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

Average Error: 10.0 → 0.1

Time: 2.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -5.04087632150993631 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{elif}\;x \le 3.8730830355980695 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x}{\frac{z}{y}}\right) - x\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r658315 = x;
        double r658316 = y;
        double r658317 = z;
        double r658318 = r658316 - r658317;
        double r658319 = 1.0;
        double r658320 = r658318 + r658319;
        double r658321 = r658315 * r658320;
        double r658322 = r658321 / r658317;
        return r658322;
}

↓

double f(double x, double y, double z) {
        double r658323 = x;
        double r658324 = -5.040876321509936e-30;
        bool r658325 = r658323 <= r658324;
        double r658326 = z;
        double r658327 = r658323 / r658326;
        double r658328 = 1.0;
        double r658329 = y;
        double r658330 = r658328 + r658329;
        double r658331 = r658327 * r658330;
        double r658332 = r658331 - r658323;
        double r658333 = 3.8730830355980695e-51;
        bool r658334 = r658323 <= r658333;
        double r658335 = r658323 * r658329;
        double r658336 = r658335 / r658326;
        double r658337 = fma(r658328, r658327, r658336);
        double r658338 = r658337 - r658323;
        double r658339 = r658326 / r658329;
        double r658340 = r658323 / r658339;
        double r658341 = fma(r658328, r658327, r658340);
        double r658342 = r658341 - r658323;
        double r658343 = r658334 ? r658338 : r658342;
        double r658344 = r658325 ? r658332 : r658343;
        return r658344;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	10.0
Target	0.5
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;x \lt -2.7148310671343599 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x \lt 3.87410881643954616 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -5.040876321509936e-30

   1. Initial program 22.5

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

   2. Taylor expanded around 0 8.4

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

   3. Simplified 8.4

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

   4. Taylor expanded around 0 8.4

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

   5. Simplified 0.1

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

   if -5.040876321509936e-30 < x < 3.8730830355980695e-51

   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 0.1

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

   if 3.8730830355980695e-51 < x

   1. Initial program 20.6

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

   2. Taylor expanded around 0 6.9

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

   3. Simplified 6.9

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

   5. Applied associate-/l* 0.3

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.04087632150993631 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{elif}\;x \le 3.8730830355980695 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x \cdot y}{z}\right) - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{z}, \frac{x}{\frac{z}{y}}\right) - x\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(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))