ab-angle->ABCF D

Percentage Accurate: 82.7% → 99.6%

Time: 5.1s

Alternatives: 6

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: 82.7% 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, 1.0× speedup

Math

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

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(Float64(b * a) * Float64(b * Float64(-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 80.4%

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

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. unswap-sqr N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-neg.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 99.6

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 95.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{-174}:\\ \;\;\;\;-\left(\left(b \cdot a\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(\left(b \cdot a\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 4.2e-174) (- (* (* (* b a) b) a)) (* (- b) (* (* b a) a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 4.2e-174) {
		tmp = -(((b * a) * 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 (a <= 4.2d-174) then
        tmp = -(((b * a) * 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 (a <= 4.2e-174) {
		tmp = -(((b * a) * b) * a);
	} else {
		tmp = -b * ((b * a) * a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 4.2e-174:
		tmp = -(((b * a) * b) * a)
	else:
		tmp = -b * ((b * a) * a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 4.2e-174)
		tmp = Float64(-Float64(Float64(Float64(b * a) * b) * a));
	else
		tmp = Float64(Float64(-b) * Float64(Float64(b * a) * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 4.2e-174)
		tmp = -(((b * a) * b) * a);
	else
		tmp = -b * ((b * a) * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 4.2e-174], (-N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]), N[((-b) * N[(N[(b * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.20000000000000021e-174

   1. Initial program 76.7%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 95.4

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

   5. Applied rewrites 95.4%

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

   if 4.20000000000000021e-174 < a

   1. Initial program 88.1%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{-174}:\\ \;\;\;\;-\left(\left(b \cdot a\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(\left(b \cdot a\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.85 \cdot 10^{-150}:\\ \;\;\;\;-\left(\left(b \cdot a\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(\left(a \cdot a\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 1.85e-150) (- (* (* (* b a) b) a)) (* (- b) (* (* a a) b))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 1.85e-150) {
		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 <= 1.85d-150) 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 <= 1.85e-150) {
		tmp = -(((b * a) * b) * a);
	} else {
		tmp = -b * ((a * a) * b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 1.85e-150:
		tmp = -(((b * a) * b) * a)
	else:
		tmp = -b * ((a * a) * b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 1.85e-150)
		tmp = Float64(-Float64(Float64(Float64(b * a) * b) * a));
	else
		tmp = Float64(Float64(-b) * Float64(Float64(a * a) * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 1.85e-150)
		tmp = -(((b * a) * b) * a);
	else
		tmp = -b * ((a * a) * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 1.85e-150], (-N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]), N[((-b) * N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.85e-150

   1. Initial program 76.6%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 95.1

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

   5. Applied rewrites 95.1%

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

   if 1.85e-150 < a

   1. Initial program 89.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.85 \cdot 10^{-150}:\\ \;\;\;\;-\left(\left(b \cdot a\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(\left(a \cdot a\right) \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% 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 80.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 95.3

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

5. Applied rewrites 95.3%

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

Alternative 5: 29.2% accurate, 1.1× 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(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]

Derivation

1. Initial program 80.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. distribute-lft-neg-in N/A

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

   8. neg-mul-1 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. unpow1 N/A

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

   12. metadata-eval N/A

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

   13. unpow-prod-down N/A

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

   14. neg-mul-1 N/A

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

   15. sqrt-pow1 N/A

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

   16. pow2 N/A

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

   17. sqr-neg-rev N/A

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

   18. sqrt-prod N/A

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

   19. rem-square-sqrt N/A

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

   20. associate-*l* N/A

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

   21. lift-*.f64 N/A

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

   22. associate-*l* N/A

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

   23. lift-*.f64 N/A

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

   24. lift-*.f64 26.9

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

4. Applied rewrites 26.9%

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

5. Final simplification 26.9%

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

Alternative 6: 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 80.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. distribute-lft-neg-in N/A

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

   8. neg-mul-1 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. unpow1 N/A

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

   12. metadata-eval N/A

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

   13. unpow-prod-down N/A

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

   14. neg-mul-1 N/A

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

   15. sqrt-pow1 N/A

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

   16. pow2 N/A

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

   17. sqr-neg-rev N/A

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

   18. sqrt-prod N/A

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

   19. rem-square-sqrt N/A

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

   20. associate-*l* N/A

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

   21. unswap-sqr N/A

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

   22. lower-*.f64 N/A

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

   23. *-commutative N/A

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

   24. lower-*.f64 N/A

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

   25. *-commutative N/A

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

   26. lower-*.f64 26.8

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

4. Applied rewrites 26.8%

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

Reproduce

herbie shell --seed 2024298 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))