Average Error: 20.5 → 20.5
Time: 19.4s
Precision: binary64
\[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}\]
↓
\[{a}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{{\left(\sqrt{180}\right)}^{1.5}} \cdot \frac{angle}{\sqrt{\sqrt{180}}}\right)\right)}^{2}\]
{\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(\frac{\pi}{{\left(\sqrt{180}\right)}^{1.5}} \cdot \frac{angle}{\sqrt{\sqrt{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 (pow (sqrt 180.0) 1.5)) (/ angle (sqrt (sqrt 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) / pow(sqrt(180.0), 1.5)) * (angle / sqrt(sqrt(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 20.5

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

  2. Taylor expanded around 0 20.5

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

  4. Applied add-sqr-sqrt_binary64_10020.7

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

  5. Applied *-un-lft-identity_binary64_7820.7

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

  6. Applied times-frac_binary64_8420.5

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

  7. Applied associate-*r*_binary64_1820.5

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

  8. Simplified20.6

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

  10. Applied add-sqr-sqrt_binary64_10020.6

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

  11. Applied *-un-lft-identity_binary64_7820.6

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

  12. Applied times-frac_binary64_8420.6

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

  13. Applied associate-*r*_binary64_1820.5

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

  14. Simplified20.5

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\frac{\pi}{{\left(\sqrt{180}\right)}^{1.5}}} \cdot \frac{angle}{\sqrt{\sqrt{180}}}\right)\right)}^{2}\]

  15. Final simplification20.5

    \[\leadsto {a}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{{\left(\sqrt{180}\right)}^{1.5}} \cdot \frac{angle}{\sqrt{\sqrt{180}}}\right)\right)}^{2}\]

Reproduce

herbie shell --seed 2021060 
(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)))