ab-angle->ABCF D

Percentage Accurate: 82.0% → 99.7%

Time: 5.8s

Alternatives: 5

Speedup: 1.0×

• 27.6% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\frac{0 + a}{0 \cdot 0 - a \cdot a}}} \cdot b\right) \cdot \left(b \cdot a\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\frac{0 + a}{0 \cdot 0 - a \cdot a}}} \cdot b\right) \cdot \left(b \cdot a\right) \]

   6. +-lft-identity N/A

      \[\leadsto \left(\frac{1}{\frac{\color{blue}{a}}{0 \cdot 0 - a \cdot a}} \cdot b\right) \cdot \left(b \cdot a\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\frac{a}{\color{blue}{0} - a \cdot a}} \cdot b\right) \cdot \left(b \cdot a\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{\frac{a}{0 - \color{blue}{a \cdot a}}} \cdot b\right) \cdot \left(b \cdot a\right) \]

   9. neg-sub0 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. lift-neg.f64 N/A

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

   14. lower-*.f64 82.3

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

6. Applied rewrites 82.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. associate-/r/ N/A

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

   9. frac-2neg N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. distribute-frac-neg2 N/A

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

   13. remove-double-neg N/A

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

   14. *-inverses N/A

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

   15. associate-/r/ N/A

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

   16. frac-2neg N/A

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

   17. metadata-eval N/A

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

   18. lift-neg.f64 N/A

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

   19. associate-/r/ N/A

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

   20. lift-neg.f64 N/A

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

   21. metadata-eval N/A

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

   22. frac-2neg N/A

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

   23. associate-/r/ N/A

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

   24. lower-/.f64 N/A

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

   25. lift-neg.f64 N/A

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

8. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. *-commutative N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 99.7

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 99.5

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

4. Applied rewrites 99.5%

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

Alternative 3: 82.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Final simplification 82.5%

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

Alternative 4: 29.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 22.7%

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

5. Final simplification 22.7%

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

Alternative 5: 29.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 22.7%

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

Reproduce

herbie shell --seed 2024267 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))