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

Average Error: 10.1 → 0.1

Time: 17.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -6.579027488067199411550511935864463917316 \cdot 10^{210} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 3.530201563568855701403158383753448655305 \cdot 10^{90}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + y\right)}{z} - x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r515374 = x;
        double r515375 = y;
        double r515376 = z;
        double r515377 = r515375 - r515376;
        double r515378 = 1.0;
        double r515379 = r515377 + r515378;
        double r515380 = r515374 * r515379;
        double r515381 = r515380 / r515376;
        return r515381;
}

↓

double f(double x, double y, double z) {
        double r515382 = x;
        double r515383 = y;
        double r515384 = z;
        double r515385 = r515383 - r515384;
        double r515386 = 1.0;
        double r515387 = r515385 + r515386;
        double r515388 = r515382 * r515387;
        double r515389 = r515388 / r515384;
        double r515390 = -6.579027488067199e+210;
        bool r515391 = r515389 <= r515390;
        double r515392 = 3.5302015635688557e+90;
        bool r515393 = r515389 <= r515392;
        double r515394 = !r515393;
        bool r515395 = r515391 || r515394;
        double r515396 = r515382 / r515384;
        double r515397 = r515386 + r515383;
        double r515398 = r515396 * r515397;
        double r515399 = r515398 - r515382;
        double r515400 = r515382 * r515397;
        double r515401 = r515400 / r515384;
        double r515402 = r515401 - r515382;
        double r515403 = r515395 ? r515399 : r515402;
        return r515403;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.1
Target	0.5
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if (/ (* x (+ (- y z) 1.0)) z) < -6.579027488067199e+210 or 3.5302015635688557e+90 < (/ (* x (+ (- y z) 1.0)) z)

   1. Initial program 27.9

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

   2. Taylor expanded around 0 10.4

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

   3. Simplified 0.1

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

   if -6.579027488067199e+210 < (/ (* x (+ (- y z) 1.0)) z) < 3.5302015635688557e+90

   1. Initial program 0.1

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

   2. Taylor expanded around 0 0.1

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

   3. Simplified 2.7

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

   5. Applied pow1 2.7

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

   6. Applied pow1 2.7

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

   7. Applied pow-prod-down 2.7

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

   8. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -6.579027488067199411550511935864463917316 \cdot 10^{210} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 3.530201563568855701403158383753448655305 \cdot 10^{90}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + y\right)}{z} - x\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))