Average Error: 36.9 → 0.4
Time: 6.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\mathsf{fma}\left(\sin x, \frac{1}{\frac{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}{{\left(\cos \varepsilon\right)}^{3} - 1}}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\mathsf{fma}\left(\sin x, \frac{1}{\frac{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}{{\left(\cos \varepsilon\right)}^{3} - 1}}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)
double f(double x, double eps) {
        double r119077 = x;
        double r119078 = eps;
        double r119079 = r119077 + r119078;
        double r119080 = sin(r119079);
        double r119081 = sin(r119077);
        double r119082 = r119080 - r119081;
        return r119082;
}
↓
double f(double x, double eps) {
        double r119083 = x;
        double r119084 = sin(r119083);
        double r119085 = 1.0;
        double r119086 = eps;
        double r119087 = cos(r119086);
        double r119088 = r119087 + r119085;
        double r119089 = fma(r119087, r119088, r119085);
        double r119090 = 3.0;
        double r119091 = pow(r119087, r119090);
        double r119092 = r119091 - r119085;
        double r119093 = r119089 / r119092;
        double r119094 = r119085 / r119093;
        double r119095 = cos(r119083);
        double r119096 = sin(r119086);
        double r119097 = r119095 * r119096;
        double r119098 = fma(r119084, r119094, r119097);
        double r119099 = -r119084;
        double r119100 = fma(r119099, r119085, r119084);
        double r119101 = r119098 + r119100;
        return r119101;
}

Error

Bits error versus x

Bits error versus eps

Target

Original36.9
Target15.1
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

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-cbrt22.2

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

  6. Applied add-sqr-sqrt43.3

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

  7. Applied prod-diff43.3

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

  8. Simplified22.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \cos x \cdot \sin \varepsilon\right)} + \mathsf{fma}\left(-\sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x} \cdot \left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)\right)\]

  9. Simplified0.4

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

  11. Applied flip3--0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \color{blue}{\frac{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]

  12. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \frac{\color{blue}{{\left(\cos \varepsilon\right)}^{3} - 1}}{\cos \varepsilon \cdot \cos \varepsilon + \left(1 \cdot 1 + \cos \varepsilon \cdot 1\right)}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]

  13. Simplified0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\color{blue}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]
  14. Using strategy rm

  15. Applied clear-num0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}{{\left(\cos \varepsilon\right)}^{3} - 1}}}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]

  16. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \frac{1}{\frac{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}{{\left(\cos \varepsilon\right)}^{3} - 1}}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

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