Average Error: 21.0 → 21.1
Time: 35.2s
Precision: binary64
\[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
↓
\[{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2} + {b}^{2} \]
{\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(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2} + {b}^{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 (* angle (* PI 0.005555555555555556)))) 2.0) (pow b 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(angle * (((double) M_PI) * 0.005555555555555556))), 2.0) + pow(b, 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 21.0

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

  2. Taylor expanded in angle around 0 21.1

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

  3. Applied div-inv_binary6421.1

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

  4. Applied associate-*l*_binary6421.1

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

  5. Simplified21.1

    \[\leadsto {\left(a \cdot \sin \left(angle \cdot \color{blue}{\left(\pi \cdot 0.005555555555555556\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

  6. Final simplification21.1

    \[\leadsto {\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2} + {b}^{2} \]

Reproduce

herbie shell --seed 2021357 
(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)))