ab-angle->ABCF D

Percentage Accurate: 82.7% → 99.7%

Time: 7.3s

Alternatives: 7

Speedup: 1.0×

• 28.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.7% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{b}{\frac{-1}{a}} \cdot \left(b \cdot a\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. unswap-sqr N/A

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

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

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

   8. lower-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. neg-sub0 N/A

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

   4. flip3-- N/A

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

   5. clear-num N/A

      \[\leadsto \left(a \cdot \color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}{{0}^{3} - {b}^{3}}}}\right) \cdot \left(a \cdot b\right) \]

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval N/A

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

   9. +-lft-identity N/A

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

   10. mul0-lft N/A

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

   11. +-rgt-identity N/A

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

   12. clear-num N/A

       \[\leadsto \frac{a}{\color{blue}{\frac{1}{\frac{{0}^{3} - {b}^{3}}{b \cdot b}}}} \cdot \left(a \cdot b\right) \]

   13. *-inverses N/A

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

   14. +-rgt-identity N/A

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

   15. mul0-lft N/A

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

   16. +-lft-identity N/A

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

   17. metadata-eval N/A

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

   18. flip3-- N/A

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

   19. neg-sub0 N/A

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

   20. lift-neg.f64 N/A

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

   21. associate-/r* N/A

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

   22. *-commutative N/A

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

6. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. *-commutative N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(b \cdot a\right) \cdot \frac{a}{\frac{-1}{b}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. unswap-sqr N/A

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

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

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

   8. lower-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. neg-sub0 N/A

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

   4. flip3-- N/A

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

   5. clear-num N/A

      \[\leadsto \left(a \cdot \color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}{{0}^{3} - {b}^{3}}}}\right) \cdot \left(a \cdot b\right) \]

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. metadata-eval N/A

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

   9. +-lft-identity N/A

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

   10. mul0-lft N/A

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

   11. +-rgt-identity N/A

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

   12. clear-num N/A

       \[\leadsto \frac{a}{\color{blue}{\frac{1}{\frac{{0}^{3} - {b}^{3}}{b \cdot b}}}} \cdot \left(a \cdot b\right) \]

   13. *-inverses N/A

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

   14. +-rgt-identity N/A

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

   15. mul0-lft N/A

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

   16. +-lft-identity N/A

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

   17. metadata-eval N/A

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

   18. flip3-- N/A

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

   19. neg-sub0 N/A

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

   20. lift-neg.f64 N/A

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

   21. associate-/r* N/A

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

   22. *-commutative N/A

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

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(Float64(b * a) * Float64(b * Float64(-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 78.7%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. unswap-sqr N/A

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

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

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

   8. lower-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 94.1% accurate, 1.0× speedup

Math

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

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 * Float64(-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 78.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 92.5

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

4. Applied rewrites 92.5%

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

5. Final simplification 92.5%

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

Alternative 5: 82.7% accurate, 1.0× speedup

Math

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

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(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 78.7%

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

3. Final simplification 78.7%

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

Alternative 6: 28.6% 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 78.7%

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

   1. lift-neg.f64 N/A

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

   2. +-lft-identity N/A

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

   3. flip3-+ N/A

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

   4. 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 rewrites 29.3%

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

5. Final simplification 29.3%

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

Alternative 7: 28.5% 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 78.7%

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

   1. lift-neg.f64 N/A

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

   2. +-lft-identity N/A

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

   3. flip3-+ N/A

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

   4. 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 rewrites 29.2%

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

5. Final simplification 29.2%

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

Reproduce

herbie shell --seed 2024237 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))