Average Error: 14.9 → 14.4
Time: 4.2s
Precision: 64
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
↓
\[\frac{e^{\log \left(1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
↓
\frac{e^{\log \left(1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}
double code(double x) {
	return (1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x))))));
}
↓
double code(double x) {
	return (exp(log(((1.0 * (1.0 - 0.5)) - (0.5 * (((cbrt(1.0) * cbrt(1.0)) / sqrt(hypot(1.0, x))) * (cbrt(1.0) / sqrt(hypot(1.0, x)))))))) / (1.0 + sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))))));
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.9

    \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
  2. Using strategy rm

  3. Applied flip--14.9

    \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

  4. Simplified14.4

    \[\leadsto \frac{\color{blue}{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt14.4

    \[\leadsto \frac{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \frac{1}{\color{blue}{\sqrt{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  7. Applied add-cube-cbrt14.4

    \[\leadsto \frac{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  8. Applied times-frac14.4

    \[\leadsto \frac{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\mathsf{hypot}\left(1, x\right)}}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
  9. Using strategy rm

  10. Applied add-exp-log14.4

    \[\leadsto \frac{\color{blue}{e^{\log \left(1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

  11. Final simplification14.4

    \[\leadsto \frac{e^{\log \left(1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))