Octave 3.8, jcobi/3

Average Error: 3.5 → 2.4

Time: 42.0s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.389187053396531788483121451916546809922 \cdot 10^{214}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}} \cdot \sqrt{\frac{1}{\sqrt{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}\right) \cdot \frac{\sqrt{\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}{1 + \mathsf{fma}\left(2, 1, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}{\mathsf{fma}\left(\sqrt{0.5}, \beta, \mathsf{fma}\left(1, \sqrt{0.5}, \sqrt{0.5} \cdot \alpha\right)\right) - \mathsf{fma}\left(0.5, \beta \cdot \sqrt{0.125}, 0.125 \cdot \frac{\alpha}{\sqrt{0.5}}\right)}}}{1 + \mathsf{fma}\left(2, 1, \alpha + \beta\right)}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r7947052 = alpha;
        double r7947053 = beta;
        double r7947054 = r7947052 + r7947053;
        double r7947055 = r7947053 * r7947052;
        double r7947056 = r7947054 + r7947055;
        double r7947057 = 1.0;
        double r7947058 = r7947056 + r7947057;
        double r7947059 = 2.0;
        double r7947060 = r7947059 * r7947057;
        double r7947061 = r7947054 + r7947060;
        double r7947062 = r7947058 / r7947061;
        double r7947063 = r7947062 / r7947061;
        double r7947064 = r7947061 + r7947057;
        double r7947065 = r7947063 / r7947064;
        return r7947065;
}

↓

double f(double alpha, double beta) {
        double r7947066 = alpha;
        double r7947067 = 3.389187053396532e+214;
        bool r7947068 = r7947066 <= r7947067;
        double r7947069 = 1.0;
        double r7947070 = beta;
        double r7947071 = r7947066 + r7947070;
        double r7947072 = fma(r7947070, r7947066, r7947071);
        double r7947073 = r7947069 + r7947072;
        double r7947074 = 2.0;
        double r7947075 = fma(r7947074, r7947069, r7947071);
        double r7947076 = r7947073 / r7947075;
        double r7947077 = r7947076 / r7947075;
        double r7947078 = sqrt(r7947077);
        double r7947079 = 1.0;
        double r7947080 = sqrt(r7947075);
        double r7947081 = r7947079 / r7947080;
        double r7947082 = sqrt(r7947081);
        double r7947083 = r7947078 * r7947082;
        double r7947084 = r7947073 / r7947080;
        double r7947085 = r7947084 / r7947075;
        double r7947086 = sqrt(r7947085);
        double r7947087 = r7947069 + r7947075;
        double r7947088 = r7947086 / r7947087;
        double r7947089 = r7947083 * r7947088;
        double r7947090 = 0.5;
        double r7947091 = sqrt(r7947090);
        double r7947092 = r7947091 * r7947066;
        double r7947093 = fma(r7947069, r7947091, r7947092);
        double r7947094 = fma(r7947091, r7947070, r7947093);
        double r7947095 = 0.125;
        double r7947096 = sqrt(r7947095);
        double r7947097 = r7947070 * r7947096;
        double r7947098 = r7947066 / r7947091;
        double r7947099 = r7947095 * r7947098;
        double r7947100 = fma(r7947090, r7947097, r7947099);
        double r7947101 = r7947094 - r7947100;
        double r7947102 = r7947075 / r7947101;
        double r7947103 = r7947081 / r7947102;
        double r7947104 = r7947103 / r7947087;
        double r7947105 = r7947068 ? r7947089 : r7947104;
        return r7947105;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 3.389187053396532e+214

   1. Initial program 2.1

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

   2. Simplified 2.1

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

   4. Applied *-un-lft-identity 2.1

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

   5. Applied add-sqr-sqrt 2.1

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

   6. Applied times-frac 2.1

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

   7. Simplified 2.1

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

   9. Applied *-un-lft-identity 2.1

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

   10. Applied *-un-lft-identity 2.1

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

   11. Applied add-sqr-sqrt 2.1

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

   12. Applied *-un-lft-identity 2.1

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

   13. Applied times-frac 2.1

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

   14. Applied times-frac 2.1

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

   15. Applied sqrt-prod 2.2

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

   16. Applied times-frac 2.2

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

   17. Applied associate-*r* 2.2

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

   18. Simplified 2.2

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

   if 3.389187053396532e+214 < alpha

   1. Initial program 16.4

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

   2. Simplified 16.4

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

   4. Applied add-sqr-sqrt 16.4

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

   5. Applied *-un-lft-identity 16.4

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

   6. Applied times-frac 16.4

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

   7. Applied associate-/l* 16.4

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

   8. Taylor expanded around 0 5.0

      \[\leadsto \frac{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}{\color{blue}{\left(\sqrt{0.5} \cdot \beta + \left(1 \cdot \sqrt{0.5} + \sqrt{0.5} \cdot \alpha\right)\right) - \left(0.5 \cdot \left(\beta \cdot \sqrt{0.125}\right) + 0.125 \cdot \frac{\alpha}{\sqrt{0.5}}\right)}}}}{1 + \mathsf{fma}\left(2, 1, \alpha + \beta\right)}\]

   9. Simplified 5.0

      \[\leadsto \frac{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}{\color{blue}{\mathsf{fma}\left(\sqrt{0.5}, \beta, \mathsf{fma}\left(1, \sqrt{0.5}, \sqrt{0.5} \cdot \alpha\right)\right) - \mathsf{fma}\left(0.5, \beta \cdot \sqrt{0.125}, 0.125 \cdot \frac{\alpha}{\sqrt{0.5}}\right)}}}}{1 + \mathsf{fma}\left(2, 1, \alpha + \beta\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.389187053396531788483121451916546809922 \cdot 10^{214}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}} \cdot \sqrt{\frac{1}{\sqrt{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}\right) \cdot \frac{\sqrt{\frac{\frac{1 + \mathsf{fma}\left(\beta, \alpha, \alpha + \beta\right)}{\sqrt{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}{1 + \mathsf{fma}\left(2, 1, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(2, 1, \alpha + \beta\right)}{\mathsf{fma}\left(\sqrt{0.5}, \beta, \mathsf{fma}\left(1, \sqrt{0.5}, \sqrt{0.5} \cdot \alpha\right)\right) - \mathsf{fma}\left(0.5, \beta \cdot \sqrt{0.125}, 0.125 \cdot \frac{\alpha}{\sqrt{0.5}}\right)}}}{1 + \mathsf{fma}\left(2, 1, \alpha + \beta\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ alpha beta) (* 2.0 1.0))) (+ (+ (+ alpha beta) (* 2.0 1.0)) 1.0)))