Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Average Error: 9.9 → 1.2

Time: 13.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.716741353213512542561167857708974654712 \cdot 10^{196}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;x \le 5.61926533510320507337176445760607473729 \cdot 10^{-243}:\\ \;\;\;\;\left(1 \cdot \frac{x}{z} + \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r767238 = x;
        double r767239 = y;
        double r767240 = z;
        double r767241 = r767239 - r767240;
        double r767242 = 1.0;
        double r767243 = r767241 + r767242;
        double r767244 = r767238 * r767243;
        double r767245 = r767244 / r767240;
        return r767245;
}

↓

double f(double x, double y, double z) {
        double r767246 = x;
        double r767247 = -2.7167413532135125e+196;
        bool r767248 = r767246 <= r767247;
        double r767249 = z;
        double r767250 = y;
        double r767251 = r767250 - r767249;
        double r767252 = 1.0;
        double r767253 = r767251 + r767252;
        double r767254 = r767249 / r767253;
        double r767255 = r767246 / r767254;
        double r767256 = 5.619265335103205e-243;
        bool r767257 = r767246 <= r767256;
        double r767258 = r767246 / r767249;
        double r767259 = r767252 * r767258;
        double r767260 = r767246 * r767250;
        double r767261 = r767260 / r767249;
        double r767262 = r767259 + r767261;
        double r767263 = r767262 - r767246;
        double r767264 = r767252 + r767250;
        double r767265 = r767258 * r767264;
        double r767266 = r767265 - r767246;
        double r767267 = r767257 ? r767263 : r767266;
        double r767268 = r767248 ? r767255 : r767267;
        return r767268;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	9.9
Target	0.4
Herbie	1.2

\[\begin{array}{l} \mathbf{if}\;x \lt -2.714831067134359919650240696134672137284 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.874108816439546156869494499878029491333 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -2.7167413532135125e+196

   1. Initial program 49.8

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

   3. Applied associate-/l* 0.1

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

   if -2.7167413532135125e+196 < x < 5.619265335103205e-243

   1. Initial program 4.4

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

   2. Taylor expanded around 0 1.5

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

   3. Simplified 2.1

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

   5. Applied *-un-lft-identity 2.1

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

   6. Applied add-cube-cbrt 2.8

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

   7. Applied times-frac 2.8

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

   8. Applied associate-*l* 3.2

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

   10. Applied distribute-lft-in 3.2

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

   11. Applied distribute-lft-in 3.2

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

   12. Simplified 2.9

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

   13. Simplified 1.5

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

   if 5.619265335103205e-243 < x

   1. Initial program 11.6

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

   2. Taylor expanded around 0 4.0

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

   3. Simplified 0.8

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.716741353213512542561167857708974654712 \cdot 10^{196}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \mathbf{elif}\;x \le 5.61926533510320507337176445760607473729 \cdot 10^{-243}:\\ \;\;\;\;\left(1 \cdot \frac{x}{z} + \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1)) z))