Average Error: 0.2 → 0.0
Time: 4.6s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\frac{1}{6}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\frac{1}{6}}
double f(double x) {
        double r907084 = 6.0;
        double r907085 = x;
        double r907086 = 1.0;
        double r907087 = r907085 - r907086;
        double r907088 = r907084 * r907087;
        double r907089 = r907085 + r907086;
        double r907090 = 4.0;
        double r907091 = sqrt(r907085);
        double r907092 = r907090 * r907091;
        double r907093 = r907089 + r907092;
        double r907094 = r907088 / r907093;
        return r907094;
}
↓
double f(double x) {
        double r907095 = x;
        double r907096 = 1.0;
        double r907097 = r907095 - r907096;
        double r907098 = sqrt(r907095);
        double r907099 = 4.0;
        double r907100 = r907095 + r907096;
        double r907101 = fma(r907098, r907099, r907100);
        double r907102 = r907097 / r907101;
        double r907103 = 1.0;
        double r907104 = 6.0;
        double r907105 = r907103 / r907104;
        double r907106 = r907102 / r907105;
        return r907106;
}

Error

Bits error versus x

Target

Original0.2
Target0.0
Herbie0.0
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

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-inv0.2

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

  5. Applied associate-/r*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020033 +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)))))