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

Average Error: 9.8 → 0.4

Time: 14.2s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -9.2290174913998737 \cdot 10^{129} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 3.0292350933407683 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r824545 = x;
        double r824546 = y;
        double r824547 = z;
        double r824548 = r824546 - r824547;
        double r824549 = 1.0;
        double r824550 = r824548 + r824549;
        double r824551 = r824545 * r824550;
        double r824552 = r824551 / r824547;
        return r824552;
}

↓

double f(double x, double y, double z) {
        double r824553 = x;
        double r824554 = y;
        double r824555 = z;
        double r824556 = r824554 - r824555;
        double r824557 = 1.0;
        double r824558 = r824556 + r824557;
        double r824559 = r824553 * r824558;
        double r824560 = r824559 / r824555;
        double r824561 = -9.229017491399874e+129;
        bool r824562 = r824560 <= r824561;
        double r824563 = 3.029235093340768e-45;
        bool r824564 = r824560 <= r824563;
        double r824565 = !r824564;
        bool r824566 = r824562 || r824565;
        double r824567 = r824553 / r824555;
        double r824568 = r824557 + r824554;
        double r824569 = r824567 * r824568;
        double r824570 = r824569 - r824553;
        double r824571 = r824555 / r824558;
        double r824572 = r824553 / r824571;
        double r824573 = r824566 ? r824570 : r824572;
        return r824573;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	9.8
Target	0.4
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (/ (* x (+ (- y z) 1.0)) z) < -9.229017491399874e+129 or 3.029235093340768e-45 < (/ (* x (+ (- y z) 1.0)) z)

   1. Initial program 18.0

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

   2. Taylor expanded around 0 6.4

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

   3. Simplified 0.1

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

   if -9.229017491399874e+129 < (/ (* x (+ (- y z) 1.0)) z) < 3.029235093340768e-45

   1. Initial program 0.1

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

   3. Applied associate-/l* 0.8

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le -9.2290174913998737 \cdot 10^{129} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \le 3.0292350933407683 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y\right) - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{\left(y - z\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))