Average Error: 36.9 → 0.5
Time: 17.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \cos \varepsilon - \sin x\right)\right)\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \cos \varepsilon - \sin x\right)\right)\right)
double f(double x, double eps) {
        double r7503299 = x;
        double r7503300 = eps;
        double r7503301 = r7503299 + r7503300;
        double r7503302 = sin(r7503301);
        double r7503303 = sin(r7503299);
        double r7503304 = r7503302 - r7503303;
        return r7503304;
}
↓
double f(double x, double eps) {
        double r7503305 = eps;
        double r7503306 = sin(r7503305);
        double r7503307 = x;
        double r7503308 = cos(r7503307);
        double r7503309 = sin(r7503307);
        double r7503310 = cos(r7503305);
        double r7503311 = r7503309 * r7503310;
        double r7503312 = r7503311 - r7503309;
        double r7503313 = fma(r7503306, r7503308, r7503312);
        double r7503314 = expm1(r7503313);
        double r7503315 = log1p(r7503314);
        return r7503315;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Target15.1
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.9

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

  3. Applied sin-sum21.7

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

  5. Applied add-cube-cbrt21.8

    \[\leadsto \left(\sin x \cdot \cos \varepsilon + \color{blue}{\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}\right)} \cdot \sin \varepsilon\right) - \sin x\]

  6. Applied associate-*l*21.8

    \[\leadsto \left(\sin x \cdot \cos \varepsilon + \color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \varepsilon\right)}\right) - \sin x\]
  7. Using strategy rm

  8. Applied log1p-expm1-u21.9

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\sin x \cdot \cos \varepsilon + \left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \left(\sqrt[3]{\cos x} \cdot \sin \varepsilon\right)\right) - \sin x\right)\right)}\]

  9. Simplified0.5

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \cos \varepsilon - \sin x\right)\right)}\right)\]

  10. Final simplification0.5

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \cos \varepsilon - \sin x\right)\right)\right)\]

Reproduce

herbie shell --seed 2019192 +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)))