ab-angle->ABCF D

Percentage Accurate: 82.2% → 99.7%

Time: 6.4s

Alternatives: 7

Speedup: 1.0×

• 27.4% 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: 82.2% 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, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ \frac{a\_m}{\frac{-1}{b}} \cdot \left(a\_m \cdot b\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b) :precision binary64 (* (/ a_m (/ -1.0 b)) (* a_m b)))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return (a_m / (-1.0 / b)) * (a_m * b);
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = (a_m / ((-1.0d0) / b)) * (a_m * b)
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	return (a_m / (-1.0 / b)) * (a_m * b);
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	return (a_m / (-1.0 / b)) * (a_m * b)

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(Float64(a_m / Float64(-1.0 / b)) * Float64(a_m * b))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = (a_m / (-1.0 / b)) * (a_m * b);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(N[(a$95$m / N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * N[(a$95$m * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.2%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. 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. unswap-sqr N/A

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

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

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

   8. lower-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-*.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. neg-sub0 N/A

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

   4. flip3-- N/A

      \[\leadsto \left(a \cdot \color{blue}{\frac{{0}^{3} - {b}^{3}}{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}}\right) \cdot \left(a \cdot b\right) \]

   5. clear-num N/A

      \[\leadsto \left(a \cdot \color{blue}{\frac{1}{\frac{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}{{0}^{3} - {b}^{3}}}}\right) \cdot \left(a \cdot b\right) \]

   6. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{a}{\frac{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}{{0}^{3} - {b}^{3}}}} \cdot \left(a \cdot b\right) \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a}{\frac{0 \cdot 0 + \left(b \cdot b + 0 \cdot b\right)}{{0}^{3} - {b}^{3}}}} \cdot \left(a \cdot b\right) \]

   8. metadata-eval N/A

      \[\leadsto \frac{a}{\frac{\color{blue}{0} + \left(b \cdot b + 0 \cdot b\right)}{{0}^{3} - {b}^{3}}} \cdot \left(a \cdot b\right) \]

   9. +-lft-identity N/A

      \[\leadsto \frac{a}{\frac{\color{blue}{b \cdot b + 0 \cdot b}}{{0}^{3} - {b}^{3}}} \cdot \left(a \cdot b\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{a}{\frac{\color{blue}{b \cdot b} + 0 \cdot b}{{0}^{3} - {b}^{3}}} \cdot \left(a \cdot b\right) \]

   11. mul0-lft N/A

       \[\leadsto \frac{a}{\frac{b \cdot b + \color{blue}{0}}{{0}^{3} - {b}^{3}}} \cdot \left(a \cdot b\right) \]

   12. +-rgt-identity N/A

       \[\leadsto \frac{a}{\frac{\color{blue}{b \cdot b}}{{0}^{3} - {b}^{3}}} \cdot \left(a \cdot b\right) \]

   13. clear-num N/A

       \[\leadsto \frac{a}{\color{blue}{\frac{1}{\frac{{0}^{3} - {b}^{3}}{b \cdot b}}}} \cdot \left(a \cdot b\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{a}{\frac{1}{\frac{{0}^{3} - {b}^{3}}{\color{blue}{b \cdot b}}}} \cdot \left(a \cdot b\right) \]

   15. metadata-eval N/A

       \[\leadsto \frac{a}{\frac{1}{\frac{\color{blue}{0} - {b}^{3}}{b \cdot b}}} \cdot \left(a \cdot b\right) \]

   16. mul0-lft N/A

       \[\leadsto \frac{a}{\frac{1}{\frac{\color{blue}{0 \cdot b} - {b}^{3}}{b \cdot b}}} \cdot \left(a \cdot b\right) \]

   17. cube-mult N/A

       \[\leadsto \frac{a}{\frac{1}{\frac{0 \cdot b - \color{blue}{b \cdot \left(b \cdot b\right)}}{b \cdot b}}} \cdot \left(a \cdot b\right) \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{a}{\frac{1}{\frac{0 \cdot b - b \cdot \color{blue}{\left(b \cdot b\right)}}{b \cdot b}}} \cdot \left(a \cdot b\right) \]

   19. frac-sub N/A

       \[\leadsto \frac{a}{\frac{1}{\color{blue}{\frac{0}{b} - \frac{b \cdot b}{b}}}} \cdot \left(a \cdot b\right) \]

   20. sub-div N/A

       \[\leadsto \frac{a}{\frac{1}{\color{blue}{\frac{0 - b \cdot b}{b}}}} \cdot \left(a \cdot b\right) \]

   21. neg-sub0 N/A

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

6. Applied rewrites 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ \left(a\_m \cdot b\right) \cdot \left(-a\_m \cdot b\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b) :precision binary64 (* (* a_m b) (- (* a_m b))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return (a_m * b) * -(a_m * b);
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = (a_m * b) * -(a_m * b)
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	return (a_m * b) * -(a_m * b);
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	return (a_m * b) * -(a_m * b)

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(Float64(a_m * b) * Float64(-Float64(a_m * b)))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = (a_m * b) * -(a_m * b);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(N[(a$95$m * b), $MachinePrecision] * (-N[(a$95$m * b), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 85.2%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. 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. unswap-sqr N/A

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

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

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

   8. lower-*.f64 N/A

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

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

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-*.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ \left(-b\right) \cdot \left(a\_m \cdot \left(a\_m \cdot b\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b) :precision binary64 (* (- b) (* a_m (* a_m b))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return -b * (a_m * (a_m * b));
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = -b * (a_m * (a_m * b))
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	return -b * (a_m * (a_m * b));
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	return -b * (a_m * (a_m * b))

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(Float64(-b) * Float64(a_m * Float64(a_m * b)))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = -b * (a_m * (a_m * b));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[((-b) * N[(a$95$m * N[(a$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 95.3

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

4. Applied rewrites 95.3%

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

5. Final simplification 95.3%

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

Alternative 4: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ -b \cdot \left(b \cdot \left(a\_m \cdot a\_m\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b) :precision binary64 (- (* b (* b (* a_m a_m)))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return -(b * (b * (a_m * a_m)));
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = -(b * (b * (a_m * a_m)))
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	return -(b * (b * (a_m * a_m)));
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	return -(b * (b * (a_m * a_m)))

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(-Float64(b * Float64(b * Float64(a_m * a_m))))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = -(b * (b * (a_m * a_m)));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := (-N[(b * N[(b * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 85.2%

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

3. Final simplification 85.2%

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

Alternative 5: 28.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ b \cdot \left(b \cdot \left(a\_m \cdot a\_m\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b) :precision binary64 (* b (* b (* a_m a_m))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return b * (b * (a_m * a_m));
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = b * (b * (a_m * a_m))
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	return b * (b * (a_m * a_m));
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	return b * (b * (a_m * a_m))

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(b * Float64(b * Float64(a_m * a_m)))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = b * (b * (a_m * a_m));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(b * N[(b * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.2%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. +-lft-identity N/A

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

   3. flip3-+ N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}}\right) \]

   4. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}\right)\right)}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}} \]

4. Applied rewrites 29.8%

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

5. Final simplification 29.8%

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

Alternative 6: 28.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ b \cdot \left(a\_m \cdot \left(a\_m \cdot b\right)\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b) :precision binary64 (* b (* a_m (* a_m b))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return b * (a_m * (a_m * b));
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = b * (a_m * (a_m * b))
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	return b * (a_m * (a_m * b));
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	return b * (a_m * (a_m * b))

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(b * Float64(a_m * Float64(a_m * b)))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = b * (a_m * (a_m * b));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(b * N[(a$95$m * N[(a$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.2%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. +-lft-identity N/A

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

   3. flip3-+ N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}}\right) \]

   4. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}\right)\right)}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}} \]

4. Applied rewrites 29.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. associate-*r* N/A

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

   8. rem-square-sqrt N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow2 N/A

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

   12. swap-sqr N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 12.4

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

6. Applied rewrites 29.8%

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

7. Final simplification 29.8%

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

Alternative 7: 28.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ [a_m, b] = \mathsf{sort}([a_m, b])\\ \\ \left(a\_m \cdot b\right) \cdot \left(a\_m \cdot b\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
(FPCore (a_m b) :precision binary64 (* (* a_m b) (* a_m b)))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return (a_m * b) * (a_m * b);
}

Fortran

a_m = abs(a)
NOTE: a_m and b should be sorted in increasing order before calling this function.
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    code = (a_m * b) * (a_m * b)
end function

Java

a_m = Math.abs(a);
assert a_m < b;
public static double code(double a_m, double b) {
	return (a_m * b) * (a_m * b);
}

Python

a_m = math.fabs(a)
[a_m, b] = sort([a_m, b])
def code(a_m, b):
	return (a_m * b) * (a_m * b)

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(Float64(a_m * b) * Float64(a_m * b))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = (a_m * b) * (a_m * b);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
NOTE: a_m and b should be sorted in increasing order before calling this function.
code[a$95$m_, b_] := N[(N[(a$95$m * b), $MachinePrecision] * N[(a$95$m * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.2%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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. +-lft-identity N/A

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

   3. flip3-+ N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}}\right) \]

   4. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} + {\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)}^{3}\right)\right)}{0 \cdot 0 + \left(\left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right) - 0 \cdot \left(\left(\left(a \cdot a\right) \cdot b\right) \cdot b\right)\right)}} \]

4. Applied rewrites 29.8%

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

Reproduce

herbie shell --seed 2024234 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))