ab-angle->ABCF D

Percentage Accurate: 82.3% → 99.7%

Time: 4.5s

Alternatives: 7

Speedup: 1.0×

• 28.0% 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: 82.3% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ -{\left(b \cdot a\right)}^{2} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (pow (* b a) 2.0)))

C

double code(double a, double b) {
	return -pow((b * a), 2.0);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = -((b * a) ** 2.0d0)
end function

Java

public static double code(double a, double b) {
	return -Math.pow((b * a), 2.0);
}

Python

def code(a, b):
	return -math.pow((b * a), 2.0)

Julia

function code(a, b)
	return Float64(-(Float64(b * a) ^ 2.0))
end

MATLAB

function tmp = code(a, b)
	tmp = -((b * a) ^ 2.0);
end

Wolfram

code[a_, b_] := (-N[Power[N[(b * a), $MachinePrecision], 2.0], $MachinePrecision])

Derivation

1. Initial program 85.4%

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

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

4. Applied rewrites 99.7%

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

Alternative 2: 88.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(a \cdot a\right) \cdot b\right) \cdot \left(-b\right)\\ \mathbf{if}\;t\_0 \leq -1.5 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-\left(\left(b \cdot a\right) \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (let* ((t_0 (* (* (* a a) b) (- b))))
   (if (<= t_0 -1.5e+70) t_0 (- (* (* (* b a) b) a)))))

C

double code(double a, double b) {
	double t_0 = ((a * a) * b) * -b;
	double tmp;
	if (t_0 <= -1.5e+70) {
		tmp = t_0;
	} else {
		tmp = -(((b * a) * b) * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((a * a) * b) * -b
    if (t_0 <= (-1.5d+70)) then
        tmp = t_0
    else
        tmp = -(((b * a) * b) * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double t_0 = ((a * a) * b) * -b;
	double tmp;
	if (t_0 <= -1.5e+70) {
		tmp = t_0;
	} else {
		tmp = -(((b * a) * b) * a);
	}
	return tmp;
}

Python

def code(a, b):
	t_0 = ((a * a) * b) * -b
	tmp = 0
	if t_0 <= -1.5e+70:
		tmp = t_0
	else:
		tmp = -(((b * a) * b) * a)
	return tmp

Julia

function code(a, b)
	t_0 = Float64(Float64(Float64(a * a) * b) * Float64(-b))
	tmp = 0.0
	if (t_0 <= -1.5e+70)
		tmp = t_0;
	else
		tmp = Float64(-Float64(Float64(Float64(b * a) * b) * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	t_0 = ((a * a) * b) * -b;
	tmp = 0.0;
	if (t_0 <= -1.5e+70)
		tmp = t_0;
	else
		tmp = -(((b * a) * b) * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := Block[{t$95$0 = N[(N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision] * (-b)), $MachinePrecision]}, If[LessEqual[t$95$0, -1.5e+70], t$95$0, (-N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 (*.f64 (*.f64 (*.f64 a a) b) b)) < -1.49999999999999988e70

   1. Initial program 83.7%

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

   if -1.49999999999999988e70 < (neg.f64 (*.f64 (*.f64 (*.f64 a a) b) b))

   1. Initial program 87.1%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 92.5

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

   5. Applied rewrites 92.5%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(a \cdot a\right) \cdot b\right) \cdot \left(-b\right) \leq -1.5 \cdot 10^{+70}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot b\right) \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\left(b \cdot a\right) \cdot b\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{+245}:\\ \;\;\;\;-\left(\left(b \cdot a\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot a\right) \cdot a\right) \cdot \left(-b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 2.9e+245) (- (* (* (* b a) b) a)) (* (* (* b a) a) (- b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.9e+245) {
		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 (b <= 2.9d+245) 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 (b <= 2.9e+245) {
		tmp = -(((b * a) * b) * a);
	} else {
		tmp = ((b * a) * a) * -b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 2.9e+245:
		tmp = -(((b * a) * b) * a)
	else:
		tmp = ((b * a) * a) * -b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 2.9e+245)
		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 (b <= 2.9e+245)
		tmp = -(((b * a) * b) * a);
	else
		tmp = ((b * a) * a) * -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 2.9e+245], (-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 b < 2.9000000000000001e245

   1. Initial program 86.6%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 92.7

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

   5. Applied rewrites 92.7%

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

   if 2.9000000000000001e245 < b

   1. Initial program 72.1%

      \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
   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 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{+245}:\\ \;\;\;\;-\left(\left(b \cdot a\right) \cdot b\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot a\right) \cdot a\right) \cdot \left(-b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% 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 85.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. 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 26.0

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

4. Applied rewrites 26.0%

   \[\leadsto \color{blue}{\left(b \cdot a\right) \cdot \left(b \cdot a\right)} \]
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 99.7

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

   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 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 94.5% 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 85.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 91.1

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

5. Applied rewrites 91.1%

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

Alternative 6: 28.5% 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 85.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. 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 26.0

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

   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 26.0

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

4. Applied rewrites 26.0%

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

Alternative 7: 28.4% 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 85.4%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. 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 26.0

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

4. Applied rewrites 26.0%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))