Average Error: 16.4 → 16.4
Time: 20.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\frac{e^{\log \left(\frac{\frac{{\left({1}^{3}\right)}^{3} + {\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}\right)}^{3}}{\mathsf{fma}\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}, {\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}, {1}^{6}\right)}}{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} - 1, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}, 1 \cdot 1\right)}\right)}}{2}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\frac{e^{\log \left(\frac{\frac{{\left({1}^{3}\right)}^{3} + {\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}\right)}^{3}}{\mathsf{fma}\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}, {\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}, {1}^{6}\right)}}{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} - 1, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}, 1 \cdot 1\right)}\right)}}{2}
double f(double alpha, double beta) {
        double r85092 = beta;
        double r85093 = alpha;
        double r85094 = r85092 - r85093;
        double r85095 = r85093 + r85092;
        double r85096 = 2.0;
        double r85097 = r85095 + r85096;
        double r85098 = r85094 / r85097;
        double r85099 = 1.0;
        double r85100 = r85098 + r85099;
        double r85101 = r85100 / r85096;
        return r85101;
}
↓
double f(double alpha, double beta) {
        double r85102 = 1.0;
        double r85103 = 3.0;
        double r85104 = pow(r85102, r85103);
        double r85105 = pow(r85104, r85103);
        double r85106 = beta;
        double r85107 = alpha;
        double r85108 = r85106 - r85107;
        double r85109 = r85107 + r85106;
        double r85110 = 2.0;
        double r85111 = r85109 + r85110;
        double r85112 = r85108 / r85111;
        double r85113 = pow(r85112, r85103);
        double r85114 = pow(r85113, r85103);
        double r85115 = r85105 + r85114;
        double r85116 = r85113 - r85104;
        double r85117 = 6.0;
        double r85118 = pow(r85102, r85117);
        double r85119 = fma(r85116, r85113, r85118);
        double r85120 = r85115 / r85119;
        double r85121 = r85112 - r85102;
        double r85122 = r85102 * r85102;
        double r85123 = fma(r85121, r85112, r85122);
        double r85124 = r85120 / r85123;
        double r85125 = log(r85124);
        double r85126 = exp(r85125);
        double r85127 = r85126 / r85110;
        return r85127;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 16.4

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

  2. Simplified16.4

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

  4. Applied add-exp-log16.4

    \[\leadsto \frac{\color{blue}{e^{\log \left(1 + \frac{\beta - \alpha}{\left(\alpha + 2\right) + \beta}\right)}}}{2}\]

  5. Simplified16.4

    \[\leadsto \frac{e^{\color{blue}{\log \left(1 + \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}}}{2}\]
  6. Using strategy rm

  7. Applied flip3-+16.4

    \[\leadsto \frac{e^{\log \color{blue}{\left(\frac{{1}^{3} + {\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}^{3}}{1 \cdot 1 + \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} - 1 \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}\right)}}}{2}\]

  8. Simplified16.4

    \[\leadsto \frac{e^{\log \left(\frac{\color{blue}{{1}^{3} + {\left(\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}\right)}^{3}}}{1 \cdot 1 + \left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2} - 1 \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}\right)}}{2}\]

  9. Simplified16.4

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

  11. Applied flip3-+16.4

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

  12. Simplified16.4

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

  13. Simplified16.4

    \[\leadsto \frac{e^{\log \left(\frac{\frac{{\left({\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}^{3}\right)}^{3} + {\left({1}^{3}\right)}^{3}}{\color{blue}{\mathsf{fma}\left({\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}^{3} - {1}^{3}, {\left(\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2}\right)}^{3}, {1}^{6}\right)}}}{\mathsf{fma}\left(\frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)} - 1, \frac{\beta - \alpha}{2 + \left(\alpha + \beta\right)}, 1 \cdot 1\right)}\right)}}{2}\]

  14. Final simplification16.4

    \[\leadsto \frac{e^{\log \left(\frac{\frac{{\left({1}^{3}\right)}^{3} + {\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}\right)}^{3}}{\mathsf{fma}\left({\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}, {\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3}, {1}^{6}\right)}}{\mathsf{fma}\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} - 1, \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2}, 1 \cdot 1\right)}\right)}}{2}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))