Average Error: 31.6 → 0.2
Time: 9.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0266096722673715806:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(1 - \cos x\right)\right)}{x}\\ \mathbf{elif}\;x \le 0.0240834512536182467:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1 - \cos x}{x}}{x}\right)\right)\\ \end{array}\]
\frac{1 - \cos x}{x \cdot x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0266096722673715806:\\
\;\;\;\;\frac{1}{x} \cdot \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(1 - \cos x\right)\right)}{x}\\

\mathbf{elif}\;x \le 0.0240834512536182467:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1 - \cos x}{x}}{x}\right)\right)\\

\end{array}
double f(double x) {
        double r25928 = 1.0;
        double r25929 = x;
        double r25930 = cos(r25929);
        double r25931 = r25928 - r25930;
        double r25932 = r25929 * r25929;
        double r25933 = r25931 / r25932;
        return r25933;
}
↓
double f(double x) {
        double r25934 = x;
        double r25935 = -0.02660967226737158;
        bool r25936 = r25934 <= r25935;
        double r25937 = 1.0;
        double r25938 = r25937 / r25934;
        double r25939 = 1.0;
        double r25940 = cos(r25934);
        double r25941 = r25939 - r25940;
        double r25942 = expm1(r25941);
        double r25943 = log1p(r25942);
        double r25944 = r25943 / r25934;
        double r25945 = r25938 * r25944;
        double r25946 = 0.024083451253618247;
        bool r25947 = r25934 <= r25946;
        double r25948 = 0.001388888888888889;
        double r25949 = 4.0;
        double r25950 = pow(r25934, r25949);
        double r25951 = 0.5;
        double r25952 = fma(r25948, r25950, r25951);
        double r25953 = 0.041666666666666664;
        double r25954 = 2.0;
        double r25955 = pow(r25934, r25954);
        double r25956 = r25953 * r25955;
        double r25957 = r25952 - r25956;
        double r25958 = r25941 / r25934;
        double r25959 = r25958 / r25934;
        double r25960 = expm1(r25959);
        double r25961 = log1p(r25960);
        double r25962 = r25947 ? r25957 : r25961;
        double r25963 = r25936 ? r25945 : r25962;
        return r25963;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.02660967226737158

    1. Initial program 0.9

      \[\frac{1 - \cos x}{x \cdot x}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.9

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

    4. Applied times-frac0.5

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]
    5. Using strategy rm

    6. Applied log1p-expm1-u0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(1 - \cos x\right)\right)}}{x}\]

    if -0.02660967226737158 < x < 0.024083451253618247

    1. Initial program 62.2

      \[\frac{1 - \cos x}{x \cdot x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]

    if 0.024083451253618247 < x

    1. Initial program 1.0

      \[\frac{1 - \cos x}{x \cdot x}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity1.0

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

    4. Applied times-frac0.5

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]
    5. Using strategy rm

    6. Applied log1p-expm1-u0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(1 - \cos x\right)\right)}}{x}\]
    7. Using strategy rm

    8. Applied log1p-expm1-u0.5

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x} \cdot \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(1 - \cos x\right)\right)}{x}\right)\right)}\]

    9. Simplified0.4

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\frac{1 - \cos x}{x}}{x}\right)}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0266096722673715806:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(1 - \cos x\right)\right)}{x}\\ \mathbf{elif}\;x \le 0.0240834512536182467:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1 - \cos x}{x}}{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))