Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.2

Time: 33.9s

Precision: 64

Math

\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]

↓

\[\left(\left(\left(x - \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(y + 0.5\right)\right) - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)\right) + y\right) - z\]

C

double f(double x, double y, double z) {
        double r238408 = x;
        double r238409 = y;
        double r238410 = 0.5;
        double r238411 = r238409 + r238410;
        double r238412 = log(r238409);
        double r238413 = r238411 * r238412;
        double r238414 = r238408 - r238413;
        double r238415 = r238414 + r238409;
        double r238416 = z;
        double r238417 = r238415 - r238416;
        return r238417;
}

↓

double f(double x, double y, double z) {
        double r238418 = x;
        double r238419 = 2.0;
        double r238420 = y;
        double r238421 = cbrt(r238420);
        double r238422 = log(r238421);
        double r238423 = r238419 * r238422;
        double r238424 = 0.5;
        double r238425 = r238420 + r238424;
        double r238426 = r238423 * r238425;
        double r238427 = r238418 - r238426;
        double r238428 = cbrt(r238421);
        double r238429 = log(r238428);
        double r238430 = r238419 * r238429;
        double r238431 = r238425 * r238430;
        double r238432 = r238425 * r238429;
        double r238433 = r238431 + r238432;
        double r238434 = r238427 - r238433;
        double r238435 = r238434 + r238420;
        double r238436 = z;
        double r238437 = r238435 - r238436;
        return r238437;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	0.1
Target	0.1
Herbie	0.2

\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y\]

Derivation

1. Initial program 0.1

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
2. Using strategy rm

3. Applied add-cube-cbrt 0.1

   \[\leadsto \left(\left(x - \left(y + 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)}\right) + y\right) - z\]

4. Applied log-prod 0.2

   \[\leadsto \left(\left(x - \left(y + 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)}\right) + y\right) - z\]

5. Applied distribute-lft-in 0.2

   \[\leadsto \left(\left(x - \color{blue}{\left(\left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)}\right) + y\right) - z\]

6. Applied associate--r+ 0.2

   \[\leadsto \left(\color{blue}{\left(\left(x - \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) - \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)} + y\right) - z\]

7. Simplified 0.2

   \[\leadsto \left(\left(\color{blue}{\left(x - \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(y + 0.5\right)\right)} - \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right) + y\right) - z\]
8. Using strategy rm

9. Applied add-cube-cbrt 0.2

   \[\leadsto \left(\left(\left(x - \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(y + 0.5\right)\right) - \left(y + 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right) + y\right) - z\]

10. Applied log-prod 0.2

    \[\leadsto \left(\left(\left(x - \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(y + 0.5\right)\right) - \left(y + 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)}\right) + y\right) - z\]

11. Applied distribute-lft-in 0.2

    \[\leadsto \left(\left(\left(x - \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(y + 0.5\right)\right) - \color{blue}{\left(\left(y + 0.5\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)}\right) + y\right) - z\]

12. Simplified 0.2

    \[\leadsto \left(\left(\left(x - \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(y + 0.5\right)\right) - \left(\color{blue}{\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)} + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)\right) + y\right) - z\]

13. Final simplification 0.2

    \[\leadsto \left(\left(\left(x - \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(y + 0.5\right)\right) - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right) + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)\right) + y\right) - z\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))