Average Error: 20.0 → 20.1
Time: 21.3s
Precision: binary64
\[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
↓
\[\begin{array}{l} t_0 := \pi \cdot \frac{\sqrt[3]{angle}}{\sqrt{180}}\\ t_1 := \sqrt[3]{angle} \cdot \sqrt[3]{angle}\\ t_2 := \sqrt[3]{\cos \left(t_0 \cdot \frac{t_1}{\sqrt{180}}\right)}\\ {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \left(t_2 \cdot \left(\sqrt[3]{\cos \left(\frac{\sqrt[3]{angle} \cdot \sqrt[3]{\sqrt[3]{angle} \cdot t_1}}{\sqrt{180}} \cdot t_0\right)} \cdot t_2\right)\right)\right)}^{2} \end{array} \]
{\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}
↓
\begin{array}{l}
t_0 := \pi \cdot \frac{\sqrt[3]{angle}}{\sqrt{180}}\\
t_1 := \sqrt[3]{angle} \cdot \sqrt[3]{angle}\\
t_2 := \sqrt[3]{\cos \left(t_0 \cdot \frac{t_1}{\sqrt{180}}\right)}\\
{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \left(t_2 \cdot \left(\sqrt[3]{\cos \left(\frac{\sqrt[3]{angle} \cdot \sqrt[3]{\sqrt[3]{angle} \cdot t_1}}{\sqrt{180}} \cdot t_0\right)} \cdot t_2\right)\right)\right)}^{2}
\end{array}

(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
 (let* ((t_0 (* PI (/ (cbrt angle) (sqrt 180.0))))
        (t_1 (* (cbrt angle) (cbrt angle)))
        (t_2 (cbrt (cos (* t_0 (/ t_1 (sqrt 180.0)))))))
   (+
    (pow (* a (sin (* 0.005555555555555556 (* angle PI)))) 2.0)
    (pow
     (*
      b
      (*
       t_2
       (*
        (cbrt
         (cos
          (*
           (/ (* (cbrt angle) (cbrt (* (cbrt angle) t_1))) (sqrt 180.0))
           t_0)))
        t_2)))
     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) {
	double t_0 = ((double) M_PI) * (cbrt(angle) / sqrt(180.0));
	double t_1 = cbrt(angle) * cbrt(angle);
	double t_2 = cbrt(cos(t_0 * (t_1 / sqrt(180.0))));
	return pow((a * sin(0.005555555555555556 * (angle * ((double) M_PI)))), 2.0) + pow((b * (t_2 * (cbrt(cos(((cbrt(angle) * cbrt(cbrt(angle) * t_1)) / sqrt(180.0)) * t_0)) * t_2))), 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 20.0

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

  2. Applied add-sqr-sqrt_binary6420.1

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

  3. Applied add-cube-cbrt_binary6420.0

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

  4. Applied times-frac_binary6420.0

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

  5. Applied associate-*l*_binary6420.0

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

  6. Applied add-cube-cbrt_binary6420.0

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

  7. Taylor expanded in angle around inf 20.1

    \[\leadsto {\left(a \cdot \color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{2} + {\left(b \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\sqrt[3]{angle} \cdot \sqrt[3]{angle}}{\sqrt{180}} \cdot \left(\frac{\sqrt[3]{angle}}{\sqrt{180}} \cdot \pi\right)\right)} \cdot \sqrt[3]{\cos \left(\frac{\sqrt[3]{angle} \cdot \sqrt[3]{angle}}{\sqrt{180}} \cdot \left(\frac{\sqrt[3]{angle}}{\sqrt{180}} \cdot \pi\right)\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\sqrt[3]{angle} \cdot \sqrt[3]{angle}}{\sqrt{180}} \cdot \left(\frac{\sqrt[3]{angle}}{\sqrt{180}} \cdot \pi\right)\right)}\right)\right)}^{2} \]

  8. Applied add-cbrt-cube_binary6420.1

    \[\leadsto {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \left(\left(\sqrt[3]{\cos \left(\frac{\color{blue}{\sqrt[3]{\left(\sqrt[3]{angle} \cdot \sqrt[3]{angle}\right) \cdot \sqrt[3]{angle}}} \cdot \sqrt[3]{angle}}{\sqrt{180}} \cdot \left(\frac{\sqrt[3]{angle}}{\sqrt{180}} \cdot \pi\right)\right)} \cdot \sqrt[3]{\cos \left(\frac{\sqrt[3]{angle} \cdot \sqrt[3]{angle}}{\sqrt{180}} \cdot \left(\frac{\sqrt[3]{angle}}{\sqrt{180}} \cdot \pi\right)\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{\sqrt[3]{angle} \cdot \sqrt[3]{angle}}{\sqrt{180}} \cdot \left(\frac{\sqrt[3]{angle}}{\sqrt{180}} \cdot \pi\right)\right)}\right)\right)}^{2} \]

  9. Final simplification20.1

    \[\leadsto {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \left(\sqrt[3]{\cos \left(\left(\pi \cdot \frac{\sqrt[3]{angle}}{\sqrt{180}}\right) \cdot \frac{\sqrt[3]{angle} \cdot \sqrt[3]{angle}}{\sqrt{180}}\right)} \cdot \left(\sqrt[3]{\cos \left(\frac{\sqrt[3]{angle} \cdot \sqrt[3]{\sqrt[3]{angle} \cdot \left(\sqrt[3]{angle} \cdot \sqrt[3]{angle}\right)}}{\sqrt{180}} \cdot \left(\pi \cdot \frac{\sqrt[3]{angle}}{\sqrt{180}}\right)\right)} \cdot \sqrt[3]{\cos \left(\left(\pi \cdot \frac{\sqrt[3]{angle}}{\sqrt{180}}\right) \cdot \frac{\sqrt[3]{angle} \cdot \sqrt[3]{angle}}{\sqrt{180}}\right)}\right)\right)\right)}^{2} \]

Reproduce

herbie shell --seed 2022081 
(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)))