ab-angle->ABCF D

Percentage Accurate: 82.5% → 99.7%

Time: 5.2s

Alternatives: 8

Speedup: 1.0×

• 28.2% 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: 82.5% 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

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

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 (/ -1.0 a_m)) (* b a_m)))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return (b / (-1.0 / a_m)) * (b * 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 / ((-1.0d0) / a_m)) * (b * 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 / (-1.0 / a_m)) * (b * a_m);
}

Python

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

Julia

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

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = (b / (-1.0 / a_m)) * (b * 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[(N[(b / N[(-1.0 / a$95$m), $MachinePrecision]), $MachinePrecision] * N[(b * a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

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

   1. lift-*.f64 N/A

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

   2. associate-*l* N/A

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

   3. lift-*.f64 N/A

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

   4. unswap-sqr N/A

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

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

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

   6. lower-*.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. neg-sub0 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. lift-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. +-lft-identity N/A

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

   11. lower-/.f64 N/A

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

   12. +-lft-identity N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-/r* N/A

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

6. Applied rewrites 99.8%

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

   1. associate-/r/ N/A

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

   2. *-commutative N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

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 (/ -1.0 b))))

C

a_m = fabs(a);
assert(a_m < b);
double code(double a_m, double b) {
	return (b * a_m) * (a_m / (-1.0 / 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 / ((-1.0d0) / 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 / (-1.0 / 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 / (-1.0 / b))

Julia

a_m = abs(a)
a_m, b = sort([a_m, b])
function code(a_m, b)
	return Float64(Float64(b * a_m) * Float64(a_m / Float64(-1.0 / 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 / (-1.0 / 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[(b * a$95$m), $MachinePrecision] * N[(a$95$m / N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

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

   1. lift-*.f64 N/A

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

   2. associate-*l* N/A

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

   3. lift-*.f64 N/A

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

   4. unswap-sqr N/A

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

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

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

   6. lower-*.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. neg-sub0 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. lift-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. +-lft-identity N/A

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

   11. lower-/.f64 N/A

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

   12. +-lft-identity N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-/r* N/A

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

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 a_m) (* b (- a_m))))

C

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

Python

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

Julia

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

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = (b * a_m) * (b * -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[(N[(b * a$95$m), $MachinePrecision] * N[(b * (-a$95$m)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

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

   1. lift-*.f64 N/A

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

   2. associate-*l* N/A

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

   3. lift-*.f64 N/A

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

   4. unswap-sqr N/A

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

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

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

   6. lower-*.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 94.3% accurate, 1.0× speedup

Math

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

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) (* a_m (* b a_m))))

C

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

Python

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

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(b * a_m)))
end

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = -b * (a_m * (b * 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[(a$95$m * N[(b * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \left({a}^{2} \cdot {b}^{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

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

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

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

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

   10. lower-*.f64 N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 93.6

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

5. Applied rewrites 93.6%

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

6. Final simplification 93.6%

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

Alternative 5: 82.5% accurate, 1.0× speedup

Math

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

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 * Float64(-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 85.0%

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

3. Final simplification 85.0%

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

Alternative 6: 28.4% 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 85.0%

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

   1. lift-*.f64 N/A

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

   2. associate-*l* N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-lft-identity N/A

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

   7. flip3-+ N/A

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

   8. 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 26.2%

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

5. Final simplification 26.2%

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

Alternative 7: 28.3% accurate, 1.1× speedup

Math

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

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 (* a_m (* b a_m))))

C

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

Python

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

Julia

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

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = b * (a_m * (b * 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[(a$95$m * N[(b * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

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

   1. lift-*.f64 N/A

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

   2. associate-*l* N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-lft-identity N/A

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

   7. flip3-+ N/A

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

   8. 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 26.2%

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

   1. swap-sqr N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 26.2

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

6. Applied rewrites 26.2%

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

7. Final simplification 26.2%

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

Alternative 8: 28.3% accurate, 1.1× speedup

Math

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

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) (* b a_m)))

C

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

Python

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

Julia

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

MATLAB

a_m = abs(a);
a_m, b = num2cell(sort([a_m, b])){:}
function tmp = code(a_m, b)
	tmp = (b * a_m) * (b * 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[(N[(b * a$95$m), $MachinePrecision] * N[(b * a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

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

   1. lift-*.f64 N/A

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

   2. associate-*l* N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-lft-identity N/A

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

   7. flip3-+ N/A

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

   8. 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 26.2%

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

5. Final simplification 26.2%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))