quadm (p42, negative)

Average Error: 33.6 → 10.3

Time: 9.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.16908657181932359 \cdot 10^{-104}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3316184968738608 \cdot 10^{61}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{b}{a}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r108440 = b;
        double r108441 = -r108440;
        double r108442 = r108440 * r108440;
        double r108443 = 4.0;
        double r108444 = a;
        double r108445 = c;
        double r108446 = r108444 * r108445;
        double r108447 = r108443 * r108446;
        double r108448 = r108442 - r108447;
        double r108449 = sqrt(r108448);
        double r108450 = r108441 - r108449;
        double r108451 = 2.0;
        double r108452 = r108451 * r108444;
        double r108453 = r108450 / r108452;
        return r108453;
}

↓

double f(double a, double b, double c) {
        double r108454 = b;
        double r108455 = -4.1690865718193236e-104;
        bool r108456 = r108454 <= r108455;
        double r108457 = -1.0;
        double r108458 = c;
        double r108459 = r108458 / r108454;
        double r108460 = r108457 * r108459;
        double r108461 = 1.3316184968738608e+61;
        bool r108462 = r108454 <= r108461;
        double r108463 = -r108454;
        double r108464 = r108454 * r108454;
        double r108465 = 4.0;
        double r108466 = a;
        double r108467 = r108466 * r108458;
        double r108468 = r108465 * r108467;
        double r108469 = r108464 - r108468;
        double r108470 = sqrt(r108469);
        double r108471 = r108463 - r108470;
        double r108472 = r108471 / r108466;
        double r108473 = 2.0;
        double r108474 = r108472 / r108473;
        double r108475 = -2.0;
        double r108476 = r108454 / r108466;
        double r108477 = r108475 * r108476;
        double r108478 = r108477 / r108473;
        double r108479 = r108462 ? r108474 : r108478;
        double r108480 = r108456 ? r108460 : r108479;
        return r108480;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.6
Target	20.6
Herbie	10.3

\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < -4.1690865718193236e-104

   1. Initial program 51.5

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

   2. Taylor expanded around -inf 11.0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

   if -4.1690865718193236e-104 < b < 1.3316184968738608e+61

   1. Initial program 12.3

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   2. Using strategy rm

   3. Applied associate-/r* 12.3

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 12.3

      \[\leadsto \frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{1 \cdot 2}}}{a}\]

   6. Applied *-un-lft-identity 12.3

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{1 \cdot 2}}{a}\]

   7. Applied times-frac 12.3

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}}{a}\]

   8. Applied associate-/l* 12.4

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{a}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}}}\]
   9. Using strategy rm

   10. Applied associate-/r/ 12.4

       \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot 2}}\]

   11. Applied associate-/r* 12.4

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{1}}{\frac{a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2}}\]

   12. Simplified 12.4

       \[\leadsto \frac{\color{blue}{\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2}\]
   13. Using strategy rm

   14. Applied associate-*l/ 12.3

       \[\leadsto \frac{\color{blue}{\frac{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{a}}}{2}\]

   15. Simplified 12.3

       \[\leadsto \frac{\frac{\color{blue}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a}}{2}\]

   if 1.3316184968738608e+61 < b

   1. Initial program 39.6

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   2. Using strategy rm

   3. Applied associate-/r* 39.5

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity 39.5

      \[\leadsto \frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{1 \cdot 2}}}{a}\]

   6. Applied *-un-lft-identity 39.5

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{1 \cdot 2}}{a}\]

   7. Applied times-frac 39.5

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}}{a}\]

   8. Applied associate-/l* 39.6

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{a}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}}}\]
   9. Using strategy rm

   10. Applied associate-/r/ 39.6

       \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot 2}}\]

   11. Applied associate-/r* 39.6

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{1}}{\frac{a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2}}\]

   12. Simplified 39.6

       \[\leadsto \frac{\color{blue}{\frac{1}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2}\]

   13. Taylor expanded around 0 4.6

       \[\leadsto \frac{\color{blue}{-2 \cdot \frac{b}{a}}}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.16908657181932359 \cdot 10^{-104}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3316184968738608 \cdot 10^{61}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{b}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))