Average Error: 30.3 → 0.3
Time: 8.3s
Precision: binary64
\[\frac{1 - \cos x}{\sin x} \]
↓
\[\frac{\sin \left(x \cdot 0.5\right)}{\cos \left(x \cdot 0.5\right)} \]
\frac{1 - \cos x}{\sin x}
↓
\frac{\sin \left(x \cdot 0.5\right)}{\cos \left(x \cdot 0.5\right)}

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (sin x)))
↓
(FPCore (x) :precision binary64 (/ (sin (* x 0.5)) (cos (* x 0.5))))
double code(double x) {
	return (1.0 - cos(x)) / sin(x);
}
↓
double code(double x) {
	return sin(x * 0.5) / cos(x * 0.5);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.3
Herbie0.3
\[\tan \left(\frac{x}{2}\right) \]

Derivation

  1. Initial program 30.3

    \[\frac{1 - \cos x}{\sin x} \]

  2. Simplified0.3

    \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right)} \]

  3. Applied add-cube-cbrt_binary640.8

    \[\leadsto \color{blue}{\left(\sqrt[3]{\tan \left(\frac{x}{2}\right)} \cdot \sqrt[3]{\tan \left(\frac{x}{2}\right)}\right) \cdot \sqrt[3]{\tan \left(\frac{x}{2}\right)}} \]

  4. Applied pow3_binary640.8

    \[\leadsto \color{blue}{{\left(\sqrt[3]{\tan \left(\frac{x}{2}\right)}\right)}^{3}} \]

  5. Taylor expanded in x around inf 0.3

    \[\leadsto \color{blue}{\frac{\sin \left(0.5 \cdot x\right)}{\cos \left(0.5 \cdot x\right)}} \]

  6. Simplified0.3

    \[\leadsto \color{blue}{\frac{\sin \left(x \cdot 0.5\right)}{\cos \left(x \cdot 0.5\right)}} \]

  7. Final simplification0.3

    \[\leadsto \frac{\sin \left(x \cdot 0.5\right)}{\cos \left(x \cdot 0.5\right)} \]

Reproduce

herbie shell --seed 1991292424 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64

  :herbie-target
  (tan (/ x 2.0))

  (/ (- 1.0 (cos x)) (sin x)))