ab-angle->ABCF D

Percentage Accurate: 82.2% → 99.7%

Time: 2.2min

Alternatives: 6

Speedup: 1.0×

• 28.3% 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: 82.2% 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

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

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

C

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

Derivation

1. Initial program 83.8%

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

4. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{-1}{{\left(b \cdot a\right)}^{-2}}} \]
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. lift-pow.f64 N/A

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

   4. sqr-pow N/A

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

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

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   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. metadata-eval N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower-/.f64 99.6

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

   22. lift-*.f64 N/A

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

   23. *-commutative N/A

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

   24. 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. Step-by-step derivation

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. sqr-pow N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. unpow1/2 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. unpow1/2 N/A

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

   15. lower-sqrt.f64 54.1

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 54.1

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

8. Applied rewrites 54.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 99.7

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ [a, b_m] = \mathsf{sort}([a, b_m])\\ \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot b\_m\\ \mathbf{if}\;t\_0 \cdot b\_m \leq 5 \cdot 10^{+200}:\\ \;\;\;\;\left(-a\right) \cdot \left(\left(b\_m \cdot a\right) \cdot b\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\_m\right) \cdot t\_0\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* a a) b_m)))
   (if (<= (* t_0 b_m) 5e+200) (* (- a) (* (* b_m a) b_m)) (* (- b_m) t_0))))

C

b_m = fabs(b);
assert(a < b_m);
double code(double a, double b_m) {
	double t_0 = (a * a) * b_m;
	double tmp;
	if ((t_0 * b_m) <= 5e+200) {
		tmp = -a * ((b_m * a) * b_m);
	} else {
		tmp = -b_m * t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * a) * b_m
    if ((t_0 * b_m) <= 5d+200) then
        tmp = -a * ((b_m * a) * b_m)
    else
        tmp = -b_m * t_0
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
assert a < b_m;
public static double code(double a, double b_m) {
	double t_0 = (a * a) * b_m;
	double tmp;
	if ((t_0 * b_m) <= 5e+200) {
		tmp = -a * ((b_m * a) * b_m);
	} else {
		tmp = -b_m * t_0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
[a, b_m] = sort([a, b_m])
def code(a, b_m):
	t_0 = (a * a) * b_m
	tmp = 0
	if (t_0 * b_m) <= 5e+200:
		tmp = -a * ((b_m * a) * b_m)
	else:
		tmp = -b_m * t_0
	return tmp

Julia

b_m = abs(b)
a, b_m = sort([a, b_m])
function code(a, b_m)
	t_0 = Float64(Float64(a * a) * b_m)
	tmp = 0.0
	if (Float64(t_0 * b_m) <= 5e+200)
		tmp = Float64(Float64(-a) * Float64(Float64(b_m * a) * b_m));
	else
		tmp = Float64(Float64(-b_m) * t_0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
a, b_m = num2cell(sort([a, b_m])){:}
function tmp_2 = code(a, b_m)
	t_0 = (a * a) * b_m;
	tmp = 0.0;
	if ((t_0 * b_m) <= 5e+200)
		tmp = -a * ((b_m * a) * b_m);
	else
		tmp = -b_m * t_0;
	end
	tmp_2 = tmp;
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_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * b$95$m), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * b$95$m), $MachinePrecision], 5e+200], N[((-a) * N[(N[(b$95$m * a), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision], N[((-b$95$m) * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 a a) b) b) < 5.00000000000000019e200

   1. Initial program 85.1%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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. associate-*l* N/A

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

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

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 97.6

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

   4. Applied rewrites 97.6%

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

   if 5.00000000000000019e200 < (*.f64 (*.f64 (*.f64 a a) b) b)

   1. Initial program 81.6%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a \cdot a\right) \cdot b\right) \cdot b \leq 5 \cdot 10^{+200}:\\ \;\;\;\;\left(-a\right) \cdot \left(\left(b \cdot a\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(\left(a \cdot a\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

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

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

C

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

Python

b_m = math.fabs(b)
[a, b_m] = sort([a, b_m])
def code(a, b_m):
	return (b_m * a) / (-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(b_m * a) / Float64(-1.0 / Float64(b_m * a)))
end

MATLAB

b_m = abs(b);
a, b_m = num2cell(sort([a, b_m])){:}
function tmp = code(a, b_m)
	tmp = (b_m * a) / (-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[(b$95$m * a), $MachinePrecision] / N[(-1.0 / N[(b$95$m * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.8%

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

4. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{-1}{{\left(b \cdot a\right)}^{-2}}} \]
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. lift-pow.f64 N/A

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

   4. sqr-pow N/A

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

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

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   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. metadata-eval N/A

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower-/.f64 99.6

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

   22. lift-*.f64 N/A

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

   23. *-commutative N/A

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

   24. 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. Final simplification 99.6%

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

Alternative 4: 82.2% 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 83.8%

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

3. Final simplification 83.8%

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

Alternative 5: 28.5% 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 83.8%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 32.8%

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

5. Final simplification 32.8%

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

Alternative 6: 28.3% accurate, 1.1× speedup

Math

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

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 (* (* (* 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(Float64(b_m * 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[(N[(N[(b$95$m * a), $MachinePrecision] * a), $MachinePrecision] * b$95$m), $MachinePrecision]

Derivation

1. Initial program 83.8%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 32.7%

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

5. Taylor expanded in a 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. lower-*.f64 32.7

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

7. Applied rewrites 32.7%

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

8. Final simplification 32.7%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))