Average Error: 37.2 → 0.3
Time: 16.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\mathsf{fma}\left(-\sin x, \sin \left(\frac{\varepsilon}{2}\right), \sin \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) + \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right), \cos x, \sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin x\right)\right)\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\mathsf{fma}\left(-\sin x, \sin \left(\frac{\varepsilon}{2}\right), \sin \left(\frac{\varepsilon}{2}\right) \cdot \sin x\right) + \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right), \cos x, \sin \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin x\right)\right)\right)\right)
double f(double x, double eps) {
        double r2091932 = x;
        double r2091933 = eps;
        double r2091934 = r2091932 + r2091933;
        double r2091935 = sin(r2091934);
        double r2091936 = sin(r2091932);
        double r2091937 = r2091935 - r2091936;
        return r2091937;
}
↓
double f(double x, double eps) {
        double r2091938 = 2.0;
        double r2091939 = eps;
        double r2091940 = r2091939 / r2091938;
        double r2091941 = sin(r2091940);
        double r2091942 = x;
        double r2091943 = sin(r2091942);
        double r2091944 = -r2091943;
        double r2091945 = r2091941 * r2091943;
        double r2091946 = fma(r2091944, r2091941, r2091945);
        double r2091947 = 0.5;
        double r2091948 = r2091947 * r2091939;
        double r2091949 = cos(r2091948);
        double r2091950 = cos(r2091942);
        double r2091951 = r2091941 * r2091944;
        double r2091952 = fma(r2091949, r2091950, r2091951);
        double r2091953 = r2091946 + r2091952;
        double r2091954 = r2091941 * r2091953;
        double r2091955 = r2091938 * r2091954;
        return r2091955;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.2
Target15.0
Herbie0.3
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 37.2

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

  3. Applied diff-sin37.5

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

  4. Simplified15.0

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

  5. Taylor expanded around inf 15.0

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

  6. Simplified15.0

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

  8. Applied fma-udef15.0

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

  9. Applied cos-sum0.3

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

  10. Simplified0.3

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

  12. Applied prod-diff0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019153 +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)))