ab-angle->ABCF D

Percentage Accurate: 82.7% → 99.7%

Time: 9.3s

Alternatives: 6

Speedup: 1.0×

• 28.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. associate-*l* N/A

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

   2. unswap-sqr N/A

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

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

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

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

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

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

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

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

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

   7. *-lowering-*.f64 99.6

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

4. Applied egg-rr 99.6%

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

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

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

   2. neg-sub0 N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

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

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

   7. +-lft-identity N/A

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

   8. clear-num N/A

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

   9. metadata-eval N/A

      \[\leadsto \left(a \cdot b\right) \cdot \frac{a}{\frac{\color{blue}{\frac{2}{2}}}{\frac{0 \cdot 0 - b \cdot b}{b}}} \]

   10. +-lft-identity N/A

       \[\leadsto \left(a \cdot b\right) \cdot \frac{a}{\frac{\frac{2}{2}}{\frac{0 \cdot 0 - b \cdot b}{\color{blue}{0 + b}}}} \]

   11. flip-- N/A

       \[\leadsto \left(a \cdot b\right) \cdot \frac{a}{\frac{\frac{2}{2}}{\color{blue}{0 - b}}} \]

   12. neg-sub0 N/A

       \[\leadsto \left(a \cdot b\right) \cdot \frac{a}{\frac{\frac{2}{2}}{\color{blue}{\mathsf{neg}\left(b\right)}}} \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \left(a \cdot b\right) \cdot \frac{a}{\color{blue}{\frac{\frac{2}{2}}{\mathsf{neg}\left(b\right)}}} \]

   14. metadata-eval N/A

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

   15. neg-lowering-neg.f64 99.7

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

6. Applied egg-rr 99.7%

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

   1. metadata-eval N/A

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

   2. frac-2neg N/A

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

   3. /-lowering-/.f64 99.7

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

8. Applied egg-rr 99.7%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. associate-*l* N/A

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

   2. unswap-sqr N/A

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

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

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

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

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

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

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

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

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

   7. *-lowering-*.f64 99.6

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 82.7% accurate, 1.0× 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(-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 78.3%

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

3. Final simplification 78.3%

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

Alternative 4: 29.4% 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 78.3%

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

   1. +-lft-identity N/A

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

   2. flip3-+ N/A

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

   3. 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 28.8%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

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

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

   7. *-lowering-*.f64 28.9

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

6. Applied egg-rr 28.9%

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

7. Final simplification 28.9%

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

Alternative 5: 29.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. +-lft-identity N/A

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

   2. flip3-+ N/A

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

   3. 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 28.8%

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

5. Final simplification 28.8%

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

Alternative 6: 29.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. +-lft-identity N/A

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

   2. flip3-+ N/A

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

   3. 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 28.8%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))