Average Error: 0.2 → 0.1
Time: 8.3s
Precision: binary64
\[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
↓
\[\frac{1}{\sin B} - \frac{x}{\tan B} \]
\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}
↓
\frac{1}{\sin B} - \frac{x}{\tan B}

(FPCore (B x)
 :precision binary64
 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))
↓
(FPCore (B x) :precision binary64 (- (/ 1.0 (sin B)) (/ x (tan 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 / sin(B)) - (x / tan(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. Simplified0.1

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

  3. Applied clear-num_binary640.2

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

  4. Applied *-un-lft-identity_binary640.2

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

  5. Applied add-sqr-sqrt_binary640.2

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

  6. Applied times-frac_binary640.2

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

  7. Applied *-un-lft-identity_binary640.2

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

  8. Applied add-sqr-sqrt_binary640.2

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

  9. Applied times-frac_binary640.2

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

  10. Applied distribute-lft-out--_binary640.2

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2022095 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))