ab-angle->ABCF D

Percentage Accurate: 82.4% → 99.7%

Time: 3.5s

Alternatives: 7

Speedup: 1.0×

• 28.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-*.f64 N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. sqrt-pow1 N/A

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

   8. pow2 N/A

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

   9. sqr-neg-rev N/A

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

   10. sqrt-prod N/A

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

   11. rem-square-sqrt N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. unswap-sqr N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 26.6

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

4. Applied rewrites 26.6%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-prod N/A

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

   3. sqr-neg-rev N/A

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

   4. pow2 N/A

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

   5. sqrt-pow1 N/A

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

   6. metadata-eval N/A

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

   7. unpow1 N/A

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

   8. lower-neg.f64 99.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. remove-double-div N/A

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

   5. unpow-1 N/A

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

   6. metadata-eval N/A

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

   7. associate-/l* N/A

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

   8. *-rgt-identity N/A

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

   9. distribute-frac-neg N/A

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

   10. frac-2neg N/A

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

   11. remove-double-neg N/A

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

   12. lower-/.f64 N/A

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

   13. metadata-eval N/A

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

   14. unpow-1 N/A

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

   15. distribute-neg-frac N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 N/A

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

   19. metadata-eval 99.7

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

8. Applied rewrites 99.7%

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

Alternative 2: 89.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-299}:\\ \;\;\;\;-\left(\left(b \cdot a\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot b\right) \cdot \left(-b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 5e-299) (- (* (* (* b a) b) a)) (* (* (* a a) b) (- b))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5e-299) {
		tmp = -(((b * a) * b) * a);
	} else {
		tmp = ((a * a) * b) * -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * a) <= 5d-299) then
        tmp = -(((b * a) * b) * a)
    else
        tmp = ((a * a) * b) * -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5e-299) {
		tmp = -(((b * a) * b) * a);
	} else {
		tmp = ((a * a) * b) * -b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a * a) <= 5e-299:
		tmp = -(((b * a) * b) * a)
	else:
		tmp = ((a * a) * b) * -b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 5e-299)
		tmp = Float64(-Float64(Float64(Float64(b * a) * b) * a));
	else
		tmp = Float64(Float64(Float64(a * a) * b) * Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a * a) <= 5e-299)
		tmp = -(((b * a) * b) * a);
	else
		tmp = ((a * a) * b) * -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 5e-299], (-N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]), N[(N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision] * (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 4.99999999999999956e-299

   1. Initial program 67.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if 4.99999999999999956e-299 < (*.f64 a a)

   1. Initial program 88.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{-299}:\\ \;\;\;\;-\left(\left(b \cdot a\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot b\right) \cdot \left(-b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{-225}:\\ \;\;\;\;-\left(\left(b \cdot a\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot a\right) \cdot a\right) \cdot \left(-b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.4e-225) (- (* (* (* b a) b) a)) (* (* (* b a) a) (- b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.4e-225) {
		tmp = -(((b * a) * b) * a);
	} else {
		tmp = ((b * a) * 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 (a <= 1.4d-225) then
        tmp = -(((b * a) * b) * a)
    else
        tmp = ((b * a) * a) * -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 1.4e-225) {
		tmp = -(((b * a) * b) * a);
	} else {
		tmp = ((b * a) * a) * -b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1.4e-225:
		tmp = -(((b * a) * b) * a)
	else:
		tmp = ((b * a) * a) * -b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.4e-225)
		tmp = Float64(-Float64(Float64(Float64(b * a) * b) * a));
	else
		tmp = Float64(Float64(Float64(b * a) * a) * Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1.4e-225)
		tmp = -(((b * a) * b) * a);
	else
		tmp = ((b * a) * a) * -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.4e-225], (-N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]), N[(N[(N[(b * a), $MachinePrecision] * a), $MachinePrecision] * (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.4e-225

   1. Initial program 80.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if 1.4e-225 < a

   1. Initial program 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 97.2

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

   4. Applied rewrites 97.2%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{-225}:\\ \;\;\;\;-\left(\left(b \cdot a\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot a\right) \cdot a\right) \cdot \left(-b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-*.f64 N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. sqrt-pow1 N/A

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

   8. pow2 N/A

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

   9. sqr-neg-rev N/A

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

   10. sqrt-prod N/A

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

   11. rem-square-sqrt N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. unswap-sqr N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 26.6

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

4. Applied rewrites 26.6%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-prod N/A

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

   3. sqr-neg-rev N/A

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

   4. pow2 N/A

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

   5. sqrt-pow1 N/A

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

   6. metadata-eval N/A

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

   7. unpow1 N/A

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

   8. lower-neg.f64 99.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 5: 94.2% accurate, 1.0× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 92.8

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

5. Applied rewrites 92.8%

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

Alternative 6: 28.8% accurate, 1.1× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. unpow1 N/A

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

   5. metadata-eval N/A

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

   6. sqrt-pow1 N/A

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

   7. pow2 N/A

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

   8. sqr-neg-rev N/A

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

   9. sqrt-prod N/A

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

   10. rem-square-sqrt N/A

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

   11. lift-*.f64 26.6

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 26.6

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

4. Applied rewrites 26.6%

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

Alternative 7: 28.8% 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 83.2%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift-*.f64 N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. sqrt-pow1 N/A

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

   8. pow2 N/A

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

   9. sqr-neg-rev N/A

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

   10. sqrt-prod N/A

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

   11. rem-square-sqrt N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. unswap-sqr N/A

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 26.6

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

4. Applied rewrites 26.6%

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

Reproduce

herbie shell --seed 2024340 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))