Average Error: 36.8 → 0.4
Time: 15.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\cos x \cdot \sin \varepsilon + \frac{\sin x}{\frac{\left(\cos \varepsilon \cdot \cos \varepsilon + 1\right) + \cos \varepsilon}{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\cos x \cdot \sin \varepsilon + \frac{\sin x}{\frac{\left(\cos \varepsilon \cdot \cos \varepsilon + 1\right) + \cos \varepsilon}{{\left(\cos \varepsilon\right)}^{3} - {1}^{3}}}
double f(double x, double eps) {
        double r70000 = x;
        double r70001 = eps;
        double r70002 = r70000 + r70001;
        double r70003 = sin(r70002);
        double r70004 = sin(r70000);
        double r70005 = r70003 - r70004;
        return r70005;
}
↓
double f(double x, double eps) {
        double r70006 = x;
        double r70007 = cos(r70006);
        double r70008 = eps;
        double r70009 = sin(r70008);
        double r70010 = r70007 * r70009;
        double r70011 = sin(r70006);
        double r70012 = cos(r70008);
        double r70013 = r70012 * r70012;
        double r70014 = 1.0;
        double r70015 = r70013 + r70014;
        double r70016 = r70015 + r70012;
        double r70017 = 3.0;
        double r70018 = pow(r70012, r70017);
        double r70019 = pow(r70014, r70017);
        double r70020 = r70018 - r70019;
        double r70021 = r70016 / r70020;
        double r70022 = r70011 / r70021;
        double r70023 = r70010 + r70022;
        return r70023;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target15.2
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Initial program 36.8

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

  3. Applied sin-sum21.5

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

  4. Applied associate--l+21.6

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

  5. Taylor expanded around inf 21.5

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

  6. Simplified0.4

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

  8. Applied flip3--0.4

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

  9. Applied associate-*r/0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019298 
(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)))