Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.1

Time: 19.1s

Precision: 64

Math

\[\frac{6.0 \cdot \left(x - 1.0\right)}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}\]

↓

\[\frac{1}{\frac{\frac{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}{x - 1.0}}{6.0}}\]

C

double f(double x) {
        double r37789138 = 6.0;
        double r37789139 = x;
        double r37789140 = 1.0;
        double r37789141 = r37789139 - r37789140;
        double r37789142 = r37789138 * r37789141;
        double r37789143 = r37789139 + r37789140;
        double r37789144 = 4.0;
        double r37789145 = sqrt(r37789139);
        double r37789146 = r37789144 * r37789145;
        double r37789147 = r37789143 + r37789146;
        double r37789148 = r37789142 / r37789147;
        return r37789148;
}

↓

double f(double x) {
        double r37789149 = 1.0;
        double r37789150 = x;
        double r37789151 = sqrt(r37789150);
        double r37789152 = 4.0;
        double r37789153 = 1.0;
        double r37789154 = r37789150 + r37789153;
        double r37789155 = fma(r37789151, r37789152, r37789154);
        double r37789156 = r37789150 - r37789153;
        double r37789157 = r37789155 / r37789156;
        double r37789158 = 6.0;
        double r37789159 = r37789157 / r37789158;
        double r37789160 = r37789149 / r37789159;
        return r37789160;
}

Error

Bits error versus x

Target

Original	0.2
Target	0.1
Herbie	0.1

\[\frac{6.0}{\frac{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}{x - 1.0}}\]

Derivation

1. Initial program 0.2

   \[\frac{6.0 \cdot \left(x - 1.0\right)}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}\]

2. Simplified 0.1

   \[\leadsto \color{blue}{\frac{6.0}{\frac{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}{x - 1.0}}}\]
3. Using strategy rm

4. Applied clear-num 0.1

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}{x - 1.0}}{6.0}}}\]

5. Final simplification 0.1

   \[\leadsto \frac{1}{\frac{\frac{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}{x - 1.0}}{6.0}}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))