ab-angle->ABCF D

Percentage Accurate: 81.7% → 99.7%

Time: 4.9s

Alternatives: 8

Speedup: 1.0×

• 26.9% 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: 81.7% 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

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

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 (* (/ a_m (/ -1.0 b_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 (a_m / (-1.0 / b_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 = (a_m / ((-1.0d0) / b_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 (a_m / (-1.0 / b_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 (a_m / (-1.0 / b_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(a_m / Float64(-1.0 / b_m)) * Float64(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 = (a_m / (-1.0 / b_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[(N[(a$95$m / N[(-1.0 / b$95$m), $MachinePrecision]), $MachinePrecision] * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.8%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 egg-rr 30.8%

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

6. Applied egg-rr 99.6%

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

Alternative 2: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ [a_m, b_m] = \mathsf{sort}([a_m, b_m])\\ \\ \begin{array}{l} t_0 := b\_m \cdot \left(a\_m \cdot a\_m\right)\\ \mathbf{if}\;b\_m \cdot t\_0 \leq 4 \cdot 10^{+182}:\\ \;\;\;\;a\_m \cdot \left(-b\_m \cdot \left(a\_m \cdot b\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\_m\right) \cdot t\_0\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* b_m (* a_m a_m))))
   (if (<= (* b_m t_0) 4e+182)
     (* a_m (- (* b_m (* a_m b_m))))
     (* (- b_m) t_0))))

C

b_m = fabs(b);
a_m = fabs(a);
assert(a_m < b_m);
double code(double a_m, double b_m) {
	double t_0 = b_m * (a_m * a_m);
	double tmp;
	if ((b_m * t_0) <= 4e+182) {
		tmp = a_m * -(b_m * (a_m * b_m));
	} else {
		tmp = -b_m * t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b_m * (a_m * a_m)
    if ((b_m * t_0) <= 4d+182) then
        tmp = a_m * -(b_m * (a_m * b_m))
    else
        tmp = -b_m * t_0
    end if
    code = tmp
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) {
	double t_0 = b_m * (a_m * a_m);
	double tmp;
	if ((b_m * t_0) <= 4e+182) {
		tmp = a_m * -(b_m * (a_m * b_m));
	} else {
		tmp = -b_m * t_0;
	}
	return tmp;
}

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):
	t_0 = b_m * (a_m * a_m)
	tmp = 0
	if (b_m * t_0) <= 4e+182:
		tmp = a_m * -(b_m * (a_m * b_m))
	else:
		tmp = -b_m * t_0
	return tmp

Julia

b_m = abs(b)
a_m = abs(a)
a_m, b_m = sort([a_m, b_m])
function code(a_m, b_m)
	t_0 = Float64(b_m * Float64(a_m * a_m))
	tmp = 0.0
	if (Float64(b_m * t_0) <= 4e+182)
		tmp = Float64(a_m * Float64(-Float64(b_m * Float64(a_m * b_m))));
	else
		tmp = Float64(Float64(-b_m) * t_0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
a_m = abs(a);
a_m, b_m = num2cell(sort([a_m, b_m])){:}
function tmp_2 = code(a_m, b_m)
	t_0 = b_m * (a_m * a_m);
	tmp = 0.0;
	if ((b_m * t_0) <= 4e+182)
		tmp = a_m * -(b_m * (a_m * b_m));
	else
		tmp = -b_m * t_0;
	end
	tmp_2 = tmp;
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_] := Block[{t$95$0 = N[(b$95$m * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(b$95$m * t$95$0), $MachinePrecision], 4e+182], N[(a$95$m * (-N[(b$95$m * N[(a$95$m * b$95$m), $MachinePrecision]), $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) < 4.0000000000000003e182

   1. Initial program 83.4%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 egg-rr 99.6%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 94.6

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

   7. Simplified 94.6%

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

   if 4.0000000000000003e182 < (*.f64 (*.f64 (*.f64 a a) b) b)

   1. Initial program 79.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(b \cdot \left(a \cdot a\right)\right) \leq 4 \cdot 10^{+182}:\\ \;\;\;\;a \cdot \left(-b \cdot \left(a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(b \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a_m = \left|a\right| \\ [a_m, b_m] = \mathsf{sort}([a_m, b_m])\\ \\ \begin{array}{l} \mathbf{if}\;a\_m \leq 3.2 \cdot 10^{-222}:\\ \;\;\;\;a\_m \cdot \left(-b\_m \cdot \left(a\_m \cdot b\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\_m\right) \cdot \left(a\_m \cdot \left(a\_m \cdot b\_m\right)\right)\\ \end{array} \end{array} \]

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
 (if (<= a_m 3.2e-222)
   (* a_m (- (* b_m (* a_m b_m))))
   (* (- 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) {
	double tmp;
	if (a_m <= 3.2e-222) {
		tmp = a_m * -(b_m * (a_m * b_m));
	} else {
		tmp = -b_m * (a_m * (a_m * b_m));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (a_m <= 3.2d-222) then
        tmp = a_m * -(b_m * (a_m * b_m))
    else
        tmp = -b_m * (a_m * (a_m * b_m))
    end if
    code = tmp
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) {
	double tmp;
	if (a_m <= 3.2e-222) {
		tmp = a_m * -(b_m * (a_m * b_m));
	} else {
		tmp = -b_m * (a_m * (a_m * b_m));
	}
	return tmp;
}

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):
	tmp = 0
	if a_m <= 3.2e-222:
		tmp = a_m * -(b_m * (a_m * b_m))
	else:
		tmp = -b_m * (a_m * (a_m * b_m))
	return tmp

Julia

b_m = abs(b)
a_m = abs(a)
a_m, b_m = sort([a_m, b_m])
function code(a_m, b_m)
	tmp = 0.0
	if (a_m <= 3.2e-222)
		tmp = Float64(a_m * Float64(-Float64(b_m * Float64(a_m * b_m))));
	else
		tmp = Float64(Float64(-b_m) * Float64(a_m * Float64(a_m * b_m)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
a_m = abs(a);
a_m, b_m = num2cell(sort([a_m, b_m])){:}
function tmp_2 = code(a_m, b_m)
	tmp = 0.0;
	if (a_m <= 3.2e-222)
		tmp = a_m * -(b_m * (a_m * b_m));
	else
		tmp = -b_m * (a_m * (a_m * b_m));
	end
	tmp_2 = tmp;
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_] := If[LessEqual[a$95$m, 3.2e-222], N[(a$95$m * (-N[(b$95$m * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[((-b$95$m) * N[(a$95$m * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.1999999999999999e-222

   1. Initial program 80.6%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 egg-rr 99.6%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 97.9

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

   7. Simplified 97.9%

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

   if 3.1999999999999999e-222 < a

   1. Initial program 83.5%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 98.9

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

   5. Simplified 98.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.2 \cdot 10^{-222}:\\ \;\;\;\;a \cdot \left(-b \cdot \left(a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(a \cdot \left(a \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

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) (* 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 (a_m * b_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 = (a_m * b_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 (a_m * b_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 (a_m * b_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(a_m * b_m) * Float64(a_m * Float64(-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 * b_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[(N[(a$95$m * b$95$m), $MachinePrecision] * N[(a$95$m * (-b$95$m)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.8%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 egg-rr 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: 93.5% accurate, 1.0× speedup

Math

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

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 (* 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 a_m * -(b_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 = a_m * -(b_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 a_m * -(b_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 a_m * -(b_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(a_m * Float64(-Float64(b_m * Float64(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 = a_m * -(b_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[(a$95$m * (-N[(b$95$m * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 81.8%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 egg-rr 99.6%

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

5. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 93.9

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

7. Simplified 93.9%

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

8. Final simplification 93.9%

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

Alternative 6: 29.6% accurate, 1.1× speedup

Math

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

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

Derivation

1. Initial program 81.8%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 egg-rr 30.9%

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

5. Final simplification 30.9%

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

Alternative 7: 29.3% accurate, 1.1× speedup

Math

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

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

Derivation

1. Initial program 81.8%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 egg-rr 30.8%

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

Alternative 8: 29.3% accurate, 1.1× speedup

Math

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

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

Derivation

1. Initial program 81.8%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 egg-rr 30.8%

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

5. Taylor expanded in a around 0

   \[\leadsto \color{blue}{{a}^{2} \cdot {b}^{2}} \]
6. Step-by-step derivation

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 30.8

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

7. Simplified 30.8%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))