Average Error: 20.4 → 20.5
Time: 24.3s
Precision: binary64
\[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
↓
\[{a}^{2} + {\left(b \cdot \sin \left(\left(\pi \cdot \frac{1}{\sqrt[3]{180} \cdot \sqrt[3]{180}}\right) \cdot \frac{angle}{\sqrt[3]{180}}\right)\right)}^{2} \]
{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}
↓
{a}^{2} + {\left(b \cdot \sin \left(\left(\pi \cdot \frac{1}{\sqrt[3]{180} \cdot \sqrt[3]{180}}\right) \cdot \frac{angle}{\sqrt[3]{180}}\right)\right)}^{2}

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* PI (/ angle 180.0)))) 2.0)
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))
↓
(FPCore (a b angle)
 :precision binary64
 (+
  (pow a 2.0)
  (pow
   (*
    b
    (sin
     (* (* PI (/ 1.0 (* (cbrt 180.0) (cbrt 180.0)))) (/ angle (cbrt 180.0)))))
   2.0)))
double code(double a, double b, double angle) {
	return pow((a * cos(((double) M_PI) * (angle / 180.0))), 2.0) + pow((b * sin(((double) M_PI) * (angle / 180.0))), 2.0);
}
↓
double code(double a, double b, double angle) {
	return pow(a, 2.0) + pow((b * sin((((double) M_PI) * (1.0 / (cbrt(180.0) * cbrt(180.0)))) * (angle / cbrt(180.0)))), 2.0);
}

Error

Bits error versus a

Bits error versus b

Bits error versus angle

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 20.4

    \[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

  2. Taylor expanded in angle around 0 20.4

    \[\leadsto {\left(a \cdot \color{blue}{1}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

  3. Applied add-cube-cbrt_binary6420.6

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{\color{blue}{\left(\sqrt[3]{180} \cdot \sqrt[3]{180}\right) \cdot \sqrt[3]{180}}}\right)\right)}^{2} \]

  4. Applied *-un-lft-identity_binary6420.6

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{\color{blue}{1 \cdot angle}}{\left(\sqrt[3]{180} \cdot \sqrt[3]{180}\right) \cdot \sqrt[3]{180}}\right)\right)}^{2} \]

  5. Applied times-frac_binary6420.5

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{180} \cdot \sqrt[3]{180}} \cdot \frac{angle}{\sqrt[3]{180}}\right)}\right)\right)}^{2} \]

  6. Applied associate-*r*_binary6420.5

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(\pi \cdot \frac{1}{\sqrt[3]{180} \cdot \sqrt[3]{180}}\right) \cdot \frac{angle}{\sqrt[3]{180}}\right)}\right)}^{2} \]

  7. Final simplification20.5

    \[\leadsto {a}^{2} + {\left(b \cdot \sin \left(\left(\pi \cdot \frac{1}{\sqrt[3]{180} \cdot \sqrt[3]{180}}\right) \cdot \frac{angle}{\sqrt[3]{180}}\right)\right)}^{2} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))