Average Error: 0.2 → 0.1
Time: 4.6s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \log \left({\left(e^{\frac{1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}\right)}^{6}\right)\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \log \left({\left(e^{\frac{1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}\right)}^{6}\right)
double f(double x) {
        double r986119 = 6.0;
        double r986120 = x;
        double r986121 = 1.0;
        double r986122 = r986120 - r986121;
        double r986123 = r986119 * r986122;
        double r986124 = r986120 + r986121;
        double r986125 = 4.0;
        double r986126 = sqrt(r986120);
        double r986127 = r986125 * r986126;
        double r986128 = r986124 + r986127;
        double r986129 = r986123 / r986128;
        return r986129;
}
↓
double f(double x) {
        double r986130 = x;
        double r986131 = sqrt(r986130);
        double r986132 = 4.0;
        double r986133 = 1.0;
        double r986134 = r986130 + r986133;
        double r986135 = fma(r986131, r986132, r986134);
        double r986136 = 6.0;
        double r986137 = r986135 / r986136;
        double r986138 = r986130 / r986137;
        double r986139 = r986133 / r986135;
        double r986140 = exp(r986139);
        double r986141 = pow(r986140, r986136);
        double r986142 = log(r986141);
        double r986143 = r986138 - r986142;
        return r986143;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
Herbie0.1
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Derivation

  1. Initial program 0.2

    \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}}\]
  3. Using strategy rm

  4. Applied div-sub0.1

    \[\leadsto \color{blue}{\frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}}\]
  5. Using strategy rm

  6. Applied div-inv0.1

    \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right) \cdot \frac{1}{6}}}\]
  7. Using strategy rm

  8. Applied add-log-exp0.1

    \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \color{blue}{\log \left(e^{\frac{1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right) \cdot \frac{1}{6}}}\right)}\]

  9. Simplified0.1

    \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \log \color{blue}{\left({\left(e^{\frac{1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}\right)}^{6}\right)}\]

  10. Final simplification0.1

    \[\leadsto \frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \log \left({\left(e^{\frac{1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}\right)}^{6}\right)\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :herbie-target
  (/ 6 (/ (+ (+ x 1) (* 4 (sqrt x))) (- x 1)))

  (/ (* 6 (- x 1)) (+ (+ x 1) (* 4 (sqrt x)))))