Average Error: 37.2 → 0.5

Time: 27.5s

Precision: 64

Internal Precision: 128

\[\sin \left(x + \varepsilon\right) - \sin x\]

↓

\[\cos x \cdot \sin \varepsilon + \sqrt[3]{\left(\left(\cos \varepsilon \cdot \sin x - \sin x\right) \cdot \left(\cos \varepsilon \cdot \sin x - \sin x\right)\right) \cdot \left(\cos \varepsilon \cdot \sin x - \sin x\right)}\]

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	37.2
Target	15.6
Herbie	0.5

\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

Initial program 37.2
\[\sin \left(x + \varepsilon\right) - \sin x\]
Using strategy rm
Applied +-commutative37.2
\[\leadsto \sin \color{blue}{\left(\varepsilon + x\right)} - \sin x\]
Applied sin-sum21.3
\[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)} - \sin x\]
Applied associate--l+0.4
\[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x + \left(\cos \varepsilon \cdot \sin x - \sin x\right)}\]
Using strategy rm
Applied add-cbrt-cube0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\sqrt[3]{\left(\left(\cos \varepsilon \cdot \sin x - \sin x\right) \cdot \left(\cos \varepsilon \cdot \sin x - \sin x\right)\right) \cdot \left(\cos \varepsilon \cdot \sin x - \sin x\right)}}\]
Final simplification0.5
\[\leadsto \cos x \cdot \sin \varepsilon + \sqrt[3]{\left(\left(\cos \varepsilon \cdot \sin x - \sin x\right) \cdot \left(\cos \varepsilon \cdot \sin x - \sin x\right)\right) \cdot \left(\cos \varepsilon \cdot \sin x - \sin x\right)}\]

Reproduce

herbie shell --seed 2019088 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))