Average Error: 1.0 → 0.6
Time: 1.2m
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\beta \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} + \left(-\alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
\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 \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} + \left(-\alpha\right) \cdot \frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}
double f(double alpha, double beta, double i) {
        double r4940087 = alpha;
        double r4940088 = beta;
        double r4940089 = r4940087 + r4940088;
        double r4940090 = r4940088 - r4940087;
        double r4940091 = r4940089 * r4940090;
        double r4940092 = 2.0;
        double r4940093 = /* ERROR: no posit support in C */;
        double r4940094 = i;
        double r4940095 = r4940093 * r4940094;
        double r4940096 = r4940089 + r4940095;
        double r4940097 = r4940091 / r4940096;
        double r4940098 = 2.0;
        double r4940099 = /* ERROR: no posit support in C */;
        double r4940100 = r4940096 + r4940099;
        double r4940101 = r4940097 / r4940100;
        double r4940102 = 1.0;
        double r4940103 = /* ERROR: no posit support in C */;
        double r4940104 = r4940101 + r4940103;
        double r4940105 = r4940104 / r4940099;
        return r4940105;
}
↓
double f(double alpha, double beta, double i) {
        double r4940106 = beta;
        double r4940107 = alpha;
        double r4940108 = r4940107 + r4940106;
        double r4940109 = 2.0;
        double r4940110 = i;
        double r4940111 = r4940109 * r4940110;
        double r4940112 = r4940108 + r4940111;
        double r4940113 = r4940108 / r4940112;
        double r4940114 = r4940106 * r4940113;
        double r4940115 = -r4940107;
        double r4940116 = r4940115 * r4940113;
        double r4940117 = r4940114 + r4940116;
        double r4940118 = 2.0;
        double r4940119 = r4940112 + r4940118;
        double r4940120 = r4940117 / r4940119;
        double r4940121 = 1.0;
        double r4940122 = r4940120 + r4940121;
        double r4940123 = r4940122 / r4940118;
        return r4940123;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 1.0

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

  3. Applied associate-/l*0.6

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

  5. Applied associate-/r/0.6

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

  7. Applied sub-neg0.6

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

  8. Applied distribute-rgt-in0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019132 +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)))