ab-angle->ABCF D

Percentage Accurate: 82.1% → 99.6%

Time: 5.3s

Alternatives: 8

Speedup: 1.0×

• 27.6% 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: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double a, double b) {
	return (b * a) * (1.0 / (-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 / ((-1.0d0) / (b * a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

   1. add-sqr-sqrt 29.3%

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

   2. sqrt-unprod 30.6%

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

   3. sqr-neg 30.6%

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

   4. sqrt-unprod 30.4%

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

   5. add-sqr-sqrt 30.4%

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

   6. associate-*l* 30.3%

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

   7. swap-sqr 30.4%

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

4. Applied egg-rr 30.4%

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

   1. add-log-exp 27.4%

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

   2. log-pow 27.4%

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

   3. add-sqr-sqrt 26.5%

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

   4. sqrt-unprod 43.6%

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

   5. sqr-neg 43.6%

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

   6. sqrt-unprod 39.4%

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

   7. add-sqr-sqrt 53.2%

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

   8. neg-sub0 53.2%

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

   9. pow-sub 53.2%

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

   10. metadata-eval 53.2%

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

   11. exp-prod 60.0%

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

   12. *-commutative 60.0%

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

   13. neg-log 60.1%

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

   14. add-log-exp 99.6%

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

6. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. distribute-rgt-neg-in 99.6%

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

8. Simplified 99.6%

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

   1. add-cube-cbrt 98.9%

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

   2. pow3 99.0%

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

10. Applied egg-rr 99.0%

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

12. Applied egg-rr 99.6%

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

13. Final simplification 99.6%

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

Alternative 2: 88.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot \left(b \cdot \left(a \cdot a\right)\right) \leq 5 \cdot 10^{+99}:\\ \;\;\;\;a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(a \cdot \left(-a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b (* b (* a a))) 5e+99)
   (* a (* b (* b (- a))))
   (* b (* b (* a (- a))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * (b * (a * a))) <= 5e+99) {
		tmp = a * (b * (b * -a));
	} 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 ((b * (b * (a * a))) <= 5d+99) then
        tmp = a * (b * (b * -a))
    else
        tmp = b * (b * (a * -a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * (b * (a * a))) <= 5e+99) {
		tmp = a * (b * (b * -a));
	} else {
		tmp = b * (b * (a * -a));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * (b * (a * a))) <= 5e+99:
		tmp = a * (b * (b * -a))
	else:
		tmp = b * (b * (a * -a))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * Float64(b * Float64(a * a))) <= 5e+99)
		tmp = Float64(a * Float64(b * Float64(b * Float64(-a))));
	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 ((b * (b * (a * a))) <= 5e+99)
		tmp = a * (b * (b * -a));
	else
		tmp = b * (b * (a * -a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * N[(b * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+99], N[(a * N[(b * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(a * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 a a) b) b) < 5.00000000000000008e99

   1. Initial program 85.0%

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

      1. associate-*l* 77.9%

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

      2. associate-*r* 83.3%

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

      3. *-commutative 83.3%

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

      4. distribute-rgt-neg-in 83.3%

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

      5. distribute-rgt-neg-in 83.3%

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

      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

   if 5.00000000000000008e99 < (*.f64 (*.f64 (*.f64 a a) b) b)

   1. Initial program 81.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot \left(b \cdot \left(a \cdot a\right)\right) \leq 5 \cdot 10^{+99}:\\ \;\;\;\;a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(a \cdot \left(-a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(-a\right)\\ \mathbf{if}\;a \leq 1.45 \cdot 10^{-234}:\\ \;\;\;\;a \cdot \left(b \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* b (- a))))
   (if (<= a 1.45e-234) (* a (* b t_0)) (* b (* a t_0)))))

C

double code(double a, double b) {
	double t_0 = b * -a;
	double tmp;
	if (a <= 1.45e-234) {
		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 = b * -a
    if (a <= 1.45d-234) 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 = b * -a;
	double tmp;
	if (a <= 1.45e-234) {
		tmp = a * (b * t_0);
	} else {
		tmp = b * (a * t_0);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = b * -a
	tmp = 0
	if a <= 1.45e-234:
		tmp = a * (b * t_0)
	else:
		tmp = b * (a * t_0)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(b * Float64(-a))
	tmp = 0.0
	if (a <= 1.45e-234)
		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 = b * -a;
	tmp = 0.0;
	if (a <= 1.45e-234)
		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[(b * (-a)), $MachinePrecision]}, If[LessEqual[a, 1.45e-234], 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 < 1.45000000000000008e-234

   1. Initial program 79.0%

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

      1. associate-*l* 76.1%

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

      2. associate-*r* 81.8%

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

      3. *-commutative 81.8%

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

      4. distribute-rgt-neg-in 81.8%

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

      5. distribute-rgt-neg-in 81.8%

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

      6. associate-*r* 94.2%

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

   3. Simplified 94.2%

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

   if 1.45000000000000008e-234 < a

   1. Initial program 90.2%

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

      1. distribute-rgt-neg-in 90.2%

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

      2. associate-*l* 98.4%

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

   3. Simplified 98.4%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.45 \cdot 10^{-234}:\\ \;\;\;\;a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a \cdot \left(b \cdot \left(-a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

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 * Float64(-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 83.4%

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

   1. add-sqr-sqrt 29.3%

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

   2. sqrt-unprod 30.6%

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

   3. sqr-neg 30.6%

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

   4. sqrt-unprod 30.4%

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

   5. add-sqr-sqrt 30.4%

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

   6. associate-*l* 30.3%

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

   7. swap-sqr 30.4%

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

4. Applied egg-rr 30.4%

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

   1. add-log-exp 27.4%

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

   2. log-pow 27.4%

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

   3. add-sqr-sqrt 26.5%

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

   4. sqrt-unprod 43.6%

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

   5. sqr-neg 43.6%

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

   6. sqrt-unprod 39.4%

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

   7. add-sqr-sqrt 53.2%

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

   8. neg-sub0 53.2%

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

   9. pow-sub 53.2%

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

   10. metadata-eval 53.2%

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

   11. exp-prod 60.0%

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

   12. *-commutative 60.0%

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

   13. neg-log 60.1%

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

   14. add-log-exp 99.6%

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

6. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. distribute-rgt-neg-in 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 5: 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 83.4%

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

3. Final simplification 83.4%

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

Alternative 6: 29.2% 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 83.4%

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

   1. distribute-rgt-neg-in 83.4%

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

   2. associate-*l* 95.1%

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

3. Simplified 95.1%

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

   1. neg-sub0 95.1%

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

   2. sub-neg 95.1%

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

   3. add-sqr-sqrt 46.5%

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

   4. sqrt-unprod 53.3%

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

   5. sqr-neg 53.3%

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

   6. sqrt-unprod 13.6%

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

   7. add-sqr-sqrt 30.4%

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

6. Applied egg-rr 30.4%

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

   1. +-lft-identity 30.4%

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

8. Simplified 30.4%

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

9. Final simplification 30.4%

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

Alternative 7: 29.1% 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 83.4%

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

   1. add-sqr-sqrt 29.3%

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

   2. sqrt-unprod 30.6%

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

   3. sqr-neg 30.6%

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

   4. sqrt-unprod 30.4%

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

   5. add-sqr-sqrt 30.4%

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

   6. associate-*l* 30.3%

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

   7. swap-sqr 30.4%

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

4. Applied egg-rr 30.4%

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

5. Final simplification 30.4%

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

Alternative 8: 29.2% accurate, 1.1× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

   1. associate-*l* 77.5%

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

   2. associate-*r* 83.3%

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

   3. *-commutative 83.3%

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

   4. distribute-rgt-neg-in 83.3%

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

   5. distribute-rgt-neg-in 83.3%

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

   6. associate-*r* 94.6%

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

3. Simplified 94.6%

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

   1. neg-sub0 94.6%

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

   2. sub-neg 94.6%

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

   3. add-sqr-sqrt 49.1%

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

   4. sqrt-unprod 56.4%

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

   5. sqr-neg 56.4%

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

   6. sqrt-prod 14.6%

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

   7. add-sqr-sqrt 30.4%

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

6. Applied egg-rr 30.4%

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

   1. +-lft-identity 30.4%

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

8. Simplified 30.4%

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

Reproduce

herbie shell --seed 2024181 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))