Average Error: 0.5 → 0.5
Time: 54.6s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\sqrt{e^{\mathsf{log1p}\left(\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{v \cdot v - 1}\right)\right)\right)}} \cdot \sqrt{e^{\mathsf{log1p}\left(\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{v \cdot v - 1}\right)\right)\right)}} - 1\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\sqrt{e^{\mathsf{log1p}\left(\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{v \cdot v - 1}\right)\right)\right)}} \cdot \sqrt{e^{\mathsf{log1p}\left(\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{v \cdot v - 1}\right)\right)\right)}} - 1
double f(double v) {
        double r18959096 = 1.0;
        double r18959097 = 5.0;
        double r18959098 = v;
        double r18959099 = r18959098 * r18959098;
        double r18959100 = r18959097 * r18959099;
        double r18959101 = r18959096 - r18959100;
        double r18959102 = r18959099 - r18959096;
        double r18959103 = r18959101 / r18959102;
        double r18959104 = acos(r18959103);
        return r18959104;
}
↓
double f(double v) {
        double r18959105 = -5.0;
        double r18959106 = v;
        double r18959107 = r18959106 * r18959106;
        double r18959108 = 1.0;
        double r18959109 = fma(r18959105, r18959107, r18959108);
        double r18959110 = r18959107 - r18959108;
        double r18959111 = r18959109 / r18959110;
        double r18959112 = acos(r18959111);
        double r18959113 = log1p(r18959112);
        double r18959114 = exp(r18959113);
        double r18959115 = sqrt(r18959114);
        double r18959116 = r18959115 * r18959115;
        double r18959117 = r18959116 - r18959108;
        return r18959117;
}

Error

Bits error versus v

Derivation

  1. Initial program 0.5

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]

  2. Simplified0.5

    \[\leadsto \color{blue}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{v \cdot v - 1}\right)}\]
  3. Using strategy rm

  4. Applied expm1-log1p-u0.5

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{v \cdot v - 1}\right)\right)\right)\right)\right)}\]
  5. Using strategy rm

  6. Applied expm1-udef0.5

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{v \cdot v - 1}\right)\right)\right)} - 1}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt0.5

    \[\leadsto \color{blue}{\sqrt{e^{\mathsf{log1p}\left(\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{v \cdot v - 1}\right)\right)\right)}} \cdot \sqrt{e^{\mathsf{log1p}\left(\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{v \cdot v - 1}\right)\right)\right)}}} - 1\]

  9. Final simplification0.5

    \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{v \cdot v - 1}\right)\right)\right)}} \cdot \sqrt{e^{\mathsf{log1p}\left(\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{v \cdot v - 1}\right)\right)\right)}} - 1\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))