Data.Approximate.Numerics:blog from approximate-0.2.2.1

Average Error: 0.2 → 0.1

Time: 15.2s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r34742551 = 6.0;
        double r34742552 = x;
        double r34742553 = 1.0;
        double r34742554 = r34742552 - r34742553;
        double r34742555 = r34742551 * r34742554;
        double r34742556 = r34742552 + r34742553;
        double r34742557 = 4.0;
        double r34742558 = sqrt(r34742552);
        double r34742559 = r34742557 * r34742558;
        double r34742560 = r34742556 + r34742559;
        double r34742561 = r34742555 / r34742560;
        return r34742561;
}

↓

double f(double x) {
        double r34742562 = 6.0;
        double r34742563 = x;
        double r34742564 = sqrt(r34742563);
        double r34742565 = 4.0;
        double r34742566 = 1.0;
        double r34742567 = r34742563 + r34742566;
        double r34742568 = fma(r34742564, r34742565, r34742567);
        double r34742569 = r34742563 - r34742566;
        double r34742570 = r34742568 / r34742569;
        double r34742571 = r34742562 / r34742570;
        double r34742572 = exp(r34742571);
        double r34742573 = log(r34742572);
        return r34742573;
}

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 add-log-exp 0.1

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

5. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019158 +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)))))