Average Error: 36.7 → 0.6
Time: 35.3s
Precision: 64
Internal Precision: 2368
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot \left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \le -0.007872812654676866:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{if}\;2 \cdot \left(\left(\sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)}\right) \le 2.4568337889173983 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot (e^{\log_* (1 + \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right))} - 1)^*\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Target14.8
Herbie0.6
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* 2 (* (* (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2))))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))))) < -0.007872812654676866 or 2.4568337889173983e-10 < (* 2 (* (* (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2))))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2))))))

    1. Initial program 29.5

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    2. Using strategy rm

    3. Applied sin-sum0.5

      \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]

    if -0.007872812654676866 < (* 2 (* (* (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2))))) (cbrt (* (sin (/ eps 2)) (cos (/ (+ x (+ eps x)) 2)))))) < 2.4568337889173983e-10

    1. Initial program 44.0

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    2. Using strategy rm

    3. Applied diff-sin44.0

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

    4. Applied simplify0.5

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)}\]
    5. Using strategy rm

    6. Applied expm1-log1p-u0.6

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{(e^{\log_* (1 + \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right))} - 1)^*}\right)\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 35.3s)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' +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)))