ab-angle->ABCF D

Percentage Accurate: 83.1% → 99.6%

Time: 3.4s

Alternatives: 2

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: 83.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b) {
	return -(((a * a) * b) * b);
}

Python

def code(a, b):
	return -(((a * a) * b) * b)

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

2. Taylor expanded in a around 0 76.1%

   \[\leadsto -\color{blue}{{a}^{2} \cdot {b}^{2}} \]
3. Step-by-step derivation

   1. unpow2 76.1%

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

   2. unpow2 76.1%

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

   3. swap-sqr 99.7%

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

   4. unpow2 99.7%

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

4. Simplified 99.7%

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

   1. unpow2 99.7%

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

   2. associate-*l* 94.9%

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

   3. *-commutative 94.9%

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

   4. associate-*l* 84.5%

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

   5. unpow2 84.5%

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

   6. add-sqr-sqrt 46.2%

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

   7. sqrt-unprod 54.5%

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

   8. sqr-neg 54.5%

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

   9. sqrt-unprod 14.0%

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

   10. add-sqr-sqrt 30.6%

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

   11. associate-*l* 30.6%

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

   12. *-commutative 30.6%

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

   13. add-sqr-sqrt 26.3%

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

   14. associate-*l* 26.3%

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

6. Applied egg-rr 53.2%

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

   1. *-commutative 53.2%

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

   2. associate-*r* 53.3%

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

   3. add-sqr-sqrt 30.3%

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

   4. pow2 30.3%

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

   5. add-sqr-sqrt 30.3%

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

   6. pow2 30.3%

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

   7. pow-prod-down 30.3%

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

   8. pow1/2 30.3%

      \[\leadsto -\left(b \cdot \sqrt{a}\right) \cdot {\left(\sqrt{a \cdot b} \cdot \sqrt{\color{blue}{{a}^{0.5}}}\right)}^{2} \]

   9. sqrt-pow1 30.3%

      \[\leadsto -\left(b \cdot \sqrt{a}\right) \cdot {\left(\sqrt{a \cdot b} \cdot \color{blue}{{a}^{\left(\frac{0.5}{2}\right)}}\right)}^{2} \]

   10. metadata-eval 30.3%

       \[\leadsto -\left(b \cdot \sqrt{a}\right) \cdot {\left(\sqrt{a \cdot b} \cdot {a}^{\color{blue}{0.25}}\right)}^{2} \]

8. Applied egg-rr 30.3%

   \[\leadsto -\left(b \cdot \sqrt{a}\right) \cdot \color{blue}{{\left(\sqrt{a \cdot b} \cdot {a}^{0.25}\right)}^{2}} \]
9. Step-by-step derivation

   1. *-commutative 30.3%

      \[\leadsto -\left(b \cdot \sqrt{a}\right) \cdot {\color{blue}{\left({a}^{0.25} \cdot \sqrt{a \cdot b}\right)}}^{2} \]

   2. unpow2 30.3%

      \[\leadsto -\left(b \cdot \sqrt{a}\right) \cdot \color{blue}{\left(\left({a}^{0.25} \cdot \sqrt{a \cdot b}\right) \cdot \left({a}^{0.25} \cdot \sqrt{a \cdot b}\right)\right)} \]

   3. swap-sqr 30.3%

      \[\leadsto -\left(b \cdot \sqrt{a}\right) \cdot \color{blue}{\left(\left({a}^{0.25} \cdot {a}^{0.25}\right) \cdot \left(\sqrt{a \cdot b} \cdot \sqrt{a \cdot b}\right)\right)} \]

   4. pow-sqr 30.3%

      \[\leadsto -\left(b \cdot \sqrt{a}\right) \cdot \left(\color{blue}{{a}^{\left(2 \cdot 0.25\right)}} \cdot \left(\sqrt{a \cdot b} \cdot \sqrt{a \cdot b}\right)\right) \]

   5. metadata-eval 30.3%

      \[\leadsto -\left(b \cdot \sqrt{a}\right) \cdot \left({a}^{\color{blue}{0.5}} \cdot \left(\sqrt{a \cdot b} \cdot \sqrt{a \cdot b}\right)\right) \]

   6. unpow1/2 30.3%

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

   7. rem-square-sqrt 53.3%

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

10. Simplified 53.3%

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

11. Final simplification 53.3%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

2. Taylor expanded in a around 0 76.1%

   \[\leadsto -\color{blue}{{a}^{2} \cdot {b}^{2}} \]
3. Step-by-step derivation

   1. unpow2 76.1%

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

   2. unpow2 76.1%

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

   3. swap-sqr 99.7%

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

   4. unpow2 99.7%

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

4. Simplified 99.7%

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

   1. unpow2 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Reproduce

herbie shell --seed 2023318 
(FPCore (a b)
  :name "ab-angle->ABCF D"
  :precision binary64
  (- (* (* (* a a) b) b)))