Average Error: 0.2 → 0.1
Time: 3.7s
Precision: binary64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}} \]
↓
\[6 \cdot \log \left(e^{\frac{-1 + x}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right) \]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \log \left(e^{\frac{-1 + x}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}}\right)

(FPCore (x)
 :precision binary64
 (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))
↓
(FPCore (x)
 :precision binary64
 (* 6.0 (log (exp (/ (+ -1.0 x) (fma 4.0 (sqrt x) (+ x 1.0)))))))
double code(double x) {
	return (6.0 * (x - 1.0)) / ((x + 1.0) + (4.0 * sqrt(x)));
}
↓
double code(double x) {
	return 6.0 * log(exp((-1.0 + x) / fma(4.0, sqrt(x), (x + 1.0))));
}

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. Applied *-un-lft-identity_binary640.2

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

  3. Applied times-frac_binary640.0

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

  4. Simplified0.0

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

  5. Simplified0.0

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

  6. Applied add-log-exp_binary640.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2021275 
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :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)))))