Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C

Average Error: 4.5 → 3.4

Time: 15.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -399267478145010466818097152 \lor \neg \left(x \le 1.211593220234172703119299050409946328088 \cdot 10^{-232}\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{-t \cdot x}{1 - z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r251462 = x;
        double r251463 = y;
        double r251464 = z;
        double r251465 = r251463 / r251464;
        double r251466 = t;
        double r251467 = 1.0;
        double r251468 = r251467 - r251464;
        double r251469 = r251466 / r251468;
        double r251470 = r251465 - r251469;
        double r251471 = r251462 * r251470;
        return r251471;
}

↓

double f(double x, double y, double z, double t) {
        double r251472 = x;
        double r251473 = -3.992674781450105e+26;
        bool r251474 = r251472 <= r251473;
        double r251475 = 1.2115932202341727e-232;
        bool r251476 = r251472 <= r251475;
        double r251477 = !r251476;
        bool r251478 = r251474 || r251477;
        double r251479 = y;
        double r251480 = z;
        double r251481 = r251479 / r251480;
        double r251482 = t;
        double r251483 = 1.0;
        double r251484 = 1.0;
        double r251485 = r251484 - r251480;
        double r251486 = r251483 / r251485;
        double r251487 = r251482 * r251486;
        double r251488 = r251481 - r251487;
        double r251489 = r251472 * r251488;
        double r251490 = r251472 * r251479;
        double r251491 = r251490 / r251480;
        double r251492 = r251482 * r251472;
        double r251493 = -r251492;
        double r251494 = r251493 / r251485;
        double r251495 = r251491 + r251494;
        double r251496 = r251478 ? r251489 : r251495;
        return r251496;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.5
Target	4.1
Herbie	3.4

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if x < -3.992674781450105e+26 or 1.2115932202341727e-232 < x

   1. Initial program 3.7

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

   3. Applied div-inv 3.7

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

   if -3.992674781450105e+26 < x < 1.2115932202341727e-232

   1. Initial program 5.7

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

   3. Applied add-cube-cbrt 6.6

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

   4. Applied associate-*l* 6.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\right)}\]
   5. Using strategy rm

   6. Applied sub-neg 6.6

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

   7. Applied distribute-lft-in 6.6

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

   8. Applied distribute-lft-in 6.6

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

   9. Simplified 3.4

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

   10. Simplified 3.1

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

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -399267478145010466818097152 \lor \neg \left(x \le 1.211593220234172703119299050409946328088 \cdot 10^{-232}\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{-t \cdot x}{1 - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1 z)))) -7.62322630331204244e-196) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 1.41339449277023022e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z)))))))

  (* x (- (/ y z) (/ t (- 1 z)))))