Average Error: 0.9 → 0.6
Time: 30.9s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\beta - \alpha}{\left(2 \cdot i + \beta\right) + \alpha} \cdot \frac{\beta + \alpha}{\beta + \left(\left(\alpha + 2.0\right) + 2 \cdot i\right)} + 1.0}{2.0}\]
\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}
↓
\frac{\frac{\beta - \alpha}{\left(2 \cdot i + \beta\right) + \alpha} \cdot \frac{\beta + \alpha}{\beta + \left(\left(\alpha + 2.0\right) + 2 \cdot i\right)} + 1.0}{2.0}
double f(double alpha, double beta, double i) {
        double r1137059 = alpha;
        double r1137060 = beta;
        double r1137061 = r1137059 + r1137060;
        double r1137062 = r1137060 - r1137059;
        double r1137063 = r1137061 * r1137062;
        double r1137064 = 2.0;
        double r1137065 = /* ERROR: no posit support in C */;
        double r1137066 = i;
        double r1137067 = r1137065 * r1137066;
        double r1137068 = r1137061 + r1137067;
        double r1137069 = r1137063 / r1137068;
        double r1137070 = 2.0;
        double r1137071 = /* ERROR: no posit support in C */;
        double r1137072 = r1137068 + r1137071;
        double r1137073 = r1137069 / r1137072;
        double r1137074 = 1.0;
        double r1137075 = /* ERROR: no posit support in C */;
        double r1137076 = r1137073 + r1137075;
        double r1137077 = r1137076 / r1137071;
        return r1137077;
}
↓
double f(double alpha, double beta, double i) {
        double r1137078 = beta;
        double r1137079 = alpha;
        double r1137080 = r1137078 - r1137079;
        double r1137081 = 2.0;
        double r1137082 = i;
        double r1137083 = r1137081 * r1137082;
        double r1137084 = r1137083 + r1137078;
        double r1137085 = r1137084 + r1137079;
        double r1137086 = r1137080 / r1137085;
        double r1137087 = r1137078 + r1137079;
        double r1137088 = 2.0;
        double r1137089 = r1137079 + r1137088;
        double r1137090 = r1137089 + r1137083;
        double r1137091 = r1137078 + r1137090;
        double r1137092 = r1137087 / r1137091;
        double r1137093 = r1137086 * r1137092;
        double r1137094 = 1.0;
        double r1137095 = r1137093 + r1137094;
        double r1137096 = r1137095 / r1137088;
        return r1137096;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 0.9

    \[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  2. Simplified1.0

    \[\leadsto \color{blue}{\frac{\left(\frac{\left(\frac{\left(\left(\beta - \alpha\right) \cdot \left(\frac{\beta}{\alpha}\right)\right)}{\left(\left(\frac{\left(\left(2\right) \cdot i\right)}{\left(\frac{\beta}{\alpha}\right)}\right) \cdot \left(\frac{\left(\frac{\beta}{\left(\frac{\alpha}{\left(2.0\right)}\right)}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}}\]
  3. Using strategy rm

  4. Applied p16-times-frac0.6

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\left(2\right) \cdot i\right)}{\left(\frac{\beta}{\alpha}\right)}\right)}\right) \cdot \left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(\frac{\left(\frac{\beta}{\left(\frac{\alpha}{\left(2.0\right)}\right)}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)\right)}}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  5. Using strategy rm

  6. Applied associate-+r+0.6

    \[\leadsto \frac{\left(\frac{\left(\left(\frac{\left(\beta - \alpha\right)}{\color{blue}{\left(\frac{\left(\frac{\left(\left(2\right) \cdot i\right)}{\beta}\right)}{\alpha}\right)}}\right) \cdot \left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(\frac{\left(\frac{\beta}{\left(\frac{\alpha}{\left(2.0\right)}\right)}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  7. Using strategy rm

  8. Applied associate-+l+0.6

    \[\leadsto \frac{\left(\frac{\left(\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\left(\left(2\right) \cdot i\right)}{\beta}\right)}{\alpha}\right)}\right) \cdot \left(\frac{\left(\frac{\beta}{\alpha}\right)}{\color{blue}{\left(\frac{\beta}{\left(\frac{\left(\frac{\alpha}{\left(2.0\right)}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}}\right)\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  9. Final simplification0.6

    \[\leadsto \frac{\frac{\beta - \alpha}{\left(2 \cdot i + \beta\right) + \alpha} \cdot \frac{\beta + \alpha}{\beta + \left(\left(\alpha + 2.0\right) + 2 \cdot i\right)} + 1.0}{2.0}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)) (>.p16 i (real->posit16 0)))
  (/.p16 (+.p16 (/.p16 (/.p16 (*.p16 (+.p16 alpha beta) (-.p16 beta alpha)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i))) (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (real->posit16 2.0))) (real->posit16 1.0)) (real->posit16 2.0)))