Given's Rotation SVD example, simplified

Average Error: 15.2 → 0.0

Time: 21.5s

Precision: 64

Math

\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.01509898314466614:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{-\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right), \left(\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right), \left(\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)\right) + \mathsf{fma}\left(1, \frac{1}{2}, \left(-\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \mathbf{elif}\;x \le 0.0112517724140347:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{-11}{128}, \left(\mathsf{fma}\left(\frac{1}{8}, \left(x \cdot x\right), \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{69}{1024} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(1, \frac{1}{8}, \left(\frac{-\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)} \cdot \frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right)}\right)\right) + \mathsf{fma}\left(\left(-\frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right)\right)}{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right), \frac{1}{4}\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \end{array}\]

C

double f(double x) {
        double r3078483 = 1.0;
        double r3078484 = 0.5;
        double r3078485 = x;
        double r3078486 = hypot(r3078483, r3078485);
        double r3078487 = r3078483 / r3078486;
        double r3078488 = r3078483 + r3078487;
        double r3078489 = r3078484 * r3078488;
        double r3078490 = sqrt(r3078489);
        double r3078491 = r3078483 - r3078490;
        return r3078491;
}

↓

double f(double x) {
        double r3078492 = x;
        double r3078493 = -0.01509898314466614;
        bool r3078494 = r3078492 <= r3078493;
        double r3078495 = 0.5;
        double r3078496 = sqrt(r3078495);
        double r3078497 = -r3078496;
        double r3078498 = 1.0;
        double r3078499 = hypot(r3078498, r3078492);
        double r3078500 = cbrt(r3078499);
        double r3078501 = r3078497 / r3078500;
        double r3078502 = r3078500 * r3078500;
        double r3078503 = r3078496 / r3078502;
        double r3078504 = r3078496 / r3078500;
        double r3078505 = r3078504 * r3078503;
        double r3078506 = fma(r3078501, r3078503, r3078505);
        double r3078507 = -r3078505;
        double r3078508 = fma(r3078498, r3078495, r3078507);
        double r3078509 = r3078506 + r3078508;
        double r3078510 = r3078495 / r3078499;
        double r3078511 = r3078510 + r3078495;
        double r3078512 = sqrt(r3078511);
        double r3078513 = r3078498 + r3078512;
        double r3078514 = r3078509 / r3078513;
        double r3078515 = 0.0112517724140347;
        bool r3078516 = r3078492 <= r3078515;
        double r3078517 = r3078492 * r3078492;
        double r3078518 = r3078517 * r3078517;
        double r3078519 = -0.0859375;
        double r3078520 = 0.125;
        double r3078521 = 0.0673828125;
        double r3078522 = r3078521 * r3078517;
        double r3078523 = r3078518 * r3078522;
        double r3078524 = fma(r3078520, r3078517, r3078523);
        double r3078525 = fma(r3078518, r3078519, r3078524);
        double r3078526 = sqrt(r3078520);
        double r3078527 = -r3078526;
        double r3078528 = r3078499 * r3078499;
        double r3078529 = r3078527 / r3078528;
        double r3078530 = r3078526 / r3078499;
        double r3078531 = r3078529 * r3078530;
        double r3078532 = fma(r3078498, r3078520, r3078531);
        double r3078533 = -r3078530;
        double r3078534 = r3078526 / r3078528;
        double r3078535 = r3078530 * r3078534;
        double r3078536 = fma(r3078533, r3078534, r3078535);
        double r3078537 = r3078532 + r3078536;
        double r3078538 = 0.25;
        double r3078539 = fma(r3078510, r3078511, r3078538);
        double r3078540 = r3078537 / r3078539;
        double r3078541 = r3078540 / r3078513;
        double r3078542 = r3078516 ? r3078525 : r3078541;
        double r3078543 = r3078494 ? r3078514 : r3078542;
        return r3078543;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.01509898314466614

   1. Initial program 1.0

      \[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

   2. Simplified 1.0

      \[\leadsto \color{blue}{1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
   3. Using strategy rm

   4. Applied flip-- 1.0

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}} \cdot \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}\]

   5. Simplified 0.1

      \[\leadsto \frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 0.1

      \[\leadsto \frac{\frac{1}{2} - \frac{\frac{1}{2}}{\color{blue}{\left(\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   8. Applied add-sqr-sqrt 0.1

      \[\leadsto \frac{\frac{1}{2} - \frac{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}}{\left(\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   9. Applied times-frac 0.1

      \[\leadsto \frac{\frac{1}{2} - \color{blue}{\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   10. Applied *-un-lft-identity 0.1

       \[\leadsto \frac{\color{blue}{1 \cdot \frac{1}{2}} - \frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   11. Applied prod-diff 0.1

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1, \frac{1}{2}, \left(-\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)\right) + \mathsf{fma}\left(\left(-\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right), \left(\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right), \left(\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   if -0.01509898314466614 < x < 0.0112517724140347

   1. Initial program 29.4

      \[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

   2. Simplified 29.4

      \[\leadsto \color{blue}{1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   3. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot {x}^{2} + \frac{69}{1024} \cdot {x}^{6}\right) - \frac{11}{128} \cdot {x}^{4}}\]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{-11}{128}, \left(\mathsf{fma}\left(\frac{1}{8}, \left(x \cdot x\right), \left(\left(\frac{69}{1024} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}\]

   if 0.0112517724140347 < x

   1. Initial program 1.0

      \[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

   2. Simplified 1.0

      \[\leadsto \color{blue}{1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
   3. Using strategy rm

   4. Applied flip-- 1.0

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}} \cdot \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}}\]

   5. Simplified 0.1

      \[\leadsto \frac{\color{blue}{\frac{1}{2} - \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
   6. Using strategy rm

   7. Applied flip3-- 0.1

      \[\leadsto \frac{\color{blue}{\frac{{\frac{1}{2}}^{3} - {\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   8. Simplified 0.1

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{8} - \frac{\frac{1}{8}}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2} \cdot \frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   9. Simplified 0.1

      \[\leadsto \frac{\frac{\frac{1}{8} - \frac{\frac{1}{8}}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\color{blue}{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right), \frac{1}{4}\right)}}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
   10. Using strategy rm

   11. Applied add-sqr-sqrt 0.1

       \[\leadsto \frac{\frac{\frac{1}{8} - \frac{\color{blue}{\sqrt{\frac{1}{8}} \cdot \sqrt{\frac{1}{8}}}}{\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}}{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right), \frac{1}{4}\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   12. Applied times-frac 0.1

       \[\leadsto \frac{\frac{\frac{1}{8} - \color{blue}{\frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)} \cdot \frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right)}}}{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right), \frac{1}{4}\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   13. Applied *-un-lft-identity 0.1

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot \frac{1}{8}} - \frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)} \cdot \frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right), \frac{1}{4}\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]

   14. Applied prod-diff 0.1

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(1, \frac{1}{8}, \left(-\frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right)\right) + \mathsf{fma}\left(\left(-\frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right)\right)}}{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right), \frac{1}{4}\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.01509898314466614:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\frac{-\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right), \left(\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right), \left(\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)\right) + \mathsf{fma}\left(1, \frac{1}{2}, \left(-\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)}} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(1, x\right)}}\right)\right)}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \mathbf{elif}\;x \le 0.0112517724140347:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \frac{-11}{128}, \left(\mathsf{fma}\left(\frac{1}{8}, \left(x \cdot x\right), \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{69}{1024} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(1, \frac{1}{8}, \left(\frac{-\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)} \cdot \frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right)}\right)\right) + \mathsf{fma}\left(\left(-\frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\sqrt{\frac{1}{8}}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right)\right)}{\mathsf{fma}\left(\left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)}\right), \left(\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}\right), \frac{1}{4}\right)}}{1 + \sqrt{\frac{\frac{1}{2}}{\mathsf{hypot}\left(1, x\right)} + \frac{1}{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))