1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-154

   1. Initial program 99.7%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Step-by-step derivation

      1. sub-neg99.7%

         \[\leadsto x - \sqrt{\color{blue}{x \cdot x + \left(-\varepsilon\right)}} \]

      2. +-commutative99.7%

         \[\leadsto x - \sqrt{\color{blue}{\left(-\varepsilon\right) + x \cdot x}} \]

      3. add-sqr-sqrt99.7%

         \[\leadsto x - \sqrt{\color{blue}{\sqrt{-\varepsilon} \cdot \sqrt{-\varepsilon}} + x \cdot x} \]

      4. hypot-def99.8%

         \[\leadsto x - \color{blue}{\mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)} \]

   3. Applied egg-rr99.8%

      \[\leadsto x - \color{blue}{\mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)} \]

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.5%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   2. Taylor expanded in x around inf 93.6%

      \[\leadsto \color{blue}{0.125 \cdot \frac{{\varepsilon}^{2}}{{x}^{3}} + 0.5 \cdot \frac{\varepsilon}{x}} \]
   3. Step-by-step derivation

      1. fma-def93.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon}{x}\right)} \]

      2. unpow293.6%

         \[\leadsto \mathsf{fma}\left(0.125, \frac{\color{blue}{\varepsilon \cdot \varepsilon}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]

   4. Simplified93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, \frac{\varepsilon \cdot \varepsilon}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon}{x}\right)} \]
   5. Step-by-step derivation

      1. unpow393.6%

         \[\leadsto \mathsf{fma}\left(0.125, \frac{\varepsilon \cdot \varepsilon}{\color{blue}{\left(x \cdot x\right) \cdot x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]

      2. times-frac99.3%

         \[\leadsto \mathsf{fma}\left(0.125, \color{blue}{\frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]

   6. Applied egg-rr99.3%

      \[\leadsto \mathsf{fma}\left(0.125, \color{blue}{\frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;x - \mathsf{hypot}\left(\sqrt{-\varepsilon}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}, \frac{\varepsilon}{x} \cdot 0.5\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-154

   1. Initial program 99.7%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.5%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   2. Taylor expanded in x around inf 93.6%

      \[\leadsto \color{blue}{0.125 \cdot \frac{{\varepsilon}^{2}}{{x}^{3}} + 0.5 \cdot \frac{\varepsilon}{x}} \]
   3. Step-by-step derivation

      1. fma-def93.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, \frac{{\varepsilon}^{2}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon}{x}\right)} \]

      2. unpow293.6%

         \[\leadsto \mathsf{fma}\left(0.125, \frac{\color{blue}{\varepsilon \cdot \varepsilon}}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]

   4. Simplified93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, \frac{\varepsilon \cdot \varepsilon}{{x}^{3}}, 0.5 \cdot \frac{\varepsilon}{x}\right)} \]
   5. Step-by-step derivation

      1. unpow393.6%

         \[\leadsto \mathsf{fma}\left(0.125, \frac{\varepsilon \cdot \varepsilon}{\color{blue}{\left(x \cdot x\right) \cdot x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]

      2. times-frac99.3%

         \[\leadsto \mathsf{fma}\left(0.125, \color{blue}{\frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]

   6. Applied egg-rr99.3%

      \[\leadsto \mathsf{fma}\left(0.125, \color{blue}{\frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}}, 0.5 \cdot \frac{\varepsilon}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{\varepsilon}{x \cdot x} \cdot \frac{\varepsilon}{x}, \frac{\varepsilon}{x} \cdot 0.5\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.9999999999999999e-154

   1. Initial program 99.7%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   if -1.9999999999999999e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 8.5%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   2. Taylor expanded in x around inf 98.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\varepsilon}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -2 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \]

1. Split input into 2 regimes

2. if x < 1.80000000000000008e-92

   1. Initial program 91.6%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   2. Taylor expanded in x around 0 90.6%

      \[\leadsto x - \sqrt{\color{blue}{-1 \cdot \varepsilon}} \]
   3. Step-by-step derivation

      1. neg-mul-190.6%

         \[\leadsto x - \sqrt{\color{blue}{-\varepsilon}} \]

   4. Simplified90.6%

      \[\leadsto x - \sqrt{\color{blue}{-\varepsilon}} \]

   if 1.80000000000000008e-92 < x

   1. Initial program 16.3%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   2. Taylor expanded in x around inf 90.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\varepsilon}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-92}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \]

1. Initial program 59.8%

   \[x - \sqrt{x \cdot x - \varepsilon} \]

2. Taylor expanded in x around inf 46.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\varepsilon}{x}} \]

3. Final simplification46.4%

   \[\leadsto \frac{\varepsilon}{x} \cdot 0.5 \]

1. Initial program 59.8%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
2. Step-by-step derivation

   1. flip--59.7%

      \[\leadsto \color{blue}{\frac{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}{x + \sqrt{x \cdot x - \varepsilon}}} \]

   2. div-inv59.6%

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

   3. add-sqr-sqrt59.4%

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

   4. sub-neg59.4%

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

   5. add-sqr-sqrt56.7%

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

   6. hypot-def56.7%

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

3. Applied egg-rr56.7%

   \[\leadsto \color{blue}{\left(x \cdot x - \left(x \cdot x - \varepsilon\right)\right) \cdot \frac{1}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/56.7%

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

   2. *-rgt-identity56.7%

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

   3. associate--r-75.8%

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

   4. +-inverses75.8%

      \[\leadsto \frac{\color{blue}{0} + \varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \]

   5. +-lft-identity75.8%

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

5. Simplified75.8%

   \[\leadsto \color{blue}{\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]
6. Step-by-step derivation

   1. *-un-lft-identity75.8%

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

   2. add-cube-cbrt74.4%

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

   3. times-frac74.3%

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

   4. pow274.3%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\right)}^{2}}} \cdot \frac{\varepsilon}{\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]

   5. add-sqr-sqrt74.3%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{\sqrt{-\varepsilon} \cdot \sqrt{-\varepsilon}}}\right)}\right)}^{2}} \cdot \frac{\varepsilon}{\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]

   6. sqrt-unprod53.0%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{\sqrt{\left(-\varepsilon\right) \cdot \left(-\varepsilon\right)}}}\right)}\right)}^{2}} \cdot \frac{\varepsilon}{\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]

   7. sqr-neg53.0%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{\sqrt{\color{blue}{\varepsilon \cdot \varepsilon}}}\right)}\right)}^{2}} \cdot \frac{\varepsilon}{\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]

   8. sqrt-prod0.0%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{\sqrt{\varepsilon} \cdot \sqrt{\varepsilon}}}\right)}\right)}^{2}} \cdot \frac{\varepsilon}{\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]

   9. add-sqr-sqrt0.0%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{\color{blue}{\varepsilon}}\right)}\right)}^{2}} \cdot \frac{\varepsilon}{\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]

7. Applied egg-rr22.5%

   \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{\varepsilon}\right)}\right)}^{2}} \cdot \frac{\varepsilon}{\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{\varepsilon}\right)}}} \]
8. Step-by-step derivation

   1. associate-*l/22.5%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\varepsilon}{\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{\varepsilon}\right)}}}{{\left(\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{\varepsilon}\right)}\right)}^{2}}} \]

   2. *-lft-identity22.5%

      \[\leadsto \frac{\color{blue}{\frac{\varepsilon}{\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{\varepsilon}\right)}}}}{{\left(\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{\varepsilon}\right)}\right)}^{2}} \]

9. Simplified22.5%

   \[\leadsto \color{blue}{\frac{\frac{\varepsilon}{\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{\varepsilon}\right)}}}{{\left(\sqrt[3]{x + \mathsf{hypot}\left(x, \sqrt{\varepsilon}\right)}\right)}^{2}}} \]

10. Taylor expanded in x around -inf 5.0%

    \[\leadsto \color{blue}{-2 \cdot x} \]
11. Step-by-step derivation

    1. *-commutative5.0%

       \[\leadsto \color{blue}{x \cdot -2} \]

12. Simplified5.0%

    \[\leadsto \color{blue}{x \cdot -2} \]

13. Final simplification5.0%

    \[\leadsto x \cdot -2 \]

1. Initial program 59.8%

   \[x - \sqrt{x \cdot x - \varepsilon} \]

2. Taylor expanded in x around 0 56.3%

   \[\leadsto x - \sqrt{\color{blue}{-1 \cdot \varepsilon}} \]
3. Step-by-step derivation

   1. neg-mul-156.3%

      \[\leadsto x - \sqrt{\color{blue}{-\varepsilon}} \]

4. Simplified56.3%

   \[\leadsto x - \sqrt{\color{blue}{-\varepsilon}} \]

5. Taylor expanded in x around inf 3.7%

   \[\leadsto \color{blue}{x} \]

6. Final simplification3.7%

   \[\leadsto x \]

herbie shell --seed 2023166 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4d"
  :precision binary64
  :pre (and (and (<= 0.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))

  :herbie-target
  (/ eps (+ x (sqrt (- (* x x) eps))))

  (- x (sqrt (- (* x x) eps))))