Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A

Average Error: 19.9 → 0.8

Time: 59.1s

Precision: 64

Math

\[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]

↓

\[\left(\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{x + y}}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{1.0 + \left(x + y\right)}{\sqrt[3]{y}}}\]

C

double f(double x, double y) {
        double r25122288 = x;
        double r25122289 = y;
        double r25122290 = r25122288 * r25122289;
        double r25122291 = r25122288 + r25122289;
        double r25122292 = r25122291 * r25122291;
        double r25122293 = 1.0;
        double r25122294 = r25122291 + r25122293;
        double r25122295 = r25122292 * r25122294;
        double r25122296 = r25122290 / r25122295;
        return r25122296;
}

↓

double f(double x, double y) {
        double r25122297 = 1.0;
        double r25122298 = x;
        double r25122299 = y;
        double r25122300 = r25122298 + r25122299;
        double r25122301 = cbrt(r25122300);
        double r25122302 = r25122301 * r25122301;
        double r25122303 = r25122297 / r25122302;
        double r25122304 = cbrt(r25122298);
        double r25122305 = cbrt(r25122299);
        double r25122306 = r25122301 / r25122305;
        double r25122307 = r25122304 / r25122306;
        double r25122308 = r25122303 * r25122307;
        double r25122309 = r25122300 / r25122305;
        double r25122310 = r25122304 / r25122309;
        double r25122311 = r25122308 * r25122310;
        double r25122312 = 1.0;
        double r25122313 = r25122312 + r25122300;
        double r25122314 = r25122313 / r25122305;
        double r25122315 = r25122304 / r25122314;
        double r25122316 = r25122311 * r25122315;
        return r25122316;
}

Error

Bits error versus x

Bits error versus y

Target

Original	19.9
Target	0.1
Herbie	0.8

\[\frac{\frac{\frac{x}{\left(y + 1.0\right) + x}}{y + x}}{\frac{1.0}{\frac{y}{y + x}}}\]

Derivation

1. Initial program 19.9

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}\]
2. Using strategy rm

3. Applied associate-/l* 11.7

   \[\leadsto \color{blue}{\frac{x}{\frac{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}{y}}}\]
4. Using strategy rm

5. Applied add-cube-cbrt 12.1

   \[\leadsto \frac{x}{\frac{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1.0\right)}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}\]

6. Applied times-frac 10.1

   \[\leadsto \frac{x}{\color{blue}{\frac{\left(x + y\right) \cdot \left(x + y\right)}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}}\]

7. Applied add-cube-cbrt 10.2

   \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\frac{\left(x + y\right) \cdot \left(x + y\right)}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

8. Applied times-frac 8.4

   \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\left(x + y\right) \cdot \left(x + y\right)}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}}\]

9. Simplified 0.9

   \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x}}{\frac{y + x}{\sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{y + x}{\sqrt[3]{y}}}\right)} \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]
10. Using strategy rm

11. Applied *-un-lft-identity 0.9

    \[\leadsto \left(\frac{\sqrt[3]{x}}{\frac{y + x}{\sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{y + x}{\color{blue}{1 \cdot \sqrt[3]{y}}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

12. Applied add-cube-cbrt 0.8

    \[\leadsto \left(\frac{\sqrt[3]{x}}{\frac{y + x}{\sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{\color{blue}{\left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right) \cdot \sqrt[3]{y + x}}}{1 \cdot \sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

13. Applied times-frac 0.8

    \[\leadsto \left(\frac{\sqrt[3]{x}}{\frac{y + x}{\sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\color{blue}{\frac{\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}}{1} \cdot \frac{\sqrt[3]{y + x}}{\sqrt[3]{y}}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

14. Applied *-un-lft-identity 0.8

    \[\leadsto \left(\frac{\sqrt[3]{x}}{\frac{y + x}{\sqrt[3]{y}}} \cdot \frac{\color{blue}{1 \cdot \sqrt[3]{x}}}{\frac{\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}}{1} \cdot \frac{\sqrt[3]{y + x}}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

15. Applied times-frac 0.8

    \[\leadsto \left(\frac{\sqrt[3]{x}}{\frac{y + x}{\sqrt[3]{y}}} \cdot \color{blue}{\left(\frac{1}{\frac{\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}}{1}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{y + x}}{\sqrt[3]{y}}}\right)}\right) \cdot \frac{\sqrt[3]{x}}{\frac{\left(x + y\right) + 1.0}{\sqrt[3]{y}}}\]

16. Final simplification 0.8

    \[\leadsto \left(\left(\frac{1}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{x + y}}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{x + y}{\sqrt[3]{y}}}\right) \cdot \frac{\sqrt[3]{x}}{\frac{1.0 + \left(x + y\right)}{\sqrt[3]{y}}}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))