Average Error: 37.1 → 0.5

Time: 23.3s

Precision: 64

Internal Precision: 2368

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

↓

\[\log \left((e^{\sin x \cdot \cos \varepsilon - \sin x} - 1)^* + 1\right) + \cos x \cdot \sin \varepsilon\]

Error

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

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Out

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Target

Original	37.1
Target	15.7
Herbie	0.5

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

Derivation

Initial program 37.1
\[\sin \left(x + \varepsilon\right) - \sin x\]
Initial simplification37.1
\[\leadsto \sin \left(\varepsilon + x\right) - \sin x\]
Using strategy rm
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 log1p-expm1-u0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\log_* (1 + (e^{\cos \varepsilon \cdot \sin x - \sin x} - 1)^*)}\]
Using strategy rm
Applied log1p-udef0.5
\[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\log \left(1 + (e^{\cos \varepsilon \cdot \sin x - \sin x} - 1)^*\right)}\]
Final simplification0.5
\[\leadsto \log \left((e^{\sin x \cdot \cos \varepsilon - \sin x} - 1)^* + 1\right) + \cos x \cdot \sin \varepsilon\]

Runtime

herbie shell --seed 2018254 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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