Average Error: 23.2 → 12.1
Time: 25.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\frac{e^{\log \left(\mathsf{fma}\left(\left(\beta - \alpha\right) \cdot \frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2.0}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1.0\right)\right)}}{2.0}\]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}
↓
\frac{e^{\log \left(\mathsf{fma}\left(\left(\beta - \alpha\right) \cdot \frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2.0}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1.0\right)\right)}}{2.0}
double f(double alpha, double beta, double i) {
        double r4006310 = alpha;
        double r4006311 = beta;
        double r4006312 = r4006310 + r4006311;
        double r4006313 = r4006311 - r4006310;
        double r4006314 = r4006312 * r4006313;
        double r4006315 = 2.0;
        double r4006316 = i;
        double r4006317 = r4006315 * r4006316;
        double r4006318 = r4006312 + r4006317;
        double r4006319 = r4006314 / r4006318;
        double r4006320 = 2.0;
        double r4006321 = r4006318 + r4006320;
        double r4006322 = r4006319 / r4006321;
        double r4006323 = 1.0;
        double r4006324 = r4006322 + r4006323;
        double r4006325 = r4006324 / r4006320;
        return r4006325;
}
↓
double f(double alpha, double beta, double i) {
        double r4006326 = beta;
        double r4006327 = alpha;
        double r4006328 = r4006326 - r4006327;
        double r4006329 = 1.0;
        double r4006330 = 2.0;
        double r4006331 = i;
        double r4006332 = r4006327 + r4006326;
        double r4006333 = fma(r4006330, r4006331, r4006332);
        double r4006334 = 2.0;
        double r4006335 = r4006333 + r4006334;
        double r4006336 = r4006329 / r4006335;
        double r4006337 = r4006328 * r4006336;
        double r4006338 = r4006332 / r4006333;
        double r4006339 = 1.0;
        double r4006340 = fma(r4006337, r4006338, r4006339);
        double r4006341 = log(r4006340);
        double r4006342 = exp(r4006341);
        double r4006343 = r4006342 / r4006334;
        return r4006343;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 23.2

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

  2. Simplified19.2

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

  4. Applied fma-udef19.3

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

  5. Simplified12.1

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

  7. Applied add-exp-log12.1

    \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta + \alpha}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{\beta - \alpha}{2.0 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} + 1.0\right)}}}{2.0}\]

  8. Simplified12.1

    \[\leadsto \frac{e^{\color{blue}{\log \left(\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2.0}, \frac{\beta + \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}, 1.0\right)\right)}}}{2.0}\]
  9. Using strategy rm

  10. Applied div-inv12.1

    \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\color{blue}{\left(\beta - \alpha\right) \cdot \frac{1}{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 2.0}}, \frac{\beta + \alpha}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}, 1.0\right)\right)}}{2.0}\]

  11. Final simplification12.1

    \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(\left(\beta - \alpha\right) \cdot \frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2.0}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1.0\right)\right)}}{2.0}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))