Average Error: 20.5 → 20.5
Time: 12.0s
Precision: binary64
\[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}\]
↓
\[{\left(a \cdot \sin \left(\frac{1}{\sqrt{180}} \cdot \left(\pi \cdot \frac{angle}{\sqrt{180}}\right)\right)\right)}^{2} + {\left(b \cdot \sqrt[3]{{\left(\sqrt[3]{{\cos \left(\pi \cdot \frac{angle}{180}\right)}^{3}}\right)}^{3}}\right)}^{2}\]
{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}
↓
{\left(a \cdot \sin \left(\frac{1}{\sqrt{180}} \cdot \left(\pi \cdot \frac{angle}{\sqrt{180}}\right)\right)\right)}^{2} + {\left(b \cdot \sqrt[3]{{\left(\sqrt[3]{{\cos \left(\pi \cdot \frac{angle}{180}\right)}^{3}}\right)}^{3}}\right)}^{2}
(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
  (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))
↓
(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* (/ 1.0 (sqrt 180.0)) (* PI (/ angle (sqrt 180.0)))))) 2.0)
  (pow
   (* b (cbrt (pow (cbrt (pow (cos (* PI (/ angle 180.0))) 3.0)) 3.0)))
   2.0)))
double code(double a, double b, double angle) {
	return pow((a * sin((angle / 180.0) * ((double) M_PI))), 2.0) + pow((b * cos((angle / 180.0) * ((double) M_PI))), 2.0);
}
↓
double code(double a, double b, double angle) {
	return pow((a * sin((1.0 / sqrt(180.0)) * (((double) M_PI) * (angle / sqrt(180.0))))), 2.0) + pow((b * cbrt(pow(cbrt(pow(cos(((double) M_PI) * (angle / 180.0)), 3.0)), 3.0))), 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 20.5

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

  3. Applied add-sqr-sqrt_binary64_44120.7

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

  4. Applied *-un-lft-identity_binary64_41920.7

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

  5. Applied times-frac_binary64_42520.5

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

  6. Applied associate-*l*_binary64_36020.5

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

  7. Simplified20.5

    \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\sqrt{180}} \cdot \color{blue}{\left(\pi \cdot \frac{angle}{\sqrt{180}}\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}\]
  8. Using strategy rm

  9. Applied add-cbrt-cube_binary64_45520.5

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

  10. Simplified20.5

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

  12. Applied add-cbrt-cube_binary64_45520.5

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

  13. Simplified20.5

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

  14. Final simplification20.5

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

Reproduce

herbie shell --seed 2020349 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))