ab-angle->ABCF D

Percentage Accurate: 82.1% → 99.7%

Time: 6.7s

Alternatives: 8

Speedup: 1.0×

• 27.7% 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: 82.1% 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, 1.0× speedup

Math

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

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 * Float64(-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 80.2%

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

3. Taylor expanded in a around 0 73.5%

   \[\leadsto \color{blue}{-1 \cdot \left({a}^{2} \cdot {b}^{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg 73.5%

      \[\leadsto \color{blue}{-{a}^{2} \cdot {b}^{2}} \]

   2. unpow2 73.5%

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

   3. unpow2 73.5%

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

   4. swap-sqr 99.6%

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

   5. unpow2 99.6%

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

5. Simplified 99.6%

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

   1. neg-mul-1 99.6%

      \[\leadsto \color{blue}{-1 \cdot {\left(a \cdot b\right)}^{2}} \]

   2. unpow2 99.6%

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

   3. associate-*r* 99.6%

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

7. Applied egg-rr 99.6%

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

   1. mul-1-neg 99.6%

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

   2. neg-sub0 99.6%

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

9. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\left(0 - a \cdot b\right)} \cdot \left(a \cdot b\right) \]
10. Step-by-step derivation

    1. neg-sub0 99.6%

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

    2. distribute-lft-neg-in 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 2: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(-b\right)\\ \mathbf{if}\;a \leq 5.4 \cdot 10^{-199}:\\ \;\;\;\;a \cdot \left(b \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* a (- b))))
   (if (<= a 5.4e-199) (* a (* b t_0)) (* b (* a t_0)))))

C

double code(double a, double b) {
	double t_0 = a * -b;
	double tmp;
	if (a <= 5.4e-199) {
		tmp = a * (b * t_0);
	} else {
		tmp = b * (a * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * -b
    if (a <= 5.4d-199) then
        tmp = a * (b * t_0)
    else
        tmp = b * (a * t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = a * -b;
	double tmp;
	if (a <= 5.4e-199) {
		tmp = a * (b * t_0);
	} else {
		tmp = b * (a * t_0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = a * -b
	tmp = 0
	if a <= 5.4e-199:
		tmp = a * (b * t_0)
	else:
		tmp = b * (a * t_0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(a * Float64(-b))
	tmp = 0.0
	if (a <= 5.4e-199)
		tmp = Float64(a * Float64(b * t_0));
	else
		tmp = Float64(b * Float64(a * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = a * -b;
	tmp = 0.0;
	if (a <= 5.4e-199)
		tmp = a * (b * t_0);
	else
		tmp = b * (a * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(a * (-b)), $MachinePrecision]}, If[LessEqual[a, 5.4e-199], N[(a * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision], N[(b * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 5.39999999999999979e-199

   1. Initial program 76.6%

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

      1. associate-*l* 70.2%

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

      2. associate-*r* 79.5%

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

      3. *-commutative 79.5%

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

      4. distribute-rgt-neg-in 79.5%

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

      5. distribute-rgt-neg-in 79.5%

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

      6. associate-*r* 96.3%

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

   3. Simplified 96.3%

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

   if 5.39999999999999979e-199 < a

   1. Initial program 85.6%

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

      1. distribute-rgt-neg-in 85.6%

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

      2. associate-*l* 96.9%

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

   3. Simplified 96.9%

      \[\leadsto \color{blue}{\left(a \cdot \left(a \cdot b\right)\right) \cdot \left(-b\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.4 \cdot 10^{-199}:\\ \;\;\;\;a \cdot \left(b \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.2 \cdot 10^{-154}:\\ \;\;\;\;a \cdot \left(b \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(a \cdot \left(-a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 2.2e-154) (* a (* b (* a (- b)))) (* b (* b (* a (- a))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 2.2e-154) {
		tmp = a * (b * (a * -b));
	} else {
		tmp = b * (b * (a * -a));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= 2.2d-154) then
        tmp = a * (b * (a * -b))
    else
        tmp = b * (b * (a * -a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 2.2e-154) {
		tmp = a * (b * (a * -b));
	} else {
		tmp = b * (b * (a * -a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 2.2e-154:
		tmp = a * (b * (a * -b))
	else:
		tmp = b * (b * (a * -a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 2.2e-154)
		tmp = Float64(a * Float64(b * Float64(a * Float64(-b))));
	else
		tmp = Float64(b * Float64(b * Float64(a * Float64(-a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 2.2e-154)
		tmp = a * (b * (a * -b));
	else
		tmp = b * (b * (a * -a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 2.2e-154], N[(a * N[(b * N[(a * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(a * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.20000000000000007e-154

   1. Initial program 76.9%

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

      1. associate-*l* 70.7%

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

      2. associate-*r* 79.6%

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

      3. *-commutative 79.6%

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

      4. distribute-rgt-neg-in 79.6%

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

      5. distribute-rgt-neg-in 79.6%

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

      6. associate-*r* 96.5%

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

   3. Simplified 96.5%

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

   if 2.20000000000000007e-154 < a

   1. Initial program 85.8%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.2 \cdot 10^{-154}:\\ \;\;\;\;a \cdot \left(b \cdot \left(a \cdot \left(-b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(a \cdot \left(-a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 1.0× speedup

Math

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

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 * Float64(-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 80.2%

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

3. Final simplification 80.2%

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

Alternative 5: 75.7% accurate, 1.0× speedup

Math

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

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(a * a) * Float64(b * Float64(-b)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

3. Taylor expanded in a around 0 73.5%

   \[\leadsto -\color{blue}{{a}^{2} \cdot {b}^{2}} \]
4. Step-by-step derivation

   1. unpow2 73.5%

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

5. Applied egg-rr 73.5%

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

   1. unpow2 73.5%

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

7. Applied egg-rr 73.5%

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

8. Final simplification 73.5%

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

Alternative 6: 29.1% 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 80.2%

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

   1. distribute-rgt-neg-in 80.2%

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

   2. associate-*l* 94.2%

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

3. Simplified 94.2%

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

   1. neg-sub0 94.2%

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

   2. sub-neg 94.2%

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

   3. add-sqr-sqrt 50.4%

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

   4. sqrt-unprod 51.4%

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

   5. sqr-neg 51.4%

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

   6. sqrt-unprod 9.9%

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

   7. add-sqr-sqrt 24.6%

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

6. Applied egg-rr 24.6%

   \[\leadsto \left(a \cdot \left(a \cdot b\right)\right) \cdot \color{blue}{\left(0 + b\right)} \]
7. Step-by-step derivation

   1. +-lft-identity 24.6%

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

8. Simplified 24.6%

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

9. Final simplification 24.6%

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

Alternative 7: 29.1% 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 80.2%

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

   1. add-sqr-sqrt 23.7%

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

   2. sqrt-unprod 24.6%

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

   3. sqr-neg 24.6%

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

   4. sqrt-unprod 24.5%

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

   5. add-sqr-sqrt 24.5%

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

   6. associate-*l* 24.4%

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

   7. swap-sqr 24.6%

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

4. Applied egg-rr 24.6%

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

Alternative 8: 29.1% accurate, 1.1× speedup

Math

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

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(a * Float64(b * Float64(a * b)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. associate-*l* 73.5%

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

   2. associate-*r* 80.4%

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

   3. *-commutative 80.4%

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

   4. distribute-rgt-neg-in 80.4%

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

   5. distribute-rgt-neg-in 80.4%

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

   6. associate-*r* 95.7%

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

3. Simplified 95.7%

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

   1. neg-sub0 95.7%

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

   2. sub-neg 95.7%

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

   3. add-sqr-sqrt 52.0%

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

   4. sqrt-unprod 53.6%

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

   5. sqr-neg 53.6%

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

   6. sqrt-prod 12.5%

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

   7. add-sqr-sqrt 24.6%

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

6. Applied egg-rr 24.6%

   \[\leadsto a \cdot \left(b \cdot \left(b \cdot \color{blue}{\left(0 + a\right)}\right)\right) \]
7. Step-by-step derivation

   1. +-lft-identity 24.6%

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

8. Simplified 24.6%

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

9. Final simplification 24.6%

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

Reproduce

herbie shell --seed 2024137 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))