ab-angle->ABCF D

Percentage Accurate: 82.8% → 99.7%

Time: 8.0s

Alternatives: 5

Speedup: 0.9×

• 28.4% 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: 82.8% 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

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

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 86.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. *-lowering-*.f64 85.3%

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

4. Applied egg-rr 85.3%

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

   1. associate-*r* N/A

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

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

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

   3. *-lowering-*.f64 95.8%

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

6. Applied egg-rr 95.8%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. /-rgt-identity N/A

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

   4. associate-/r/ N/A

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

   5. associate-*r* N/A

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

   6. associate-/l/ N/A

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

   7. associate-/r/ N/A

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

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

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

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

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

   10. /-lowering-/.f64 N/A

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

   11. *-lowering-*.f64 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

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

Alternative 3: 29.6% 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 86.4%

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

   1. associate-*l* N/A

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

   2. associate-*r* N/A

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

   3. +-lft-identity N/A

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

   4. flip3-+ N/A

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

   5. distribute-neg-frac N/A

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

4. Applied egg-rr 30.3%

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

5. Final simplification 30.3%

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

Alternative 4: 29.5% 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 86.4%

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

   1. associate-*l* N/A

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

   2. associate-*r* N/A

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

   3. +-lft-identity N/A

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

   4. flip3-+ N/A

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

   5. distribute-neg-frac N/A

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

4. Applied egg-rr 30.2%

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

   1. associate-*r* N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. *-lowering-*.f64 30.2%

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

6. Applied egg-rr 30.2%

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

7. Final simplification 30.2%

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

Alternative 5: 29.5% 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 86.4%

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

   1. associate-*l* N/A

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

   2. associate-*r* N/A

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

   3. +-lft-identity N/A

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

   4. flip3-+ N/A

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

   5. distribute-neg-frac N/A

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

4. Applied egg-rr 30.2%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))