Average Error: 0.2 → 0.2
Time: 8.8s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
↓
\[\frac{1 - \cos B \cdot x}{\sin B} \]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1 - \cos B \cdot x}{\sin B}

(FPCore (B x)
 :precision binary64
 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))
↓
(FPCore (B x) :precision binary64 (/ (- 1.0 (* (cos B) x)) (sin B)))
double code(double B, double x) {
	return -(x * (1.0 / tan(B))) + (1.0 / sin(B));
}
↓
double code(double B, double x) {
	return (1.0 - (cos(B) * x)) / sin(B);
}

Error

Bits error versus B

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]

  3. Taylor expanded in x around 0 0.2

    \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

  4. Applied div-inv_binary640.3

    \[\leadsto \frac{1}{\sin B} - \color{blue}{\left(x \cdot \cos B\right) \cdot \frac{1}{\sin B}} \]

  5. Applied *-un-lft-identity_binary640.3

    \[\leadsto \frac{1}{\color{blue}{1 \cdot \sin B}} - \left(x \cdot \cos B\right) \cdot \frac{1}{\sin B} \]

  6. Applied *-un-lft-identity_binary640.3

    \[\leadsto \frac{\color{blue}{1 \cdot 1}}{1 \cdot \sin B} - \left(x \cdot \cos B\right) \cdot \frac{1}{\sin B} \]

  7. Applied times-frac_binary640.3

    \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{1}{\sin B}} - \left(x \cdot \cos B\right) \cdot \frac{1}{\sin B} \]

  8. Applied distribute-rgt-out--_binary640.3

    \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \left(\frac{1}{1} - x \cdot \cos B\right)} \]

  9. Taylor expanded in B around inf 0.2

    \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]

  10. Final simplification0.2

    \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]

Reproduce

herbie shell --seed 2022097 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))