ab-angle->ABCF D

Percentage Accurate: 82.5% → 99.7%

Time: 4.8s

Alternatives: 7

Speedup: 1.0×

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.5% 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

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

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 (/ -1.0 b_m)) (* a b_m)))

C

b_m = fabs(b);
assert(a < b_m);
double code(double a, double b_m) {
	return (a / (-1.0 / 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 / ((-1.0d0) / 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 / (-1.0 / 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 / (-1.0 / 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 / Float64(-1.0 / 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 / (-1.0 / 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 / N[(-1.0 / b$95$m), $MachinePrecision]), $MachinePrecision] * N[(a * b$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.4%

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

   1. neg-sub0 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. lift-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. +-lft-identity N/A

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

   9. clear-num N/A

      \[\leadsto \left(a \cdot \color{blue}{\frac{1}{\frac{0 + b}{b \cdot \left(\mathsf{neg}\left(b\right)\right)}}}\right) \cdot \left(a \cdot b\right) \]

   10. un-div-inv N/A

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

   11. lower-/.f64 N/A

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

   12. +-lft-identity N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-/r* N/A

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

6. Applied egg-rr 99.7%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

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(-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 81.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. Final simplification 99.6%

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

Alternative 3: 94.2% accurate, 1.0× speedup

Math

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

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(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 81.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 94.4

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

5. Simplified 94.4%

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

6. Final simplification 94.4%

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

Alternative 4: 82.5% accurate, 1.0× speedup

Math

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

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 (* b_m (* a (- a)))))

C

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

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 * (b_m * (a * -a))
end function

Java

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

Python

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

Julia

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

MATLAB

b_m = abs(b);
a, b_m = num2cell(sort([a, b_m])){:}
function tmp = code(a, b_m)
	tmp = b_m * (b_m * (a * -a));
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[(b$95$m * N[(a * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.4%

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

3. Final simplification 81.4%

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

Alternative 5: 28.8% accurate, 1.1× speedup

Math

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

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 (* b_m (* a a))))

C

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

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 * (b_m * (a * a))
end function

Java

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

Python

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

Julia

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

MATLAB

b_m = abs(b);
a, b_m = num2cell(sort([a, b_m])){:}
function tmp = code(a, b_m)
	tmp = b_m * (b_m * (a * a));
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[(b$95$m * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. 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 31.6%

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

5. Final simplification 31.6%

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

Alternative 6: 28.7% 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 81.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. 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 31.5%

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

   1. swap-sqr N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 31.5

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

6. Applied egg-rr 31.5%

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

7. Final simplification 31.5%

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

Alternative 7: 28.6% 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 81.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. 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 31.5%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))