Average Error: 0.2 → 0.1
Time: 36.2s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{\frac{6}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\sqrt{x} + \sqrt{1}}}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\sqrt{x} - \sqrt{1}}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{\frac{6}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\sqrt{x} + \sqrt{1}}}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\sqrt{x} - \sqrt{1}}}
double f(double x) {
        double r594202 = 6.0;
        double r594203 = x;
        double r594204 = 1.0;
        double r594205 = r594203 - r594204;
        double r594206 = r594202 * r594205;
        double r594207 = r594203 + r594204;
        double r594208 = 4.0;
        double r594209 = sqrt(r594203);
        double r594210 = r594208 * r594209;
        double r594211 = r594207 + r594210;
        double r594212 = r594206 / r594211;
        return r594212;
}
↓
double f(double x) {
        double r594213 = 6.0;
        double r594214 = x;
        double r594215 = sqrt(r594214);
        double r594216 = 4.0;
        double r594217 = 1.0;
        double r594218 = r594214 + r594217;
        double r594219 = fma(r594215, r594216, r594218);
        double r594220 = sqrt(r594219);
        double r594221 = sqrt(r594217);
        double r594222 = r594215 + r594221;
        double r594223 = r594220 / r594222;
        double r594224 = r594213 / r594223;
        double r594225 = r594215 - r594221;
        double r594226 = r594220 / r594225;
        double r594227 = r594224 / r594226;
        return r594227;
}

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. Simplified0.1

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

  4. Applied add-sqr-sqrt0.1

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

  5. Applied add-sqr-sqrt0.3

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

  6. Applied difference-of-squares0.3

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied times-frac0.1

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

  9. Applied associate-/r*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019322 +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)))))