ab-angle->ABCF D

Percentage Accurate: 82.2% → 99.7%

Time: 8.6s

Alternatives: 7

Speedup: 0.9×

• 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: 82.2% 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

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

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 82.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 99.6%

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 31.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+110}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;0 - a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2e+110) (* (* b b) (* a a)) (- 0.0 (* a b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2e+110) {
		tmp = (b * b) * (a * a);
	} else {
		tmp = 0.0 - (a * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 2d+110) then
        tmp = (b * b) * (a * a)
    else
        tmp = 0.0d0 - (a * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 2e+110) {
		tmp = (b * b) * (a * a);
	} else {
		tmp = 0.0 - (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2e+110:
		tmp = (b * b) * (a * a)
	else:
		tmp = 0.0 - (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2e+110)
		tmp = Float64(Float64(b * b) * Float64(a * a));
	else
		tmp = Float64(0.0 - Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 2e+110)
		tmp = (b * b) * (a * a);
	else
		tmp = 0.0 - (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2e+110], N[(N[(b * b), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2e110

   1. Initial program 82.9%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 29.1%

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

   if 2e110 < b

   1. Initial program 79.9%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 73.8%

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

   4. Applied egg-rr 73.8%

      \[\leadsto -\color{blue}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a} \]
   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 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. *-commutative N/A

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

      2. rem-exp-log N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot e^{\log \left(a \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2}\right)}\right), a\right)\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot e^{\log \left(a \cdot b\right) \cdot \frac{1}{2} + \log \left(a \cdot b\right) \cdot \frac{1}{2}}\right), a\right)\right) \]

      6. count-2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot e^{2 \cdot \left(\log \left(a \cdot b\right) \cdot \frac{1}{2}\right)}\right), a\right)\right) \]

      7. exp-prod N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\log \left(a \cdot b\right) \cdot \frac{1}{2}\right)}\right), a\right)\right) \]

      8. rem-log-exp N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left(e^{\log \left(a \cdot b\right) \cdot \frac{1}{2}}\right)}\right), a\right)\right) \]

      9. pow-to-exp N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left({\left(a \cdot b\right)}^{\frac{1}{2}}\right)}\right), a\right)\right) \]

      10. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right), a\right)\right) \]

      11. log-prod N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) + \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right)}\right), a\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\frac{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) \cdot \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) - \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) \cdot \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) - \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right)}\right), a\right)\right) \]

   8. Applied egg-rr 23.0%

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

      1. *-rgt-identity N/A

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

      2. *-lowering-*.f64 23.0%

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

   10. Applied egg-rr 23.0%

       \[\leadsto -\color{blue}{b \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+110}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;0 - a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 31.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.25 \cdot 10^{+109}:\\ \;\;\;\;b \cdot \left(a \cdot \left(a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.25e+109) (* b (* a (* a b))) (- 0.0 (* a b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.25e+109) {
		tmp = b * (a * (a * b));
	} else {
		tmp = 0.0 - (a * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 3.25d+109) then
        tmp = b * (a * (a * b))
    else
        tmp = 0.0d0 - (a * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.25e+109) {
		tmp = b * (a * (a * b));
	} else {
		tmp = 0.0 - (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.25e+109:
		tmp = b * (a * (a * b))
	else:
		tmp = 0.0 - (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.25e+109)
		tmp = Float64(b * Float64(a * Float64(a * b)));
	else
		tmp = Float64(0.0 - Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.25e+109)
		tmp = b * (a * (a * b));
	else
		tmp = 0.0 - (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.25e+109], N[(b * N[(a * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.25e109

   1. Initial program 82.9%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 29.2%

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

   if 3.25e109 < b

   1. Initial program 79.9%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 73.8%

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

   4. Applied egg-rr 73.8%

      \[\leadsto -\color{blue}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a} \]
   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 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. *-commutative N/A

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

      2. rem-exp-log N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot e^{\log \left(a \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2}\right)}\right), a\right)\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot e^{\log \left(a \cdot b\right) \cdot \frac{1}{2} + \log \left(a \cdot b\right) \cdot \frac{1}{2}}\right), a\right)\right) \]

      6. count-2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot e^{2 \cdot \left(\log \left(a \cdot b\right) \cdot \frac{1}{2}\right)}\right), a\right)\right) \]

      7. exp-prod N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\log \left(a \cdot b\right) \cdot \frac{1}{2}\right)}\right), a\right)\right) \]

      8. rem-log-exp N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left(e^{\log \left(a \cdot b\right) \cdot \frac{1}{2}}\right)}\right), a\right)\right) \]

      9. pow-to-exp N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left({\left(a \cdot b\right)}^{\frac{1}{2}}\right)}\right), a\right)\right) \]

      10. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right), a\right)\right) \]

      11. log-prod N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) + \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right)}\right), a\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\frac{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) \cdot \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) - \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) \cdot \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) - \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right)}\right), a\right)\right) \]

   8. Applied egg-rr 23.0%

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

      1. *-rgt-identity N/A

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

      2. *-lowering-*.f64 23.0%

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

   10. Applied egg-rr 23.0%

       \[\leadsto -\color{blue}{b \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.25 \cdot 10^{+109}:\\ \;\;\;\;b \cdot \left(a \cdot \left(a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 31.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{+109}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;0 - a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3.8e+109) (* (* a b) (* a b)) (- 0.0 (* a b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3.8e+109) {
		tmp = (a * b) * (a * b);
	} else {
		tmp = 0.0 - (a * b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 3.8d+109) then
        tmp = (a * b) * (a * b)
    else
        tmp = 0.0d0 - (a * b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3.8e+109) {
		tmp = (a * b) * (a * b);
	} else {
		tmp = 0.0 - (a * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3.8e+109:
		tmp = (a * b) * (a * b)
	else:
		tmp = 0.0 - (a * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3.8e+109)
		tmp = Float64(Float64(a * b) * Float64(a * b));
	else
		tmp = Float64(0.0 - Float64(a * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3.8e+109)
		tmp = (a * b) * (a * b);
	else
		tmp = 0.0 - (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3.8e+109], N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.80000000000000039e109

   1. Initial program 82.9%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 29.1%

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

   if 3.80000000000000039e109 < b

   1. Initial program 79.9%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 73.8%

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

   4. Applied egg-rr 73.8%

      \[\leadsto -\color{blue}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a} \]
   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 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. *-commutative N/A

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

      2. rem-exp-log N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot e^{\log \left(a \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2}\right)}\right), a\right)\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot e^{\log \left(a \cdot b\right) \cdot \frac{1}{2} + \log \left(a \cdot b\right) \cdot \frac{1}{2}}\right), a\right)\right) \]

      6. count-2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot e^{2 \cdot \left(\log \left(a \cdot b\right) \cdot \frac{1}{2}\right)}\right), a\right)\right) \]

      7. exp-prod N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\log \left(a \cdot b\right) \cdot \frac{1}{2}\right)}\right), a\right)\right) \]

      8. rem-log-exp N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left(e^{\log \left(a \cdot b\right) \cdot \frac{1}{2}}\right)}\right), a\right)\right) \]

      9. pow-to-exp N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left({\left(a \cdot b\right)}^{\frac{1}{2}}\right)}\right), a\right)\right) \]

      10. sqr-pow N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right), a\right)\right) \]

      11. log-prod N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) + \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right)}\right), a\right)\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\frac{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) \cdot \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) - \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) \cdot \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) - \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right)}\right), a\right)\right) \]

   8. Applied egg-rr 23.0%

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

      1. *-rgt-identity N/A

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

      2. *-lowering-*.f64 23.0%

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

   10. Applied egg-rr 23.0%

       \[\leadsto -\color{blue}{b \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{+109}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(a \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;0 - a \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 18.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-15}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 1.65e-15) (* a b) -1.0))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.65e-15) {
		tmp = a * b;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 1.65d-15) then
        tmp = a * b
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 1.65e-15) {
		tmp = a * b;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 1.65e-15:
		tmp = a * b
	else:
		tmp = -1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.65e-15)
		tmp = Float64(a * b);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 1.65e-15)
		tmp = a * b;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.65e-15], N[(a * b), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if b < 1.65e-15

   1. Initial program 81.2%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 31.8%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 31.7%

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

   6. Applied egg-rr 31.7%

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

      1. *-commutative N/A

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

      2. rem-exp-log N/A

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

      3. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b \cdot e^{\log \left(a \cdot b\right) \cdot 1}\right), a\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b \cdot e^{\log \left(a \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2}\right)}\right), a\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b \cdot e^{\log \left(a \cdot b\right) \cdot \frac{1}{2} + \log \left(a \cdot b\right) \cdot \frac{1}{2}}\right), a\right) \]

      6. count-2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b \cdot e^{2 \cdot \left(\log \left(a \cdot b\right) \cdot \frac{1}{2}\right)}\right), a\right) \]

      7. exp-prod N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\log \left(a \cdot b\right) \cdot \frac{1}{2}\right)}\right), a\right) \]

      8. rem-log-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left(e^{\log \left(a \cdot b\right) \cdot \frac{1}{2}}\right)}\right), a\right) \]

      9. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left({\left(a \cdot b\right)}^{\frac{1}{2}}\right)}\right), a\right) \]

      10. sqr-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right), a\right) \]

      11. log-prod N/A

          \[\leadsto \mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) + \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right)}\right), a\right) \]

      12. flip-+ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\frac{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) \cdot \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) - \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) \cdot \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) - \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right)}\right), a\right) \]

   8. Applied egg-rr 25.1%

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

      1. *-rgt-identity N/A

         \[\leadsto b \cdot a \]

      2. *-lowering-*.f64 25.1%

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

   10. Applied egg-rr 25.1%

       \[\leadsto \color{blue}{b \cdot a} \]

   if 1.65e-15 < b

   1. Initial program 85.3%

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

      1. associate-*l* N/A

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

      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. Step-by-step derivation

      1. unpow1 N/A

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

      2. sqr-pow N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left({\left(a \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot b\right)}^{\left(\frac{1}{2}\right)}\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left({\left({\left(a \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)}^{2}\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({\left(a \cdot b\right)}^{\left(\frac{1}{2}\right)}\right), 2\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot b\right), \left(\frac{1}{2}\right)\right), 2\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\frac{1}{2}\right)\right), 2\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

      7. metadata-eval 45.5%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, b\right), \frac{1}{2}\right), 2\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   6. Applied egg-rr 45.5%

      \[\leadsto -\color{blue}{{\left({\left(a \cdot b\right)}^{0.5}\right)}^{2}} \cdot \left(a \cdot b\right) \]

   8. Applied egg-rr 4.4%

      \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-15}:\\ \;\;\;\;a \cdot b\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 21.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ 0 - a \cdot b \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 82.1%

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

4. Applied egg-rr 82.1%

   \[\leadsto -\color{blue}{\left(a \cdot \left(b \cdot b\right)\right) \cdot a} \]
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 95.4%

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

6. Applied egg-rr 95.4%

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

   1. *-commutative N/A

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

   2. rem-exp-log N/A

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

   3. *-rgt-identity N/A

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

   4. metadata-eval N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot e^{\log \left(a \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2}\right)}\right), a\right)\right) \]

   5. distribute-lft-out N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot e^{\log \left(a \cdot b\right) \cdot \frac{1}{2} + \log \left(a \cdot b\right) \cdot \frac{1}{2}}\right), a\right)\right) \]

   6. count-2 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot e^{2 \cdot \left(\log \left(a \cdot b\right) \cdot \frac{1}{2}\right)}\right), a\right)\right) \]

   7. exp-prod N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\log \left(a \cdot b\right) \cdot \frac{1}{2}\right)}\right), a\right)\right) \]

   8. rem-log-exp N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left(e^{\log \left(a \cdot b\right) \cdot \frac{1}{2}}\right)}\right), a\right)\right) \]

   9. pow-to-exp N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left({\left(a \cdot b\right)}^{\frac{1}{2}}\right)}\right), a\right)\right) \]

   10. sqr-pow N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right), a\right)\right) \]

   11. log-prod N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) + \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right)}\right), a\right)\right) \]

   12. flip-+ N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(b \cdot {\left(e^{2}\right)}^{\left(\frac{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) \cdot \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) - \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) \cdot \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}{\log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) - \log \left({\left(a \cdot b\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}\right)}\right), a\right)\right) \]

8. Applied egg-rr 23.6%

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

   1. *-rgt-identity N/A

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

   2. *-lowering-*.f64 23.6%

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

10. Applied egg-rr 23.6%

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

11. Final simplification 23.6%

    \[\leadsto 0 - a \cdot b \]
12. Add Preprocessing

Alternative 7: 4.2% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return -1.0

Julia

function code(a, b)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, b_] := -1.0

Derivation

1. Initial program 82.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 99.6%

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

4. Applied egg-rr 99.6%

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

   1. unpow1 N/A

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

   2. sqr-pow N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left({\left(a \cdot b\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(a \cdot b\right)}^{\left(\frac{1}{2}\right)}\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   3. pow2 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left({\left({\left(a \cdot b\right)}^{\left(\frac{1}{2}\right)}\right)}^{2}\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({\left(a \cdot b\right)}^{\left(\frac{1}{2}\right)}\right), 2\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\left(a \cdot b\right), \left(\frac{1}{2}\right)\right), 2\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

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

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, b\right), \left(\frac{1}{2}\right)\right), 2\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

   7. metadata-eval 56.7%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(a, b\right), \frac{1}{2}\right), 2\right), \mathsf{*.f64}\left(a, b\right)\right)\right) \]

6. Applied egg-rr 56.7%

   \[\leadsto -\color{blue}{{\left({\left(a \cdot b\right)}^{0.5}\right)}^{2}} \cdot \left(a \cdot b\right) \]

8. Applied egg-rr 4.3%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))