Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, D

Average Error: 22.9 → 0.2

Time: 3.3s

Precision: binary64

Math

\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -98382367.3054202348 \lor \neg \left(y \le 107573633.8387101\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y \cdot \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1}\right) - \left(\frac{1 - x}{\sqrt[3]{y \cdot y - 1 \cdot 1} \cdot \sqrt[3]{y \cdot y - 1 \cdot 1}} \cdot \frac{y}{\sqrt[3]{y \cdot y - 1 \cdot 1}}\right) \cdot \left(-1\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Target

Original	22.9
Target	0.2
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -98382367.3054202348 or 107573633.8387101 < y

   1. Initial program 46.0

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

   2. Taylor expanded around inf 0.1

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

   3. Simplified 0.1

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

   if -98382367.3054202348 < y < 107573633.8387101

   1. Initial program 0.2

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]
   2. Using strategy rm

   3. Applied flip-+ 0.2

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

   4. Applied associate-/r/ 0.2

      \[\leadsto 1 - \color{blue}{\frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)}\]
   5. Using strategy rm

   6. Applied sub-neg 0.2

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

   7. Applied distribute-lft-in 0.2

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

   8. Applied associate--r+ 0.2

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

   9. Simplified 0.2

      \[\leadsto \color{blue}{\left(1 - y \cdot \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1}\right)} - \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1} \cdot \left(-1\right)\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 0.2

       \[\leadsto \left(1 - y \cdot \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1}\right) - \frac{\left(1 - x\right) \cdot y}{\color{blue}{\left(\sqrt[3]{y \cdot y - 1 \cdot 1} \cdot \sqrt[3]{y \cdot y - 1 \cdot 1}\right) \cdot \sqrt[3]{y \cdot y - 1 \cdot 1}}} \cdot \left(-1\right)\]

   12. Applied times-frac 0.2

       \[\leadsto \left(1 - y \cdot \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1}\right) - \color{blue}{\left(\frac{1 - x}{\sqrt[3]{y \cdot y - 1 \cdot 1} \cdot \sqrt[3]{y \cdot y - 1 \cdot 1}} \cdot \frac{y}{\sqrt[3]{y \cdot y - 1 \cdot 1}}\right)} \cdot \left(-1\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -98382367.3054202348 \lor \neg \left(y \le 107573633.8387101\right):\\ \;\;\;\;1 \cdot \left(\frac{1}{y} - \frac{x}{y}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y \cdot \frac{\left(1 - x\right) \cdot y}{y \cdot y - 1 \cdot 1}\right) - \left(\frac{1 - x}{\sqrt[3]{y \cdot y - 1 \cdot 1} \cdot \sqrt[3]{y \cdot y - 1 \cdot 1}} \cdot \frac{y}{\sqrt[3]{y \cdot y - 1 \cdot 1}}\right) \cdot \left(-1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))