ab-angle->ABCF D

Percentage Accurate: 81.8% → 99.7%

Time: 6.5s

Alternatives: 6

Speedup: 1.0×

• 28.1% 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: 81.8% 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 80.9%

   \[-\left(\left(a \cdot a\right) \cdot b\right) \cdot b \]
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 99.6

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

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. 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. mul0-lft N/A

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

   11. +-rgt-identity N/A

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

   12. clear-num N/A

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

   13. *-inverses N/A

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

   14. +-rgt-identity N/A

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

   15. mul0-lft N/A

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

   16. +-lft-identity N/A

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

   17. metadata-eval N/A

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

   18. flip3-- N/A

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

   19. neg-sub0 N/A

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

   20. lift-neg.f64 N/A

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

   21. associate-/r* N/A

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

   22. *-commutative N/A

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

6. Applied rewrites 99.7%

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

Alternative 2: 99.7% 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 82.0%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))