Average Error: 39.6 → 1.2
Time: 22.7s
Precision: 64
Internal Precision: 128
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.8770758529849264 \cdot 10^{-06} \lor \neg \left(\varepsilon \le 6.378520904994828 \cdot 10^{-05}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\left(\left(\sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*} \cdot \sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*}\right) \cdot \left(\sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*} \cdot \sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*}\right)\right) \cdot \left(\sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*} \cdot \sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*}\right)} \cdot \sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*}\right)\right) \cdot -2\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.8770758529849264e-06 or 6.378520904994828e-05 < eps

    1. Initial program 29.9

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

    3. Applied cos-sum1.0

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

    if -1.8770758529849264e-06 < eps < 6.378520904994828e-05

    1. Initial program 49.4

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

    3. Applied diff-cos37.8

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

    4. Simplified0.5

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

    6. Applied expm1-log1p-u0.5

      \[\leadsto -2 \cdot \left(\color{blue}{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
    7. Using strategy rm

    8. Applied add-cube-cbrt1.4

      \[\leadsto -2 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*} \cdot \sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*}\right) \cdot \sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*}\right)} \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]
    9. Using strategy rm

    10. Applied add-cbrt-cube1.5

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.8770758529849264 \cdot 10^{-06} \lor \neg \left(\varepsilon \le 6.378520904994828 \cdot 10^{-05}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\sqrt[3]{\left(\left(\sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*} \cdot \sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*}\right) \cdot \left(\sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*} \cdot \sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*}\right)\right) \cdot \left(\sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*} \cdot \sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*}\right)} \cdot \sqrt[3]{(e^{\log_* (1 + \sin \left(\frac{\left(x + x\right) + \varepsilon}{2}\right))} - 1)^*}\right)\right) \cdot -2\\ \end{array}\]

Runtime

Time bar (total: 22.7s)Debug logProfile

herbie shell --seed 2018348 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))