Error

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Target

Derivation

1. Split input into 4 regimes

2. if y < -1.326101464362353e+154

   1. Initial program 63.6

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
   2. Using strategy rm

   3. Applied add-log-exp63.6

      \[\leadsto \color{blue}{\log \left(e^{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right)}\]

   4. Taylor expanded around 0 0

      \[\leadsto \color{blue}{-1}\]

   if -1.326101464362353e+154 < y < -7.866138089097476e-160

   1. Initial program 0.0

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
   2. Using strategy rm

   3. Applied add-log-exp0.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right)}\]

   if -7.866138089097476e-160 < y < 2.903536397889742e-204

   1. Initial program 29.9

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
   2. Using strategy rm

   3. Applied add-log-exp29.9

      \[\leadsto \color{blue}{\log \left(e^{\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}}\right)}\]

   4. Taylor expanded around inf 14.9

      \[\leadsto \color{blue}{1}\]

   if 2.903536397889742e-204 < y

   1. Initial program 5.9

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt5.9

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

   4. Applied times-frac6.3

      \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.326101464362353 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -7.866138089097476 \cdot 10^{-160}:\\ \;\;\;\;\log \left(e^{\frac{\left(x - y\right) \cdot \left(y + x\right)}{y \cdot y + x \cdot x}}\right)\\ \mathbf{elif}\;y \le 2.903536397889742 \cdot 10^{-204}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\sqrt{y \cdot y + x \cdot x}} \cdot \frac{y + x}{\sqrt{y \cdot y + x \cdot x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(FPCore (x y)
  :name "Kahan p9 Example"
  :pre (and (< 0 x 1) (< y 1))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

Details

Time bar (total: 17.7s)Debug log