Average Error: 0.0 → 0.0
Time: 20.3s
Precision: 64
\[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
↓
\[\tan^{-1} \left(\sqrt{(e^{\log_* (1 + \frac{1 - x}{1 + x})} - 1)^*}\right) \cdot 2\]
2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)
↓
\tan^{-1} \left(\sqrt{(e^{\log_* (1 + \frac{1 - x}{1 + x})} - 1)^*}\right) \cdot 2
double f(double x) {
        double r505202 = 2.0;
        double r505203 = 1.0;
        double r505204 = x;
        double r505205 = r505203 - r505204;
        double r505206 = r505203 + r505204;
        double r505207 = r505205 / r505206;
        double r505208 = sqrt(r505207);
        double r505209 = atan(r505208);
        double r505210 = r505202 * r505209;
        return r505210;
}
↓
double f(double x) {
        double r505211 = 1.0;
        double r505212 = x;
        double r505213 = r505211 - r505212;
        double r505214 = r505211 + r505212;
        double r505215 = r505213 / r505214;
        double r505216 = log1p(r505215);
        double r505217 = expm1(r505216);
        double r505218 = sqrt(r505217);
        double r505219 = atan(r505218);
        double r505220 = 2.0;
        double r505221 = r505219 * r505220;
        return r505221;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 0.0

    \[2 \cdot \tan^{-1} \left(\sqrt{\frac{1 - x}{1 + x}}\right)\]
  2. Using strategy rm

  3. Applied expm1-log1p-u0.0

    \[\leadsto 2 \cdot \tan^{-1} \left(\sqrt{\color{blue}{(e^{\log_* (1 + \frac{1 - x}{1 + x})} - 1)^*}}\right)\]

  4. Final simplification0.0

    \[\leadsto \tan^{-1} \left(\sqrt{(e^{\log_* (1 + \frac{1 - x}{1 + x})} - 1)^*}\right) \cdot 2\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x)
  :name "arccos"
  (* 2 (atan (sqrt (/ (- 1 x) (+ 1 x))))))