Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2

Average Error: 25.5 → 6.0

Time: 5.4s

Precision: binary64

Math

\[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.4109285273931826 \cdot 10^{134}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 7.3819289826067232 \cdot 10^{116}:\\ \;\;\;\;\left(x \cdot \left(y \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{z}}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}\right)\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	25.5
Target	8.0
Herbie	6.0

\[\begin{array}{l} \mathbf{if}\;z \lt -3.1921305903852764 \cdot 10^{46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z \lt 5.9762681209208942 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -1.4109285273931826e134

   1. Initial program 49.5

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]

   2. Simplified 48.0

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

   3. Taylor expanded around -inf 1.6

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

   4. Simplified 1.6

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

   if -1.4109285273931826e134 < z < 7.3819289826067232e116

   1. Initial program 11.6

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]

   2. Simplified 9.5

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)}\]
   3. Using strategy rm

   4. Applied associate-*r* 9.6

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

   6. Applied add-cube-cbrt 10.0

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

   7. Applied sqrt-prod 10.0

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

   8. Applied add-cube-cbrt 10.3

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

   9. Applied times-frac 10.3

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

   10. Applied associate-*r* 9.4

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

   11. Simplified 8.5

       \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{z}}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}\right)\right)\right)} \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\]

   if 7.3819289826067232e116 < z

   1. Initial program 47.3

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}}\]

   2. Simplified 45.1

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

   3. Taylor expanded around inf 1.8

      \[\leadsto \color{blue}{x \cdot y}\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.4109285273931826 \cdot 10^{134}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;z \le 7.3819289826067232 \cdot 10^{116}:\\ \;\;\;\;\left(x \cdot \left(y \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{z}}{\left|\sqrt[3]{z \cdot z - t \cdot a}\right|}\right)\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt{\sqrt[3]{z \cdot z - t \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z -3.1921305903852764e+46) (neg (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))