ab-angle->ABCF D

Percentage Accurate: 82.3% → 99.7%

Time: 6.3s

Alternatives: 5

Speedup: 1.0×

• 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: 82.3% 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

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

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
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 (/ -1.0 b_m))))

C

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

Fortran

b_m = abs(b)
a_m = abs(a)
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 / ((-1.0d0) / b_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $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[(b$95$m * a$95$m), $MachinePrecision] * N[(a$95$m / N[(-1.0 / b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

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

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

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

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

   5. neg-sub0 N/A

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

   6. flip-- N/A

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

   7. metadata-eval N/A

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

   8. neg-sub0 N/A

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

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

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

   10. lift-neg.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. clear-num N/A

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

   13. un-div-inv N/A

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

   14. lower-/.f64 N/A

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

   15. +-lft-identity N/A

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

   16. lift-*.f64 N/A

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

   17. associate-/r* N/A

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

   18. +-lft-identity N/A

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

   19. metadata-eval N/A

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

   20. associate--r- N/A

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

   21. neg-sub0 N/A

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

   22. lift-neg.f64 N/A

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

   23. sub0-neg N/A

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

   24. lift-neg.f64 N/A

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

   25. frac-2neg N/A

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

   26. lift-neg.f64 N/A

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

   27. distribute-frac-neg N/A

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

   28. *-inverses N/A

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

   29. metadata-eval N/A

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
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) (* b_m a_m)))

C

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

Fortran

b_m = abs(b)
a_m = abs(a)
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) * (b_m * a_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $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[(b$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 3: 82.3% accurate, 1.0× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
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

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

Fortran

b_m = abs(b)
a_m = abs(a)
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

b_m = Math.abs(b);
a_m = Math.abs(a);
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

b_m = math.fabs(b)
a_m = math.fabs(a)
[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

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

MATLAB

b_m = abs(b);
a_m = abs(a);
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

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $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[(N[(a$95$m * a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

3. Final simplification 83.5%

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

Alternative 4: 28.3% accurate, 1.1× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
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 a_m) b_m) b_m))

C

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

Fortran

b_m = abs(b)
a_m = abs(a)
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 * a_m) * b_m) * b_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $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[(N[(a$95$m * a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]

Derivation

1. Initial program 83.5%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 38.8%

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

5. Final simplification 38.8%

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

Alternative 5: 28.0% accurate, 1.1× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
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) (* b_m a_m)))

C

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

Fortran

b_m = abs(b)
a_m = abs(a)
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) * (b_m * a_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $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[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.5%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 38.7%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))