \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   9. unpow1/2 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(\sqrt{b}\right)\right), \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sqrt.f64}\left(b\right)\right), \left({b}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sqrt.f64}\left(b\right)\right), \left({b}^{\left(\frac{1}{2}\right)}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sqrt.f64}\left(b\right)\right), \left({b}^{\frac{1}{2}}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   13. unpow1/2 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sqrt.f64}\left(b\right)\right), \left(\sqrt{b}\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   14. sqrt-lowering-sqrt.f64 45.8%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sqrt.f64}\left(b\right)\right), \mathsf{sqrt.f64}\left(b\right)\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

6. Applied egg-rr 45.8%

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

8. Applied egg-rr 99.5%

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

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\frac{\frac{-1}{a}}{\color{blue}{b}}\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\frac{-1 \cdot \frac{1}{a}}{b}\right)\right) \]

   3. neg-mul-1 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\frac{\mathsf{neg}\left(\frac{1}{a}\right)}{b}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right), \color{blue}{b}\right)\right) \]

   5. neg-mul-1 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{/.f64}\left(\left(-1 \cdot \frac{1}{a}\right), b\right)\right) \]

   6. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{/.f64}\left(\left(\frac{-1}{a}\right), b\right)\right) \]

   7. /-lowering-/.f64 99.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\right)\right) \]

10. Applied egg-rr 99.6%

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

Alternative 3: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ [a_m, b_m] = \mathsf{sort}([a_m, b_m])\\ \\ \left(a\_m \cdot b\_m\right) \cdot \left(a\_m \cdot \left(0 - b\_m\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
(FPCore (a_m b_m) :precision binary64 (* (* a_m b_m) (* a_m (- 0.0 b_m))))

C

a_m = fabs(a);
b_m = fabs(b);
assert(a_m < b_m);
double code(double a_m, double b_m) {
	return (a_m * b_m) * (a_m * (0.0 - b_m));
}

Fortran

a_m = abs(a)
b_m = abs(b)
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
real(8) function code(a_m, b_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    code = (a_m * b_m) * (a_m * (0.0d0 - b_m))
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
assert a_m < b_m;
public static double code(double a_m, double b_m) {
	return (a_m * b_m) * (a_m * (0.0 - b_m));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
[a_m, b_m] = sort([a_m, b_m])
def code(a_m, b_m):
	return (a_m * b_m) * (a_m * (0.0 - b_m))

Julia

a_m = abs(a)
b_m = abs(b)
a_m, b_m = sort([a_m, b_m])
function code(a_m, b_m)
	return Float64(Float64(a_m * b_m) * Float64(a_m * Float64(0.0 - b_m)))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
a_m, b_m = num2cell(sort([a_m, b_m])){:}
function tmp = code(a_m, b_m)
	tmp = (a_m * b_m) * (a_m * (0.0 - b_m));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
NOTE: a_m and b_m should be sorted in increasing order before calling this function.
code[a$95$m_, b$95$m_] := N[(N[(a$95$m * b$95$m), $MachinePrecision] * N[(a$95$m * N[(0.0 - b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right) \]

   2. unswap-sqr N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(a \cdot b\right), \left(a \cdot b\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(a \cdot b\right)\right)\right) \]

   5. *-lowering-*.f64 99.6%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

   \[\leadsto \left(a \cdot b\right) \cdot \left(a \cdot \left(0 - b\right)\right) \]
6. Add Preprocessing

Alternative 4: 28.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l* N/A

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

   2. associate-*r* N/A

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

   3. +-lft-identity N/A

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

   4. flip3-+ N/A

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

   5. distribute-neg-frac N/A

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

4. Applied egg-rr 28.7%

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

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(a \cdot a\right) \cdot b\right), b\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), b\right), b\right) \]

   3. *-lowering-*.f64 28.6%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right), b\right) \]

6. Applied egg-rr 28.6%

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

7. Final simplification 28.6%

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

Alternative 5: 28.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l* N/A

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

   2. associate-*r* N/A

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

   3. +-lft-identity N/A

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

   4. flip3-+ N/A

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

   5. distribute-neg-frac N/A

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

4. Applied egg-rr 28.7%

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

5. Final simplification 28.7%

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

Alternative 6: 28.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*l* N/A

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

   2. associate-*r* N/A

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

   3. +-lft-identity N/A

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

   4. flip3-+ N/A

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

   5. distribute-neg-frac N/A

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

4. Applied egg-rr 28.7%

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

Reproduce

herbie shell --seed 2024148 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))