Average Error: 36.8 → 0.3
Time: 21.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\left(2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(-\sin x, \sin \left(\varepsilon \cdot \frac{1}{2}\right), \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) + \left(2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos x, \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(-\sin x\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\left(2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(-\sin x, \sin \left(\varepsilon \cdot \frac{1}{2}\right), \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x\right) + \left(2 \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos x, \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(-\sin x\right)\right)
double f(double x, double eps) {
        double r3648008 = x;
        double r3648009 = eps;
        double r3648010 = r3648008 + r3648009;
        double r3648011 = sin(r3648010);
        double r3648012 = sin(r3648008);
        double r3648013 = r3648011 - r3648012;
        return r3648013;
}
↓
double f(double x, double eps) {
        double r3648014 = 2.0;
        double r3648015 = eps;
        double r3648016 = 0.5;
        double r3648017 = r3648015 * r3648016;
        double r3648018 = sin(r3648017);
        double r3648019 = r3648014 * r3648018;
        double r3648020 = x;
        double r3648021 = sin(r3648020);
        double r3648022 = -r3648021;
        double r3648023 = r3648018 * r3648021;
        double r3648024 = fma(r3648022, r3648018, r3648023);
        double r3648025 = r3648019 * r3648024;
        double r3648026 = cos(r3648017);
        double r3648027 = cos(r3648020);
        double r3648028 = r3648018 * r3648022;
        double r3648029 = fma(r3648026, r3648027, r3648028);
        double r3648030 = r3648019 * r3648029;
        double r3648031 = r3648025 + r3648030;
        return r3648031;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.8
Target14.9
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.8

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

  3. Applied diff-sin37.1

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

  4. Simplified14.9

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

  5. Taylor expanded around inf 14.9

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

  6. Simplified14.9

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

  8. Applied fma-udef14.9

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

  9. Applied cos-sum0.3

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

  11. Applied prod-diff0.3

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

  12. Applied distribute-rgt-in0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 +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)))