System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2

Average Error: 25.4 → 8.1

Time: 27.0s

Precision: 64

• could not determine a ground truth for program body

  1. x = -1.1425440199203244e+248
  2. y = 2.000111729313626e+44
  3. z = 2.931459133389821e+208
  4. t = 3.625484546781197e-95

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{z} \le 0.9999999353825346215529634719132445752621:\\ \;\;\;\;x - \left(\frac{\sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}\right) \cdot \frac{\sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}}{\sqrt[3]{t}}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(1, y \cdot \frac{z}{t}, \mathsf{fma}\left(\frac{\left(y \cdot z\right) \cdot z}{t}, 0.5, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r217634 = x;
        double r217635 = 1.0;
        double r217636 = y;
        double r217637 = r217635 - r217636;
        double r217638 = z;
        double r217639 = exp(r217638);
        double r217640 = r217636 * r217639;
        double r217641 = r217637 + r217640;
        double r217642 = log(r217641);
        double r217643 = t;
        double r217644 = r217642 / r217643;
        double r217645 = r217634 - r217644;
        return r217645;
}

↓

double f(double x, double y, double z, double t) {
        double r217646 = z;
        double r217647 = exp(r217646);
        double r217648 = 0.9999999353825346;
        bool r217649 = r217647 <= r217648;
        double r217650 = x;
        double r217651 = y;
        double r217652 = 1.0;
        double r217653 = r217652 - r217651;
        double r217654 = fma(r217651, r217647, r217653);
        double r217655 = log(r217654);
        double r217656 = cbrt(r217655);
        double r217657 = t;
        double r217658 = cbrt(r217657);
        double r217659 = r217658 * r217658;
        double r217660 = r217656 / r217659;
        double r217661 = r217660 * r217656;
        double r217662 = r217656 / r217658;
        double r217663 = r217661 * r217662;
        double r217664 = r217650 - r217663;
        double r217665 = r217646 / r217657;
        double r217666 = r217651 * r217665;
        double r217667 = r217651 * r217646;
        double r217668 = r217667 * r217646;
        double r217669 = r217668 / r217657;
        double r217670 = 0.5;
        double r217671 = log(r217652);
        double r217672 = r217671 / r217657;
        double r217673 = fma(r217669, r217670, r217672);
        double r217674 = fma(r217652, r217666, r217673);
        double r217675 = r217650 - r217674;
        double r217676 = r217649 ? r217664 : r217675;
        return r217676;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	25.4
Target	16.4
Herbie	8.1

\[\begin{array}{l} \mathbf{if}\;z \lt -2.887462308820794658905265984545350618896 \cdot 10^{119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (exp z) < 0.9999999353825346

   1. Initial program 11.2

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

   2. Simplified 11.2

      \[\leadsto \color{blue}{x - \frac{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}{t}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 11.4

      \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\]

   5. Applied add-cube-cbrt 11.4

      \[\leadsto x - \frac{\color{blue}{\left(\sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)} \cdot \sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}\right) \cdot \sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\]

   6. Applied times-frac 11.4

      \[\leadsto x - \color{blue}{\frac{\sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)} \cdot \sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}}{\sqrt[3]{t}}}\]

   7. Simplified 11.5

      \[\leadsto x - \color{blue}{\left(\frac{\sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}\right)} \cdot \frac{\sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}}{\sqrt[3]{t}}\]

   if 0.9999999353825346 < (exp z)

   1. Initial program 31.8

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

   2. Simplified 31.8

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

   3. Taylor expanded around 0 7.2

      \[\leadsto x - \color{blue}{\left(1 \cdot \frac{z \cdot y}{t} + \left(0.5 \cdot \frac{{z}^{2} \cdot y}{t} + \frac{\log 1}{t}\right)\right)}\]

   4. Simplified 6.5

      \[\leadsto x - \color{blue}{\mathsf{fma}\left(1, \frac{z}{t} \cdot y, \mathsf{fma}\left(\frac{\left(y \cdot z\right) \cdot z}{t}, 0.5, \frac{\log 1}{t}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 0.9999999353825346215529634719132445752621:\\ \;\;\;\;x - \left(\frac{\sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}\right) \cdot \frac{\sqrt[3]{\log \left(\mathsf{fma}\left(y, e^{z}, 1 - y\right)\right)}}{\sqrt[3]{t}}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(1, y \cdot \frac{z}{t}, \mathsf{fma}\left(\frac{\left(y \cdot z\right) \cdot z}{t}, 0.5, \frac{\log 1}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"

  :herbie-target
  (if (< z -2.8874623088207947e+119) (- (- x (/ (/ (- 0.5) (* y t)) (* z z))) (* (/ (- 0.5) (* y t)) (/ (/ 2.0 z) (* z z)))) (- x (/ (log (+ 1.0 (* z y))) t)))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))