Average Error: 24.0 → 12.6
Time: 20.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)\right)}^{3}}}{2}\]
\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} + 1}{2}
↓
\frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)\right)}^{3}}}{2}
double f(double alpha, double beta, double i) {
        double r71085 = alpha;
        double r71086 = beta;
        double r71087 = r71085 + r71086;
        double r71088 = r71086 - r71085;
        double r71089 = r71087 * r71088;
        double r71090 = 2.0;
        double r71091 = i;
        double r71092 = r71090 * r71091;
        double r71093 = r71087 + r71092;
        double r71094 = r71089 / r71093;
        double r71095 = r71093 + r71090;
        double r71096 = r71094 / r71095;
        double r71097 = 1.0;
        double r71098 = r71096 + r71097;
        double r71099 = r71098 / r71090;
        return r71099;
}
↓
double f(double alpha, double beta, double i) {
        double r71100 = beta;
        double r71101 = alpha;
        double r71102 = r71100 - r71101;
        double r71103 = 2.0;
        double r71104 = i;
        double r71105 = r71101 + r71100;
        double r71106 = fma(r71103, r71104, r71105);
        double r71107 = r71106 + r71103;
        double r71108 = sqrt(r71107);
        double r71109 = r71102 / r71108;
        double r71110 = 1.0;
        double r71111 = r71110 / r71107;
        double r71112 = sqrt(r71111);
        double r71113 = r71109 * r71112;
        double r71114 = r71105 / r71106;
        double r71115 = 1.0;
        double r71116 = fma(r71113, r71114, r71115);
        double r71117 = 3.0;
        double r71118 = pow(r71116, r71117);
        double r71119 = cbrt(r71118);
        double r71120 = r71119 / r71103;
        return r71120;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 24.0

    \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]

  2. Simplified12.6

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}}\]
  3. Using strategy rm

  4. Applied div-inv12.6

    \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\beta - \alpha\right) \cdot \frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt12.7

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

  7. Applied associate-*r*12.7

    \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\left(\beta - \alpha\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\]
  8. Using strategy rm

  9. Applied sqrt-div12.6

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

  10. Applied associate-*r/12.6

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

  11. Simplified12.6

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\color{blue}{\beta - \alpha}}{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}{2}\]
  12. Using strategy rm

  13. Applied add-cbrt-cube12.6

    \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right) \cdot \mathsf{fma}\left(\frac{\beta - \alpha}{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)\right) \cdot \mathsf{fma}\left(\frac{\beta - \alpha}{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)}}}{2}\]

  14. Simplified12.6

    \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)\right)}^{3}}}}{2}\]

  15. Final simplification12.6

    \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(\frac{\beta - \alpha}{\sqrt{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, i, \alpha + \beta\right) + 2}}, \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}, 1\right)\right)}^{3}}}{2}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))