ab-angle->ABCF D

Percentage Accurate: 81.7% → 99.7%

Time: 8.4s

Alternatives: 7

Speedup: 0.9×

• 27.8% 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: 81.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{b \cdot a}{\frac{\frac{-1}{b}}{a}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (* b a) (/ (/ -1.0 b) a)))

C

double code(double a, double b) {
	return (b * a) / ((-1.0 / b) / a);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (b * a) / (((-1.0d0) / b) / a)
end function

Java

public static double code(double a, double b) {
	return (b * a) / ((-1.0 / b) / a);
}

Python

def code(a, b):
	return (b * a) / ((-1.0 / b) / a)

Julia

function code(a, b)
	return Float64(Float64(b * a) / Float64(Float64(-1.0 / b) / a))
end

MATLAB

function tmp = code(a, b)
	tmp = (b * a) / ((-1.0 / b) / a);
end

Wolfram

code[a_, b_] := N[(N[(b * a), $MachinePrecision] / N[(N[(-1.0 / b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.1%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\left(b \cdot b\right) \cdot \left(a \cdot a\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right) \]

   5. *-lowering-*.f64 77.4%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right) \]

4. Applied egg-rr 77.4%

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

   1. *-commutative N/A

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

   2. swap-sqr N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. neg-mul-1 N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. associate-/r/ N/A

      \[\leadsto \frac{a \cdot b}{\color{blue}{\frac{-1}{a \cdot b}}} \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(a \cdot b\right), \color{blue}{\left(\frac{-1}{a \cdot b}\right)}\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\left(b \cdot a\right), \left(\frac{\color{blue}{-1}}{a \cdot b}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, a\right), \left(\frac{\color{blue}{-1}}{a \cdot b}\right)\right) \]

   12. associate-/l/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, a\right), \left(\frac{\frac{-1}{b}}{\color{blue}{a}}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, a\right), \left(\frac{\frac{\mathsf{neg}\left(1\right)}{b}}{a}\right)\right) \]

   14. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, a\right), \left(\frac{\mathsf{neg}\left(\frac{1}{b}\right)}{a}\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, a\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{b}\right)\right), \color{blue}{a}\right)\right) \]

   16. distribute-neg-frac N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, a\right), \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{b}\right), a\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, a\right), \mathsf{/.f64}\left(\left(\frac{-1}{b}\right), a\right)\right) \]

   18. /-lowering-/.f64 99.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, a\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, b\right), a\right)\right) \]

6. Applied egg-rr 99.8%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{b \cdot a}{\frac{\frac{-1}{a}}{b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (* b a) (/ (/ -1.0 a) b)))

C

double code(double a, double b) {
	return (b * a) / ((-1.0 / a) / b);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (b * a) / (((-1.0d0) / a) / b)
end function

Java

public static double code(double a, double b) {
	return (b * a) / ((-1.0 / a) / b);
}

Python

def code(a, b):
	return (b * a) / ((-1.0 / a) / b)

Julia

function code(a, b)
	return Float64(Float64(b * a) / Float64(Float64(-1.0 / a) / b))
end

MATLAB

function tmp = code(a, b)
	tmp = (b * a) / ((-1.0 / a) / b);
end

Wolfram

code[a_, b_] := N[(N[(b * a), $MachinePrecision] / N[(N[(-1.0 / a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.1%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(a \cdot \left(a \cdot b\right)\right), b\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\left(a \cdot b\right) \cdot a\right), b\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot b\right), a\right), b\right)\right) \]

   4. *-lowering-*.f64 94.5%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), a\right), b\right)\right) \]

4. Applied egg-rr 94.5%

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

   1. associate-*r* N/A

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

   2. distribute-lft-neg-in N/A

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

   3. remove-double-div N/A

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

   4. un-div-inv N/A

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

   5. distribute-neg-frac N/A

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

   6. distribute-neg-frac2 N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(a \cdot b\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{a \cdot b}\right)\right)}\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{a \cdot b}}\right)\right)\right) \]

   9. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{a \cdot b}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\frac{-1}{\color{blue}{a} \cdot b}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{/.f64}\left(-1, \color{blue}{\left(a \cdot b\right)}\right)\right) \]

   12. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(a, \color{blue}{b}\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{a \cdot b}{\frac{-1}{a \cdot b}}} \]
7. Step-by-step derivation

   1. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\frac{\frac{-1}{a}}{\color{blue}{b}}\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\frac{-1 \cdot \frac{1}{a}}{b}\right)\right) \]

   3. neg-mul-1 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\frac{\mathsf{neg}\left(\frac{1}{a}\right)}{b}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{a}\right)\right), \color{blue}{b}\right)\right) \]

   5. neg-mul-1 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{/.f64}\left(\left(-1 \cdot \frac{1}{a}\right), b\right)\right) \]

   6. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{/.f64}\left(\left(\frac{-1}{a}\right), b\right)\right) \]

   7. /-lowering-/.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(-1, a\right), b\right)\right) \]

8. Applied egg-rr 99.7%

   \[\leadsto \frac{a \cdot b}{\color{blue}{\frac{\frac{-1}{a}}{b}}} \]

9. Final simplification 99.7%

   \[\leadsto \frac{b \cdot a}{\frac{\frac{-1}{a}}{b}} \]
10. Add Preprocessing

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(b \cdot a\right) \cdot \left(0 - b \cdot a\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (* b a) (- 0.0 (* b a))))

C

double code(double a, double b) {
	return (b * a) * (0.0 - (b * a));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (b * a) * (0.0d0 - (b * a))
end function

Java

public static double code(double a, double b) {
	return (b * a) * (0.0 - (b * a));
}

Python

def code(a, b):
	return (b * a) * (0.0 - (b * a))

Julia

function code(a, b)
	return Float64(Float64(b * a) * Float64(0.0 - Float64(b * a)))
end

MATLAB

function tmp = code(a, b)
	tmp = (b * a) * (0.0 - (b * a));
end

Wolfram

code[a_, b_] := N[(N[(b * a), $MachinePrecision] * N[(0.0 - N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.1%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\left(a \cdot a\right) \cdot \left(b \cdot b\right)\right)\right) \]

   2. unswap-sqr N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(a \cdot b\right), \left(a \cdot b\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(a \cdot b\right)\right)\right) \]

   5. *-lowering-*.f64 99.7%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, b\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

   \[\leadsto \left(b \cdot a\right) \cdot \left(0 - b \cdot a\right) \]
6. Add Preprocessing

Alternative 4: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(a \cdot \left(b \cdot b\right)\right) \cdot \left(0 - a\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (* a (* b b)) (- 0.0 a)))

C

double code(double a, double b) {
	return (a * (b * b)) * (0.0 - a);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * (b * b)) * (0.0d0 - a)
end function

Java

public static double code(double a, double b) {
	return (a * (b * b)) * (0.0 - a);
}

Python

def code(a, b):
	return (a * (b * b)) * (0.0 - a)

Julia

function code(a, b)
	return Float64(Float64(a * Float64(b * b)) * Float64(0.0 - a))
end

MATLAB

function tmp = code(a, b)
	tmp = (a * (b * b)) * (0.0 - a);
end

Wolfram

code[a_, b_] := N[(N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.0 - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.1%

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

3. Taylor expanded in a around 0

   \[\leadsto \mathsf{neg.f64}\left(\color{blue}{\left({a}^{2} \cdot {b}^{2}\right)}\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\left(a \cdot a\right) \cdot {b}^{2}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(a \cdot \left(a \cdot {b}^{2}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(a \cdot \left({b}^{2} \cdot a\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left({b}^{2} \cdot a\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \left(a \cdot {b}^{2}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left({b}^{2}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \left(b \cdot b\right)\right)\right)\right) \]

   8. *-lowering-*.f64 83.4%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(b, b\right)\right)\right)\right) \]

5. Simplified 83.4%

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

6. Final simplification 83.4%

   \[\leadsto \left(a \cdot \left(b \cdot b\right)\right) \cdot \left(0 - a\right) \]
7. Add Preprocessing

Alternative 5: 28.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

   1. +-lft-identity N/A

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

   2. flip3-+ N/A

      \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}\right) \]

   3. distribute-neg-frac N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left({0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}\right)\right)}{\color{blue}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}} \]

4. Applied egg-rr 32.3%

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

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(a \cdot a\right) \cdot b\right), b\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(a \cdot a\right), b\right), b\right) \]

   3. *-lowering-*.f64 32.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right), b\right) \]

6. Applied egg-rr 32.3%

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

7. Final simplification 32.3%

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

Alternative 6: 28.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

   1. +-lft-identity N/A

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

   2. flip3-+ N/A

      \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}\right) \]

   3. distribute-neg-frac N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left({0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}\right)\right)}{\color{blue}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}} \]

4. Applied egg-rr 32.3%

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

5. Final simplification 32.3%

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

Alternative 7: 28.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

   1. +-lft-identity N/A

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

   2. flip3-+ N/A

      \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}\right) \]

   3. distribute-neg-frac N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left({0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}\right)\right)}{\color{blue}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}} \]

4. Applied egg-rr 32.2%

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

5. Final simplification 32.2%

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

Reproduce

herbie shell --seed 2024185 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))