Average Error: 20.4 → 20.5
Time: 17.8s
Precision: binary64
\[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
↓
\[{a}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\pi}{\sqrt[3]{180} \cdot \sqrt[3]{180}} \cdot \frac{angle}{\sqrt[3]{180}}\right)\right)\right)\right)\right)\right)}^{2} \]
{\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 \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\pi}{\sqrt[3]{180} \cdot \sqrt[3]{180}} \cdot \frac{angle}{\sqrt[3]{180}}\right)\right)\right)\right)\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
    (log1p
     (expm1
      (expm1
       (log1p
        (sin
         (* (/ PI (* (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 * log1p(expm1(expm1(log1p(sin((((double) M_PI) / (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 log1p-expm1-u_binary6420.4

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} \]

  4. Applied add-cube-cbrt_binary6420.6

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \frac{angle}{\color{blue}{\left(\sqrt[3]{180} \cdot \sqrt[3]{180}\right) \cdot \sqrt[3]{180}}}\right)\right)\right)\right)}^{2} \]

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

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \frac{\color{blue}{1 \cdot angle}}{\left(\sqrt[3]{180} \cdot \sqrt[3]{180}\right) \cdot \sqrt[3]{180}}\right)\right)\right)\right)}^{2} \]

  6. Applied times-frac_binary6420.5

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\pi \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{180} \cdot \sqrt[3]{180}} \cdot \frac{angle}{\sqrt[3]{180}}\right)}\right)\right)\right)\right)}^{2} \]

  7. Applied associate-*r*_binary6420.5

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \color{blue}{\left(\left(\pi \cdot \frac{1}{\sqrt[3]{180} \cdot \sqrt[3]{180}}\right) \cdot \frac{angle}{\sqrt[3]{180}}\right)}\right)\right)\right)}^{2} \]

  8. Simplified20.5

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\color{blue}{\frac{\pi}{\sqrt[3]{180} \cdot \sqrt[3]{180}}} \cdot \frac{angle}{\sqrt[3]{180}}\right)\right)\right)\right)}^{2} \]

  9. Applied expm1-log1p-u_binary6420.5

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\pi}{\sqrt[3]{180} \cdot \sqrt[3]{180}} \cdot \frac{angle}{\sqrt[3]{180}}\right)\right)\right)}\right)\right)\right)}^{2} \]

  10. Final simplification20.5

    \[\leadsto {a}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\pi}{\sqrt[3]{180} \cdot \sqrt[3]{180}} \cdot \frac{angle}{\sqrt[3]{180}}\right)\right)\right)\right)\right)\right)}^{2} \]

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)))