ab-angle->ABCF D

Percentage Accurate: 83.4% → 99.6%

Time: 5.6s

Alternatives: 7

Speedup: 1.0×

• 28.2% 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: 83.4% 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: 99.6% accurate, 0.0× speedup

Math

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

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
 (* (* a_m b_m) (* (sqrt a_m) (* b_m (- (sqrt 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 (a_m * b_m) * (sqrt(a_m) * (b_m * -sqrt(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 = (a_m * b_m) * (sqrt(a_m) * (b_m * -sqrt(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 (a_m * b_m) * (Math.sqrt(a_m) * (b_m * -Math.sqrt(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 (a_m * b_m) * (math.sqrt(a_m) * (b_m * -math.sqrt(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(Float64(a_m * b_m) * Float64(sqrt(a_m) * Float64(b_m * Float64(-sqrt(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 = (a_m * b_m) * (sqrt(a_m) * (b_m * -sqrt(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[(N[(a$95$m * b$95$m), $MachinePrecision] * N[(N[Sqrt[a$95$m], $MachinePrecision] * N[(b$95$m * (-N[Sqrt[a$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

3. Taylor expanded in a around 0 71.0%

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

   1. mul-1-neg 71.0%

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

   2. unpow2 71.0%

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

   3. unpow2 71.0%

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

   4. swap-sqr 99.6%

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

   5. unpow2 99.6%

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

5. Simplified 99.6%

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

   1. unpow2 99.6%

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

   2. distribute-rgt-neg-in 99.6%

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

7. Applied egg-rr 99.6%

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

   1. distribute-rgt-neg-in 99.6%

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

   2. add-sqr-sqrt 54.4%

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

   3. associate-*l* 54.4%

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

9. Applied egg-rr 54.4%

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

10. Final simplification 54.4%

    \[\leadsto \left(a \cdot b\right) \cdot \left(\sqrt{a} \cdot \left(b \cdot \left(-\sqrt{a}\right)\right)\right) \]
11. Add Preprocessing

Alternative 2: 97.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ [a_m, b_m] = \mathsf{sort}([a_m, b_m])\\ \\ \begin{array}{l} t_0 := b\_m \cdot \left(b\_m \cdot \left(a\_m \cdot \left(-a\_m\right)\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(b\_m \cdot \left(a\_m \cdot \left(-b\_m\right)\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* b_m (* b_m (* a_m (- a_m))))))
   (if (<= t_0 -2e+69) t_0 (* 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) {
	double t_0 = b_m * (b_m * (a_m * -a_m));
	double tmp;
	if (t_0 <= -2e+69) {
		tmp = t_0;
	} else {
		tmp = a_m * (b_m * (a_m * -b_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) :: t_0
    real(8) :: tmp
    t_0 = b_m * (b_m * (a_m * -a_m))
    if (t_0 <= (-2d+69)) then
        tmp = t_0
    else
        tmp = a_m * (b_m * (a_m * -b_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 t_0 = b_m * (b_m * (a_m * -a_m));
	double tmp;
	if (t_0 <= -2e+69) {
		tmp = t_0;
	} else {
		tmp = a_m * (b_m * (a_m * -b_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):
	t_0 = b_m * (b_m * (a_m * -a_m))
	tmp = 0
	if t_0 <= -2e+69:
		tmp = t_0
	else:
		tmp = a_m * (b_m * (a_m * -b_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)
	t_0 = Float64(b_m * Float64(b_m * Float64(a_m * Float64(-a_m))))
	tmp = 0.0
	if (t_0 <= -2e+69)
		tmp = t_0;
	else
		tmp = Float64(a_m * Float64(b_m * Float64(a_m * Float64(-b_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)
	t_0 = b_m * (b_m * (a_m * -a_m));
	tmp = 0.0;
	if (t_0 <= -2e+69)
		tmp = t_0;
	else
		tmp = a_m * (b_m * (a_m * -b_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_] := Block[{t$95$0 = N[(b$95$m * N[(b$95$m * N[(a$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+69], t$95$0, N[(a$95$m * N[(b$95$m * N[(a$95$m * (-b$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 (*.f64 (*.f64 (*.f64 a a) b) b)) < -2.0000000000000001e69

   1. Initial program 77.8%

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

   if -2.0000000000000001e69 < (neg.f64 (*.f64 (*.f64 (*.f64 a a) b) b))

   1. Initial program 82.2%

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

      1. associate-*l* 76.6%

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

      2. associate-*r* 82.2%

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

      3. *-commutative 82.2%

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

      4. distribute-rgt-neg-in 82.2%

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

      5. distribute-rgt-neg-in 82.2%

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

      6. associate-*r* 94.8%

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

   3. Simplified 94.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(b \cdot \left(a \cdot \left(-a\right)\right)\right) \leq -2 \cdot 10^{+69}:\\ \;\;\;\;b \cdot \left(b \cdot \left(a \cdot \left(-a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(b \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

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(Float64(a_m * b_m) * Float64(a_m * Float64(-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 80.1%

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

3. Taylor expanded in a around 0 71.0%

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

   1. mul-1-neg 71.0%

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

   2. unpow2 71.0%

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

   3. unpow2 71.0%

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

   4. swap-sqr 99.6%

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

   5. unpow2 99.6%

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

5. Simplified 99.6%

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

   1. unpow2 99.6%

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

   2. distribute-rgt-neg-in 99.6%

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

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

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

Alternative 4: 83.4% 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 80.1%

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

3. Final simplification 80.1%

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

Alternative 5: 29.2% 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 80.1%

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

   1. distribute-rgt-neg-in 80.1%

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

   2. associate-*l* 93.1%

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

3. Simplified 93.1%

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

   1. neg-sub0 93.1%

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

   2. sub-neg 93.1%

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

   3. add-sqr-sqrt 44.9%

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

   4. sqrt-unprod 50.8%

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

   5. sqr-neg 50.8%

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

   6. sqrt-unprod 13.0%

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

   7. add-sqr-sqrt 26.6%

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

6. Applied egg-rr 26.6%

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

   1. +-lft-identity 26.6%

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

8. Simplified 26.6%

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

9. Final simplification 26.6%

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

Alternative 6: 29.1% 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 80.1%

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

   1. add-sqr-sqrt 25.7%

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

   2. sqrt-unprod 26.7%

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

   3. sqr-neg 26.7%

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

   4. sqrt-unprod 26.6%

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

   5. add-sqr-sqrt 26.6%

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

   6. associate-*l* 26.3%

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

   7. swap-sqr 26.5%

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

4. Applied egg-rr 26.5%

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

Alternative 7: 29.1% 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 80.1%

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

   1. associate-*l* 71.0%

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

   2. associate-*r* 77.8%

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

   3. *-commutative 77.8%

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

   4. distribute-rgt-neg-in 77.8%

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

   5. distribute-rgt-neg-in 77.8%

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

   6. associate-*r* 89.9%

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

3. Simplified 89.9%

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

   1. neg-sub0 89.9%

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

   2. sub-neg 89.9%

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

   3. add-sqr-sqrt 42.2%

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

   4. sqrt-unprod 49.3%

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

   5. sqr-neg 49.3%

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

   6. sqrt-prod 15.3%

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

   7. add-sqr-sqrt 26.6%

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

6. Applied egg-rr 26.6%

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

   1. +-lft-identity 26.6%

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

8. Simplified 26.6%

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

9. Final simplification 26.6%

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

Reproduce

herbie shell --seed 2024177 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))