Rosa's FloatVsDoubleBenchmark

Average Error: 0.5 → 0.5

Time: 3.3m

Precision: 64

Math

\[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

↓

\[\left(x1 \cdot x1 + 1\right) \cdot \left(x1 + \left(\left(x1 \cdot x1\right) \cdot \left(\frac{x2 \cdot 2 - \left(x1 - x1 \cdot \left(x1 \cdot 3\right)\right)}{x1 \cdot x1 + 1} \cdot 4 - 6\right) + \frac{x2 \cdot 2 - \left(x1 - x1 \cdot \left(x1 \cdot 3\right)\right)}{x1 \cdot x1 + 1} \cdot \left(\left(x1 \cdot \left(\frac{x2 \cdot 2 - \left(x1 - x1 \cdot \left(x1 \cdot 3\right)\right)}{x1 \cdot x1 + 1} - 3\right)\right) \cdot 2\right)\right)\right) + \left(\left(\frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - x1 \cdot \left(x1 \cdot 3\right)\right)}}{\sqrt[3]{\sqrt{x1 \cdot x1 + 1}}} \cdot \left(\frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - x1 \cdot \left(x1 \cdot 3\right)\right)}}{\sqrt[3]{\sqrt{x1 \cdot x1 + 1}} \cdot \sqrt[3]{\sqrt{x1 \cdot x1 + 1}}} \cdot \left(x1 \cdot \left(x1 \cdot 3\right)\right)\right)\right) \cdot \frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - x1 \cdot \left(x1 \cdot 3\right)\right)}}{\sqrt{x1 \cdot x1 + 1}} + \left(3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - x1\right) - x2 \cdot 2}{x1 \cdot x1 + 1} + x1\right)\right)\]

C

double f(double x1, double x2) {
        double r25120538 = x1;
        double r25120539 = 2.0;
        double r25120540 = r25120539 * r25120538;
        double r25120541 = 3.0;
        double r25120542 = r25120541 * r25120538;
        double r25120543 = r25120542 * r25120538;
        double r25120544 = x2;
        double r25120545 = r25120539 * r25120544;
        double r25120546 = r25120543 + r25120545;
        double r25120547 = r25120546 - r25120538;
        double r25120548 = r25120538 * r25120538;
        double r25120549 = 1.0;
        double r25120550 = r25120548 + r25120549;
        double r25120551 = r25120547 / r25120550;
        double r25120552 = r25120540 * r25120551;
        double r25120553 = r25120551 - r25120541;
        double r25120554 = r25120552 * r25120553;
        double r25120555 = 4.0;
        double r25120556 = r25120555 * r25120551;
        double r25120557 = 6.0;
        double r25120558 = r25120556 - r25120557;
        double r25120559 = r25120548 * r25120558;
        double r25120560 = r25120554 + r25120559;
        double r25120561 = r25120560 * r25120550;
        double r25120562 = r25120543 * r25120551;
        double r25120563 = r25120561 + r25120562;
        double r25120564 = r25120548 * r25120538;
        double r25120565 = r25120563 + r25120564;
        double r25120566 = r25120565 + r25120538;
        double r25120567 = r25120543 - r25120545;
        double r25120568 = r25120567 - r25120538;
        double r25120569 = r25120568 / r25120550;
        double r25120570 = r25120541 * r25120569;
        double r25120571 = r25120566 + r25120570;
        double r25120572 = r25120538 + r25120571;
        return r25120572;
}

↓

double f(double x1, double x2) {
        double r25120573 = x1;
        double r25120574 = r25120573 * r25120573;
        double r25120575 = 1.0;
        double r25120576 = r25120574 + r25120575;
        double r25120577 = x2;
        double r25120578 = 2.0;
        double r25120579 = r25120577 * r25120578;
        double r25120580 = 3.0;
        double r25120581 = r25120573 * r25120580;
        double r25120582 = r25120573 * r25120581;
        double r25120583 = r25120573 - r25120582;
        double r25120584 = r25120579 - r25120583;
        double r25120585 = r25120584 / r25120576;
        double r25120586 = 4.0;
        double r25120587 = r25120585 * r25120586;
        double r25120588 = 6.0;
        double r25120589 = r25120587 - r25120588;
        double r25120590 = r25120574 * r25120589;
        double r25120591 = r25120585 - r25120580;
        double r25120592 = r25120573 * r25120591;
        double r25120593 = r25120592 * r25120578;
        double r25120594 = r25120585 * r25120593;
        double r25120595 = r25120590 + r25120594;
        double r25120596 = r25120573 + r25120595;
        double r25120597 = r25120576 * r25120596;
        double r25120598 = cbrt(r25120584);
        double r25120599 = sqrt(r25120576);
        double r25120600 = cbrt(r25120599);
        double r25120601 = r25120598 / r25120600;
        double r25120602 = r25120600 * r25120600;
        double r25120603 = r25120598 / r25120602;
        double r25120604 = r25120603 * r25120582;
        double r25120605 = r25120601 * r25120604;
        double r25120606 = r25120598 / r25120599;
        double r25120607 = r25120605 * r25120606;
        double r25120608 = r25120582 - r25120573;
        double r25120609 = r25120608 - r25120579;
        double r25120610 = r25120609 / r25120576;
        double r25120611 = r25120580 * r25120610;
        double r25120612 = r25120611 + r25120573;
        double r25120613 = r25120607 + r25120612;
        double r25120614 = r25120597 + r25120613;
        return r25120614;
}

Error

Bits error versus x1

Bits error versus x2

Derivation

1. Initial program 0.5

   \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

2. Simplified 0.5

   \[\leadsto \color{blue}{\left(\left(x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x1\right) - x2 \cdot 2}{1 + x1 \cdot x1}\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1}\right) + \left(1 + x1 \cdot x1\right) \cdot \left(\left(\left(4 \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\left(\left(\frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 3\right) \cdot x1\right) \cdot 2\right) \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1}\right) + x1\right)}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 0.5

   \[\leadsto \left(\left(x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x1\right) - x2 \cdot 2}{1 + x1 \cdot x1}\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{\color{blue}{\sqrt{1 + x1 \cdot x1} \cdot \sqrt{1 + x1 \cdot x1}}}\right) + \left(1 + x1 \cdot x1\right) \cdot \left(\left(\left(4 \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\left(\left(\frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 3\right) \cdot x1\right) \cdot 2\right) \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1}\right) + x1\right)\]

5. Applied add-cube-cbrt 0.5

   \[\leadsto \left(\left(x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x1\right) - x2 \cdot 2}{1 + x1 \cdot x1}\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)} \cdot \sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}\right) \cdot \sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}}}{\sqrt{1 + x1 \cdot x1} \cdot \sqrt{1 + x1 \cdot x1}}\right) + \left(1 + x1 \cdot x1\right) \cdot \left(\left(\left(4 \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\left(\left(\frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 3\right) \cdot x1\right) \cdot 2\right) \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1}\right) + x1\right)\]

6. Applied times-frac 0.5

   \[\leadsto \left(\left(x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x1\right) - x2 \cdot 2}{1 + x1 \cdot x1}\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)} \cdot \sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}}{\sqrt{1 + x1 \cdot x1}} \cdot \frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}}{\sqrt{1 + x1 \cdot x1}}\right)}\right) + \left(1 + x1 \cdot x1\right) \cdot \left(\left(\left(4 \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\left(\left(\frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 3\right) \cdot x1\right) \cdot 2\right) \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1}\right) + x1\right)\]

7. Applied associate-*r* 0.5

   \[\leadsto \left(\left(x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x1\right) - x2 \cdot 2}{1 + x1 \cdot x1}\right) + \color{blue}{\left(\left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)} \cdot \sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}}{\sqrt{1 + x1 \cdot x1}}\right) \cdot \frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}}{\sqrt{1 + x1 \cdot x1}}}\right) + \left(1 + x1 \cdot x1\right) \cdot \left(\left(\left(4 \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\left(\left(\frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 3\right) \cdot x1\right) \cdot 2\right) \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1}\right) + x1\right)\]
8. Using strategy rm

9. Applied add-cube-cbrt 0.5

   \[\leadsto \left(\left(x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x1\right) - x2 \cdot 2}{1 + x1 \cdot x1}\right) + \left(\left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)} \cdot \sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}}{\color{blue}{\left(\sqrt[3]{\sqrt{1 + x1 \cdot x1}} \cdot \sqrt[3]{\sqrt{1 + x1 \cdot x1}}\right) \cdot \sqrt[3]{\sqrt{1 + x1 \cdot x1}}}}\right) \cdot \frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}}{\sqrt{1 + x1 \cdot x1}}\right) + \left(1 + x1 \cdot x1\right) \cdot \left(\left(\left(4 \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\left(\left(\frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 3\right) \cdot x1\right) \cdot 2\right) \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1}\right) + x1\right)\]

10. Applied times-frac 0.5

    \[\leadsto \left(\left(x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x1\right) - x2 \cdot 2}{1 + x1 \cdot x1}\right) + \left(\left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}}{\sqrt[3]{\sqrt{1 + x1 \cdot x1}} \cdot \sqrt[3]{\sqrt{1 + x1 \cdot x1}}} \cdot \frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}}{\sqrt[3]{\sqrt{1 + x1 \cdot x1}}}\right)}\right) \cdot \frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}}{\sqrt{1 + x1 \cdot x1}}\right) + \left(1 + x1 \cdot x1\right) \cdot \left(\left(\left(4 \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\left(\left(\frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 3\right) \cdot x1\right) \cdot 2\right) \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1}\right) + x1\right)\]

11. Applied associate-*r* 0.5

    \[\leadsto \left(\left(x1 + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - x1\right) - x2 \cdot 2}{1 + x1 \cdot x1}\right) + \color{blue}{\left(\left(\left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}}{\sqrt[3]{\sqrt{1 + x1 \cdot x1}} \cdot \sqrt[3]{\sqrt{1 + x1 \cdot x1}}}\right) \cdot \frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}}{\sqrt[3]{\sqrt{1 + x1 \cdot x1}}}\right)} \cdot \frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}}{\sqrt{1 + x1 \cdot x1}}\right) + \left(1 + x1 \cdot x1\right) \cdot \left(\left(\left(4 \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 6\right) \cdot \left(x1 \cdot x1\right) + \left(\left(\left(\frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1} - 3\right) \cdot x1\right) \cdot 2\right) \cdot \frac{x2 \cdot 2 - \left(x1 - \left(3 \cdot x1\right) \cdot x1\right)}{1 + x1 \cdot x1}\right) + x1\right)\]

12. Final simplification 0.5

    \[\leadsto \left(x1 \cdot x1 + 1\right) \cdot \left(x1 + \left(\left(x1 \cdot x1\right) \cdot \left(\frac{x2 \cdot 2 - \left(x1 - x1 \cdot \left(x1 \cdot 3\right)\right)}{x1 \cdot x1 + 1} \cdot 4 - 6\right) + \frac{x2 \cdot 2 - \left(x1 - x1 \cdot \left(x1 \cdot 3\right)\right)}{x1 \cdot x1 + 1} \cdot \left(\left(x1 \cdot \left(\frac{x2 \cdot 2 - \left(x1 - x1 \cdot \left(x1 \cdot 3\right)\right)}{x1 \cdot x1 + 1} - 3\right)\right) \cdot 2\right)\right)\right) + \left(\left(\frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - x1 \cdot \left(x1 \cdot 3\right)\right)}}{\sqrt[3]{\sqrt{x1 \cdot x1 + 1}}} \cdot \left(\frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - x1 \cdot \left(x1 \cdot 3\right)\right)}}{\sqrt[3]{\sqrt{x1 \cdot x1 + 1}} \cdot \sqrt[3]{\sqrt{x1 \cdot x1 + 1}}} \cdot \left(x1 \cdot \left(x1 \cdot 3\right)\right)\right)\right) \cdot \frac{\sqrt[3]{x2 \cdot 2 - \left(x1 - x1 \cdot \left(x1 \cdot 3\right)\right)}}{\sqrt{x1 \cdot x1 + 1}} + \left(3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - x1\right) - x2 \cdot 2}{x1 \cdot x1 + 1} + x1\right)\right)\]

Reproduce

herbie shell --seed 2019107 
(FPCore (x1 x2)
  :name "Rosa's FloatVsDoubleBenchmark"
  (+ x1 (+ (+ (+ (+ (* (+ (* (* (* 2 x1) (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1))) (- (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1)) 3)) (* (* x1 x1) (- (* 4 (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1))) 6))) (+ (* x1 x1) 1)) (* (* (* 3 x1) x1) (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1)))) (* (* x1 x1) x1)) x1) (* 3 (/ (- (- (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1))))))