Average Error: 37.3 → 0.5

Time: 25.9s

Precision: 64

Internal Precision: 128

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

↓

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

Error

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

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Target

Original	37.3
Target	15.1
Herbie	0.5

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

Derivation

Initial program 37.3
\[\sin \left(x + \varepsilon\right) - \sin x\]
Initial simplification37.3
\[\leadsto \sin \left(\varepsilon + x\right) - \sin x\]
Using strategy rm
Applied sin-sum22.0
\[\leadsto \color{blue}{\left(\sin \varepsilon \cdot \cos x + \cos \varepsilon \cdot \sin x\right)} - \sin x\]
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-cube-cbrt0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \left(\color{blue}{\left(\left(\sqrt[3]{\cos \varepsilon} \cdot \sqrt[3]{\cos \varepsilon}\right) \cdot \sqrt[3]{\cos \varepsilon}\right)} \cdot \sin x - \sin x\right)\]
Applied associate-*l*0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \left(\color{blue}{\left(\sqrt[3]{\cos \varepsilon} \cdot \sqrt[3]{\cos \varepsilon}\right) \cdot \left(\sqrt[3]{\cos \varepsilon} \cdot \sin x\right)} - \sin x\right)\]
Final simplification0.5
\[\leadsto \cos x \cdot \sin \varepsilon + \left(\left(\sqrt[3]{\cos \varepsilon} \cdot \sin x\right) \cdot \left(\sqrt[3]{\cos \varepsilon} \cdot \sqrt[3]{\cos \varepsilon}\right) - \sin x\right)\]

Runtime

herbie shell --seed 2018304 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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