ab-angle->ABCF D

Percentage Accurate: 83.2% → 99.7%

Time: 6.5s

Alternatives: 5

Speedup: 1.0×

• 28.0% 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: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -\left(\left(a \cdot a\right) \cdot b\right) \cdot b \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* (* (* a a) b) b)))

C

double code(double a, double b) {
	return -(((a * a) * b) * b);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -(((a * a) * b) * b)
end function

Java

public static double code(double a, double b) {
	return -(((a * a) * b) * b);
}

Python

def code(a, b):
	return -(((a * a) * b) * b)

Julia

function code(a, b)
	return Float64(-Float64(Float64(Float64(a * a) * b) * b))
end

MATLAB

function tmp = code(a, b)
	tmp = -(((a * a) * b) * b);
end

Wolfram

code[a_, b_] := (-N[(N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision])

Alternative 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in a around 0 78.2%

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

   1. unpow2 78.2%

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

   2. unpow2 78.2%

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

   3. swap-sqr 99.7%

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

   4. unpow2 99.7%

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

5. Simplified 99.7%

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

7. Applied egg-rr 48.2%

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

   1. add-sqr-sqrt 10.9%

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

   2. sqrt-unprod 11.5%

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

   3. sqr-neg 11.5%

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

   4. sqrt-unprod 11.4%

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

   5. add-sqr-sqrt 11.4%

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

   6. associate-*r* 11.4%

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

   7. *-commutative 11.4%

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

   8. associate-*l* 11.4%

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

   9. *-commutative 11.4%

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

   10. associate-*l* 11.4%

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

   11. add-sqr-sqrt 27.3%

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

   12. *-commutative 27.3%

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

   13. associate-*r* 27.4%

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

   14. *-commutative 27.4%

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

   15. associate-*r* 27.4%

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

   16. remove-double-div 27.4%

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

   17. div-inv 27.4%

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

9. Applied egg-rr 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in a around 0 78.2%

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

   1. unpow2 78.2%

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

   2. unpow2 78.2%

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

   3. swap-sqr 99.7%

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

   4. 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. unpow2 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 28.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

   1. distribute-rgt-neg-in 85.6%

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

   2. associate-*l* 94.6%

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

3. Simplified 94.6%

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

   1. neg-sub0 94.6%

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

   2. sub-neg 94.6%

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

   3. add-sqr-sqrt 49.4%

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

   4. sqrt-unprod 53.3%

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

   5. sqr-neg 53.3%

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

   6. sqrt-unprod 11.4%

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

   7. add-sqr-sqrt 27.4%

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

6. Applied egg-rr 27.4%

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

   1. +-lft-identity 27.4%

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

8. Simplified 27.4%

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

9. Final simplification 27.4%

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

Alternative 4: 28.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

   1. add-sqr-sqrt 26.5%

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

   2. sqrt-unprod 27.5%

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

   3. sqr-neg 27.5%

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

   4. sqrt-unprod 27.4%

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

   5. add-sqr-sqrt 27.4%

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

   6. associate-*l* 27.2%

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

   7. swap-sqr 27.3%

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

4. Applied egg-rr 27.3%

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

Alternative 5: 28.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

   1. associate-*l* 78.2%

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

   2. associate-*r* 82.1%

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

   3. *-commutative 82.1%

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

   4. distribute-rgt-neg-in 82.1%

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

   5. distribute-rgt-neg-in 82.1%

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

   6. associate-*r* 93.6%

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

3. Simplified 93.6%

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

   1. neg-sub0 93.6%

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

   2. sub-neg 93.6%

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

   3. add-sqr-sqrt 41.1%

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

   4. sqrt-unprod 51.0%

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

   5. sqr-neg 51.0%

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

   6. sqrt-prod 14.6%

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

   7. add-sqr-sqrt 27.4%

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

6. Applied egg-rr 27.4%

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

   1. +-lft-identity 27.4%

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

8. Simplified 27.4%

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

9. Final simplification 27.4%

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

Reproduce

herbie shell --seed 2024123 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))