Average Error: 37.1 → 0.4
Time: 47.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \log \left(e^{\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \log \left(e^{\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)
double f(double x, double eps) {
        double r7149457 = x;
        double r7149458 = eps;
        double r7149459 = r7149457 + r7149458;
        double r7149460 = sin(r7149459);
        double r7149461 = sin(r7149457);
        double r7149462 = r7149460 - r7149461;
        return r7149462;
}
↓
double f(double x, double eps) {
        double r7149463 = 2.0;
        double r7149464 = x;
        double r7149465 = cos(r7149464);
        double r7149466 = eps;
        double r7149467 = 0.5;
        double r7149468 = r7149466 * r7149467;
        double r7149469 = cos(r7149468);
        double r7149470 = r7149465 * r7149469;
        double r7149471 = sin(r7149464);
        double r7149472 = sin(r7149468);
        double r7149473 = r7149471 * r7149472;
        double r7149474 = exp(r7149473);
        double r7149475 = log(r7149474);
        double r7149476 = r7149470 - r7149475;
        double r7149477 = r7149476 * r7149472;
        double r7149478 = r7149463 * r7149477;
        return r7149478;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.0
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.1

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

  3. Applied diff-sin37.4

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

  4. Simplified15.0

    \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)}\]

  5. Taylor expanded around inf 15.0

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)}\right)\]

  6. Simplified15.0

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \color{blue}{\cos \left(\mathsf{fma}\left(\varepsilon, \frac{1}{2}, x\right)\right)}\right)\]
  7. Using strategy rm

  8. Applied fma-udef15.0

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \color{blue}{\left(\varepsilon \cdot \frac{1}{2} + x\right)}\right)\]

  9. Applied cos-sum0.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \color{blue}{\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x - \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right)}\right)\]
  10. Using strategy rm

  11. Applied add-log-exp0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos x - \color{blue}{\log \left(e^{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)}\right)\right)\]

  12. Final simplification0.4

    \[\leadsto 2 \cdot \left(\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \log \left(e^{\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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