ab-angle->ABCF D

Percentage Accurate: 82.8% → 96.7%

Time: 5.9s

Alternatives: 6

Speedup: 1.0×

• 28.8% 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: 82.8% 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: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ [a_m, b_m] = \mathsf{sort}([a_m, b_m])\\ \\ \begin{array}{l} \mathbf{if}\;a\_m \cdot a\_m \leq 10^{-298}:\\ \;\;\;\;a\_m \cdot \left(b\_m \cdot \left(a\_m \cdot \left(-b\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b\_m \cdot \left(b\_m \cdot \left(a\_m \cdot \left(-a\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
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 a_m) 1e-298)
   (* a_m (* b_m (* a_m (- b_m))))
   (* b_m (* b_m (* a_m (- a_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
assert(a_m < b_m);
double code(double a_m, double b_m) {
	double tmp;
	if ((a_m * a_m) <= 1e-298) {
		tmp = a_m * (b_m * (a_m * -b_m));
	} else {
		tmp = b_m * (b_m * (a_m * -a_m));
	}
	return tmp;
}

Fortran

a_m = abs(a)
b_m = abs(b)
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 * a_m) <= 1d-298) then
        tmp = a_m * (b_m * (a_m * -b_m))
    else
        tmp = b_m * (b_m * (a_m * -a_m))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
assert a_m < b_m;
public static double code(double a_m, double b_m) {
	double tmp;
	if ((a_m * a_m) <= 1e-298) {
		tmp = a_m * (b_m * (a_m * -b_m));
	} else {
		tmp = b_m * (b_m * (a_m * -a_m));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
[a_m, b_m] = sort([a_m, b_m])
def code(a_m, b_m):
	tmp = 0
	if (a_m * a_m) <= 1e-298:
		tmp = a_m * (b_m * (a_m * -b_m))
	else:
		tmp = b_m * (b_m * (a_m * -a_m))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
a_m, b_m = sort([a_m, b_m])
function code(a_m, b_m)
	tmp = 0.0
	if (Float64(a_m * a_m) <= 1e-298)
		tmp = Float64(a_m * Float64(b_m * Float64(a_m * Float64(-b_m))));
	else
		tmp = Float64(b_m * Float64(b_m * Float64(a_m * Float64(-a_m))));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
a_m, b_m = num2cell(sort([a_m, b_m])){:}
function tmp_2 = code(a_m, b_m)
	tmp = 0.0;
	if ((a_m * a_m) <= 1e-298)
		tmp = a_m * (b_m * (a_m * -b_m));
	else
		tmp = b_m * (b_m * (a_m * -a_m));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $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[N[(a$95$m * a$95$m), $MachinePrecision], 1e-298], N[(a$95$m * N[(b$95$m * N[(a$95$m * (-b$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b$95$m * N[(b$95$m * N[(a$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 9.99999999999999912e-299

   1. Initial program 80.7%

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

      1. associate-*l* 79.3%

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

      2. associate-*r* 88.5%

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

      3. *-commutative 88.5%

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

      4. distribute-rgt-neg-in 88.5%

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

      5. distribute-rgt-neg-in 88.5%

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

      6. associate-*r* 99.9%

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

   3. Simplified 99.9%

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

   if 9.99999999999999912e-299 < (*.f64 a a)

   1. Initial program 86.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 10^{-298}:\\ \;\;\;\;a \cdot \left(b \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(a \cdot \left(-a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

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

Fortran

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

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

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

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

MATLAB

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

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $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 84.8%

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

3. Taylor expanded in a around 0 77.6%

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

   1. mul-1-neg 77.6%

      \[\leadsto \color{blue}{-{a}^{2} \cdot {b}^{2}} \]

   2. unpow2 77.6%

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

   3. unpow2 77.6%

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

   4. swap-sqr 99.7%

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

   5. unpow2 99.7%

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

5. Simplified 99.7%

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

   1. neg-mul-1 99.7%

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

   2. unpow2 99.7%

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

   3. associate-*r* 99.7%

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

7. Applied egg-rr 99.7%

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

   1. mul-1-neg 99.7%

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

   2. neg-sub0 99.7%

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

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left(0 - a \cdot b\right)} \cdot \left(a \cdot b\right) \]
10. Step-by-step derivation

    1. neg-sub0 99.7%

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

    2. distribute-rgt-neg-in 99.7%

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

11. Simplified 99.7%

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

Alternative 3: 82.8% accurate, 1.0× speedup

Math

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

FPCore

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

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

Fortran

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

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

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

a_m = abs(a)
b_m = abs(b)
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 * Float64(-a_m))))
end

MATLAB

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

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $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 84.8%

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

3. Final simplification 84.8%

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

Alternative 4: 28.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

   1. distribute-rgt-neg-in 84.8%

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

   2. associate-*l* 94.0%

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

3. Simplified 94.0%

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

   1. neg-sub0 94.0%

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

   2. sub-neg 94.0%

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

   3. add-sqr-sqrt 48.4%

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

   4. sqrt-unprod 57.7%

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

   5. sqr-neg 57.7%

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

   6. sqrt-unprod 17.7%

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

   7. add-sqr-sqrt 32.8%

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

6. Applied egg-rr 32.8%

   \[\leadsto \left(a \cdot \left(a \cdot b\right)\right) \cdot \color{blue}{\left(0 + b\right)} \]
7. Step-by-step derivation

   1. +-lft-identity 32.8%

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

8. Simplified 32.8%

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

9. Final simplification 32.8%

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

Alternative 5: 28.3% accurate, 1.1× speedup

Math

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

FPCore

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

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

Fortran

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

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

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

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

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

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $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 84.8%

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

   1. add-sqr-sqrt 31.9%

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

   2. sqrt-unprod 32.8%

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

   3. sqr-neg 32.8%

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

   4. sqrt-unprod 32.7%

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

   5. add-sqr-sqrt 32.7%

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

   6. associate-*l* 32.6%

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

   7. swap-sqr 32.7%

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

4. Applied egg-rr 32.7%

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

Alternative 6: 28.3% accurate, 1.1× speedup

Math

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

FPCore

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

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

Fortran

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

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

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

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

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

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $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 84.8%

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

   1. associate-*l* 77.6%

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

   2. associate-*r* 85.0%

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

   3. *-commutative 85.0%

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

   4. distribute-rgt-neg-in 85.0%

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

   5. distribute-rgt-neg-in 85.0%

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

   6. associate-*r* 94.9%

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

3. Simplified 94.9%

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

   1. neg-sub0 94.9%

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

   2. sub-neg 94.9%

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

   3. add-sqr-sqrt 47.5%

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

   4. sqrt-unprod 58.6%

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

   5. sqr-neg 58.6%

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

   6. sqrt-prod 17.4%

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

   7. add-sqr-sqrt 32.8%

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

6. Applied egg-rr 32.8%

   \[\leadsto a \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(0 + a\right)}\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 32.8%

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

8. Simplified 32.8%

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

9. Final simplification 32.8%

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

Reproduce

herbie shell --seed 2024146 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))