ab-angle->ABCF D

Percentage Accurate: 81.8% → 99.7%

Time: 5.7s

Alternatives: 7

Speedup: 1.0×

• 28.1% 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: 81.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(N[(a$95$m / N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(a$95$m * b), $MachinePrecision]), $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-*.f64 N/A

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

   2. associate-*l* N/A

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

   3. lift-*.f64 N/A

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

   4. unswap-sqr N/A

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

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

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

   6. lower-*.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. neg-sub0 N/A

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

   2. flip3-- N/A

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

   3. clear-num N/A

      \[\leadsto \left(a \cdot \color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}{{0}^{3} - {b}^{3}}}}\right) \cdot \left(a \cdot b\right) \]

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. metadata-eval N/A

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

   7. +-lft-identity N/A

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

   8. mul0-lft N/A

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

   9. +-rgt-identity N/A

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

   10. clear-num N/A

       \[\leadsto \frac{a}{\color{blue}{\frac{1}{\frac{{0}^{3} - {b}^{3}}{b \cdot b}}}} \cdot \left(a \cdot b\right) \]

   11. *-inverses N/A

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

   12. +-rgt-identity N/A

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

   13. mul0-lft N/A

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

   14. +-lft-identity N/A

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

   15. metadata-eval N/A

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

   16. flip3-- N/A

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

   17. neg-sub0 N/A

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

   18. lift-neg.f64 N/A

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

   19. associate-/r* N/A

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

   20. *-commutative N/A

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

   21. associate-/r* N/A

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

6. Applied rewrites 99.7%

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

Alternative 2: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ \begin{array}{l} t_0 := a\_m \cdot \left(-b\right)\\ \mathbf{if}\;a\_m \leq 3.2 \cdot 10^{-214}:\\ \;\;\;\;a\_m \cdot \left(b \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a\_m \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b)
 :precision binary64
 (let* ((t_0 (* a_m (- b))))
   (if (<= a_m 3.2e-214) (* a_m (* b t_0)) (* b (* a_m t_0)))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	double t_0 = a_m * -b;
	double tmp;
	if (a_m <= 3.2e-214) {
		tmp = a_m * (b * t_0);
	} else {
		tmp = b * (a_m * t_0);
	}
	return tmp;
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a_m * -b
    if (a_m <= 3.2d-214) then
        tmp = a_m * (b * t_0)
    else
        tmp = b * (a_m * t_0)
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	double t_0 = a_m * -b;
	double tmp;
	if (a_m <= 3.2e-214) {
		tmp = a_m * (b * t_0);
	} else {
		tmp = b * (a_m * t_0);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	t_0 = a_m * -b
	tmp = 0
	if a_m <= 3.2e-214:
		tmp = a_m * (b * t_0)
	else:
		tmp = b * (a_m * t_0)
	return tmp

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	t_0 = Float64(a_m * Float64(-b))
	tmp = 0.0
	if (a_m <= 3.2e-214)
		tmp = Float64(a_m * Float64(b * t_0));
	else
		tmp = Float64(b * Float64(a_m * t_0));
	end
	return tmp
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp_2 = code(a_m, b)
	t_0 = a_m * -b;
	tmp = 0.0;
	if (a_m <= 3.2e-214)
		tmp = a_m * (b * t_0);
	else
		tmp = b * (a_m * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := Block[{t$95$0 = N[(a$95$m * (-b)), $MachinePrecision]}, If[LessEqual[a$95$m, 3.2e-214], N[(a$95$m * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision], N[(b * N[(a$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.20000000000000013e-214

   1. Initial program 83.6%

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

      1. lift-*.f64 N/A

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

      2. associate-*l* N/A

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

      3. lift-*.f64 N/A

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

      4. unswap-sqr N/A

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

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

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. neg-sub0 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

         \[\leadsto \left(a \cdot \color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}{{0}^{3} - {b}^{3}}}}\right) \cdot \left(a \cdot b\right) \]

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. +-lft-identity N/A

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

      8. mul0-lft N/A

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

      9. +-rgt-identity N/A

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

      10. clear-num N/A

          \[\leadsto \frac{a}{\color{blue}{\frac{1}{\frac{{0}^{3} - {b}^{3}}{b \cdot b}}}} \cdot \left(a \cdot b\right) \]

      11. *-inverses N/A

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

      12. +-rgt-identity N/A

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

      13. mul0-lft N/A

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

      14. +-lft-identity N/A

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

      15. metadata-eval N/A

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

      16. flip3-- N/A

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

      17. neg-sub0 N/A

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

      18. lift-neg.f64 N/A

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

      19. associate-/r* N/A

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

      20. *-commutative N/A

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

      21. associate-/r* N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 95.2

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

   9. Applied rewrites 95.2%

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

   if 3.20000000000000013e-214 < a

   1. Initial program 84.0%

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

   3. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

      10. lower-*.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 96.3

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

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{b \cdot \left(a \cdot \left(a \cdot \left(-b\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 1.9 \cdot 10^{-161}:\\ \;\;\;\;a\_m \cdot \left(b \cdot \left(a\_m \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(a\_m \cdot \left(-a\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 1.9e-161)
   (* a_m (* b (* a_m (- b))))
   (* b (* b (* a_m (- a_m))))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	double tmp;
	if (a_m <= 1.9e-161) {
		tmp = a_m * (b * (a_m * -b));
	} else {
		tmp = b * (b * (a_m * -a_m));
	}
	return tmp;
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a_m <= 1.9d-161) then
        tmp = a_m * (b * (a_m * -b))
    else
        tmp = b * (b * (a_m * -a_m))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	double tmp;
	if (a_m <= 1.9e-161) {
		tmp = a_m * (b * (a_m * -b));
	} else {
		tmp = b * (b * (a_m * -a_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	tmp = 0
	if a_m <= 1.9e-161:
		tmp = a_m * (b * (a_m * -b))
	else:
		tmp = b * (b * (a_m * -a_m))
	return tmp

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	tmp = 0.0
	if (a_m <= 1.9e-161)
		tmp = Float64(a_m * Float64(b * Float64(a_m * Float64(-b))));
	else
		tmp = Float64(b * Float64(b * Float64(a_m * Float64(-a_m))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if (a_m <= 1.9e-161)
		tmp = a_m * (b * (a_m * -b));
	else
		tmp = b * (b * (a_m * -a_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := If[LessEqual[a$95$m, 1.9e-161], N[(a$95$m * N[(b * N[(a$95$m * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(a$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.9000000000000001e-161

   1. Initial program 83.6%

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

      1. lift-*.f64 N/A

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

      2. associate-*l* N/A

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

      3. lift-*.f64 N/A

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

      4. unswap-sqr N/A

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

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

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

      6. lower-*.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. neg-sub0 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

         \[\leadsto \left(a \cdot \color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}{{0}^{3} - {b}^{3}}}}\right) \cdot \left(a \cdot b\right) \]

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. +-lft-identity N/A

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

      8. mul0-lft N/A

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

      9. +-rgt-identity N/A

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

      10. clear-num N/A

          \[\leadsto \frac{a}{\color{blue}{\frac{1}{\frac{{0}^{3} - {b}^{3}}{b \cdot b}}}} \cdot \left(a \cdot b\right) \]

      11. *-inverses N/A

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

      12. +-rgt-identity N/A

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

      13. mul0-lft N/A

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

      14. +-lft-identity N/A

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

      15. metadata-eval N/A

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

      16. flip3-- N/A

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

      17. neg-sub0 N/A

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

      18. lift-neg.f64 N/A

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

      19. associate-/r* N/A

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

      20. *-commutative N/A

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

      21. associate-/r* N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 95.4

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

   9. Applied rewrites 95.4%

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

   if 1.9000000000000001e-161 < a

   1. Initial program 84.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.9 \cdot 10^{-161}:\\ \;\;\;\;a \cdot \left(b \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(a \cdot \left(-a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(N[(a$95$m * b), $MachinePrecision] * N[(a$95$m * (-b)), $MachinePrecision]), $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-*.f64 N/A

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

   2. associate-*l* N/A

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

   3. lift-*.f64 N/A

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

   4. unswap-sqr N/A

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

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

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

   6. lower-*.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 5: 81.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 6: 28.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(b * N[(b * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $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-*.f64 N/A

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

   2. associate-*l* N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-lft-identity N/A

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

   7. flip3-+ N/A

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

   8. distribute-neg-frac N/A

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

4. Applied rewrites 29.7%

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

5. Final simplification 29.7%

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

Alternative 7: 28.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(N[(a$95$m * b), $MachinePrecision] * N[(a$95$m * b), $MachinePrecision]), $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-*.f64 N/A

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

   2. associate-*l* N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-lft-identity N/A

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

   7. flip3-+ N/A

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

   8. distribute-neg-frac N/A

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

4. Applied rewrites 29.7%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))