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

Average Error: 25.6 → 14.3

Time: 8.2s

Precision: 64

• could not determine a ground truth for program body

  1. x = -7.133308535761891e-224
  2. y = 5.56035114616897e-248
  3. z = 8.52518331542308e+67
  4. t = 1.5054925628386541e-83

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.152892260263348607747194563956545567434 \cdot 10^{210}:\\ \;\;\;\;x - \frac{\log \left(1 + \left({z}^{2} \cdot \left(\frac{1}{2} + z \cdot \frac{1}{6}\right) + z\right) \cdot y\right)}{t}\\ \mathbf{elif}\;y \le -2.469031949062149442410480885508201558103 \cdot 10^{131}:\\ \;\;\;\;x - \left(\frac{\log \left(1 - e^{z}\right)}{t} - \left(1 \cdot \frac{1}{\left(1 - e^{z}\right) \cdot \left(y \cdot t\right)} + \frac{\log \left(\frac{-1}{y}\right)}{t}\right)\right)\\ \mathbf{elif}\;y \le -1.297671033110611467296589409782087111976 \cdot 10^{105}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}}\\ \mathbf{elif}\;y \le -6.569041928042183231455829851483351473071 \cdot 10^{103}:\\ \;\;\;\;x - \left(\frac{\log \left(1 - e^{z}\right)}{t} - \left(1 \cdot \frac{1}{\left(1 - e^{z}\right) \cdot \left(y \cdot t\right)} + \frac{\log \left(\frac{-1}{y}\right)}{t}\right)\right)\\ \mathbf{elif}\;y \le 2.445855841223627624785485148522430782767 \cdot 10^{-124}:\\ \;\;\;\;x - \left(\frac{\log \left({1}^{3} + {\left(\left(e^{z} - 1\right) \cdot y\right)}^{3}\right)}{t} - \frac{\log \left(1 \cdot 1 + \left(\left(\left(e^{z} - 1\right) \cdot y\right) \cdot \left(\left(e^{z} - 1\right) \cdot y\right) - 1 \cdot \left(\left(e^{z} - 1\right) \cdot y\right)\right)\right)}{t}\right)\\ \mathbf{elif}\;y \le 1.449609952745778752399845107288990046227 \cdot 10^{165}:\\ \;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \frac{\log 1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + \left({z}^{2} \cdot \left(\frac{1}{2} + z \cdot \frac{1}{6}\right) + z\right) \cdot y\right)}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r363480 = x;
        double r363481 = 1.0;
        double r363482 = y;
        double r363483 = r363481 - r363482;
        double r363484 = z;
        double r363485 = exp(r363484);
        double r363486 = r363482 * r363485;
        double r363487 = r363483 + r363486;
        double r363488 = log(r363487);
        double r363489 = t;
        double r363490 = r363488 / r363489;
        double r363491 = r363480 - r363490;
        return r363491;
}

↓

double f(double x, double y, double z, double t) {
        double r363492 = y;
        double r363493 = -1.1528922602633486e+210;
        bool r363494 = r363492 <= r363493;
        double r363495 = x;
        double r363496 = 1.0;
        double r363497 = z;
        double r363498 = 2.0;
        double r363499 = pow(r363497, r363498);
        double r363500 = 0.5;
        double r363501 = 0.16666666666666666;
        double r363502 = r363497 * r363501;
        double r363503 = r363500 + r363502;
        double r363504 = r363499 * r363503;
        double r363505 = r363504 + r363497;
        double r363506 = r363505 * r363492;
        double r363507 = r363496 + r363506;
        double r363508 = log(r363507);
        double r363509 = t;
        double r363510 = r363508 / r363509;
        double r363511 = r363495 - r363510;
        double r363512 = -2.4690319490621494e+131;
        bool r363513 = r363492 <= r363512;
        double r363514 = 1.0;
        double r363515 = exp(r363497);
        double r363516 = r363514 - r363515;
        double r363517 = log(r363516);
        double r363518 = r363517 / r363509;
        double r363519 = r363492 * r363509;
        double r363520 = r363516 * r363519;
        double r363521 = r363514 / r363520;
        double r363522 = r363496 * r363521;
        double r363523 = -1.0;
        double r363524 = r363523 / r363492;
        double r363525 = log(r363524);
        double r363526 = r363525 / r363509;
        double r363527 = r363522 + r363526;
        double r363528 = r363518 - r363527;
        double r363529 = r363495 - r363528;
        double r363530 = -1.2976710331106115e+105;
        bool r363531 = r363492 <= r363530;
        double r363532 = log(r363496);
        double r363533 = 0.5;
        double r363534 = r363533 * r363499;
        double r363535 = r363496 * r363497;
        double r363536 = r363534 + r363535;
        double r363537 = r363492 * r363536;
        double r363538 = r363532 + r363537;
        double r363539 = r363509 / r363538;
        double r363540 = r363514 / r363539;
        double r363541 = r363495 - r363540;
        double r363542 = -6.569041928042183e+103;
        bool r363543 = r363492 <= r363542;
        double r363544 = 2.4458558412236276e-124;
        bool r363545 = r363492 <= r363544;
        double r363546 = 3.0;
        double r363547 = pow(r363496, r363546);
        double r363548 = r363515 - r363514;
        double r363549 = r363548 * r363492;
        double r363550 = pow(r363549, r363546);
        double r363551 = r363547 + r363550;
        double r363552 = log(r363551);
        double r363553 = r363552 / r363509;
        double r363554 = r363496 * r363496;
        double r363555 = r363549 * r363549;
        double r363556 = r363496 * r363549;
        double r363557 = r363555 - r363556;
        double r363558 = r363554 + r363557;
        double r363559 = log(r363558);
        double r363560 = r363559 / r363509;
        double r363561 = r363553 - r363560;
        double r363562 = r363495 - r363561;
        double r363563 = 1.4496099527457788e+165;
        bool r363564 = r363492 <= r363563;
        double r363565 = r363497 * r363492;
        double r363566 = r363565 / r363509;
        double r363567 = r363496 * r363566;
        double r363568 = r363532 / r363509;
        double r363569 = r363567 + r363568;
        double r363570 = r363495 - r363569;
        double r363571 = r363564 ? r363570 : r363511;
        double r363572 = r363545 ? r363562 : r363571;
        double r363573 = r363543 ? r363529 : r363572;
        double r363574 = r363531 ? r363541 : r363573;
        double r363575 = r363513 ? r363529 : r363574;
        double r363576 = r363494 ? r363511 : r363575;
        return r363576;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	25.6
Target	16.0
Herbie	14.3

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

Derivation

1. Split input into 5 regimes

2. if y < -1.1528922602633486e+210 or 1.4496099527457788e+165 < y

   1. Initial program 46.7

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

   3. Applied sub-neg 46.7

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

   4. Applied associate-+l+ 21.4

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

   5. Simplified 21.3

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

   6. Taylor expanded around 0 20.7

      \[\leadsto x - \frac{\log \left(1 + \color{blue}{\left(z + \left(\frac{1}{6} \cdot {z}^{3} + \frac{1}{2} \cdot {z}^{2}\right)\right)} \cdot y\right)}{t}\]

   7. Simplified 20.7

      \[\leadsto x - \frac{\log \left(1 + \color{blue}{\left({z}^{2} \cdot \left(\frac{1}{2} + z \cdot \frac{1}{6}\right) + z\right)} \cdot y\right)}{t}\]

   if -1.1528922602633486e+210 < y < -2.4690319490621494e+131 or -1.2976710331106115e+105 < y < -6.569041928042183e+103

   1. Initial program 38.0

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

   3. Applied sub-neg 38.0

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

   4. Applied associate-+l+ 17.9

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

   5. Simplified 17.9

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

   6. Taylor expanded around -inf 38.0

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

   if -2.4690319490621494e+131 < y < -1.2976710331106115e+105

   1. Initial program 41.8

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

   2. Taylor expanded around 0 25.5

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

   3. Simplified 25.5

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

   5. Applied clear-num 25.5

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

   if -6.569041928042183e+103 < y < 2.4458558412236276e-124

   1. Initial program 15.3

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

   3. Applied sub-neg 15.3

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

   4. Applied associate-+l+ 11.0

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

   5. Simplified 11.0

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

   7. Applied flip3-+ 11.1

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

   8. Applied log-div 11.1

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

   9. Applied div-sub 11.1

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

   if 2.4458558412236276e-124 < y < 1.4496099527457788e+165

   1. Initial program 34.9

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

   2. Taylor expanded around 0 13.0

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

   3. Simplified 13.0

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

   4. Taylor expanded around 0 10.9

      \[\leadsto \color{blue}{x - \left(1 \cdot \frac{z \cdot y}{t} + \frac{\log 1}{t}\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 14.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.152892260263348607747194563956545567434 \cdot 10^{210}:\\ \;\;\;\;x - \frac{\log \left(1 + \left({z}^{2} \cdot \left(\frac{1}{2} + z \cdot \frac{1}{6}\right) + z\right) \cdot y\right)}{t}\\ \mathbf{elif}\;y \le -2.469031949062149442410480885508201558103 \cdot 10^{131}:\\ \;\;\;\;x - \left(\frac{\log \left(1 - e^{z}\right)}{t} - \left(1 \cdot \frac{1}{\left(1 - e^{z}\right) \cdot \left(y \cdot t\right)} + \frac{\log \left(\frac{-1}{y}\right)}{t}\right)\right)\\ \mathbf{elif}\;y \le -1.297671033110611467296589409782087111976 \cdot 10^{105}:\\ \;\;\;\;x - \frac{1}{\frac{t}{\log 1 + y \cdot \left(0.5 \cdot {z}^{2} + 1 \cdot z\right)}}\\ \mathbf{elif}\;y \le -6.569041928042183231455829851483351473071 \cdot 10^{103}:\\ \;\;\;\;x - \left(\frac{\log \left(1 - e^{z}\right)}{t} - \left(1 \cdot \frac{1}{\left(1 - e^{z}\right) \cdot \left(y \cdot t\right)} + \frac{\log \left(\frac{-1}{y}\right)}{t}\right)\right)\\ \mathbf{elif}\;y \le 2.445855841223627624785485148522430782767 \cdot 10^{-124}:\\ \;\;\;\;x - \left(\frac{\log \left({1}^{3} + {\left(\left(e^{z} - 1\right) \cdot y\right)}^{3}\right)}{t} - \frac{\log \left(1 \cdot 1 + \left(\left(\left(e^{z} - 1\right) \cdot y\right) \cdot \left(\left(e^{z} - 1\right) \cdot y\right) - 1 \cdot \left(\left(e^{z} - 1\right) \cdot y\right)\right)\right)}{t}\right)\\ \mathbf{elif}\;y \le 1.449609952745778752399845107288990046227 \cdot 10^{165}:\\ \;\;\;\;x - \left(1 \cdot \frac{z \cdot y}{t} + \frac{\log 1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + \left({z}^{2} \cdot \left(\frac{1}{2} + z \cdot \frac{1}{6}\right) + z\right) \cdot y\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (if (< z -2.8874623088207947e+119) (- (- x (/ (/ (- 0.5) (* y t)) (* z z))) (* (/ (- 0.5) (* y t)) (/ (/ 2 z) (* z z)))) (- x (/ (log (+ 1 (* z y))) t)))

  (- x (/ (log (+ (- 1 y) (* y (exp z)))) t)))