The quadratic formula (r2)

Average Error: 33.9 → 10.5

Time: 27.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.5961406266953245 \cdot 10^{-58}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.1115579814291686 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r3200023 = b;
        double r3200024 = -r3200023;
        double r3200025 = r3200023 * r3200023;
        double r3200026 = 4.0;
        double r3200027 = a;
        double r3200028 = c;
        double r3200029 = r3200027 * r3200028;
        double r3200030 = r3200026 * r3200029;
        double r3200031 = r3200025 - r3200030;
        double r3200032 = sqrt(r3200031);
        double r3200033 = r3200024 - r3200032;
        double r3200034 = 2.0;
        double r3200035 = r3200034 * r3200027;
        double r3200036 = r3200033 / r3200035;
        return r3200036;
}

↓

double f(double a, double b, double c) {
        double r3200037 = b;
        double r3200038 = -1.5961406266953245e-58;
        bool r3200039 = r3200037 <= r3200038;
        double r3200040 = c;
        double r3200041 = r3200040 / r3200037;
        double r3200042 = -r3200041;
        double r3200043 = 3.1115579814291686e+29;
        bool r3200044 = r3200037 <= r3200043;
        double r3200045 = 1.0;
        double r3200046 = 2.0;
        double r3200047 = a;
        double r3200048 = r3200046 * r3200047;
        double r3200049 = -r3200037;
        double r3200050 = r3200037 * r3200037;
        double r3200051 = r3200047 * r3200040;
        double r3200052 = 4.0;
        double r3200053 = r3200051 * r3200052;
        double r3200054 = r3200050 - r3200053;
        double r3200055 = sqrt(r3200054);
        double r3200056 = r3200049 - r3200055;
        double r3200057 = r3200048 / r3200056;
        double r3200058 = r3200045 / r3200057;
        double r3200059 = r3200049 / r3200047;
        double r3200060 = r3200044 ? r3200058 : r3200059;
        double r3200061 = r3200039 ? r3200042 : r3200060;
        return r3200061;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.9
Target	20.6
Herbie	10.5

\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < -1.5961406266953245e-58

   1. Initial program 53.4

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

   2. Taylor expanded around -inf 53.4

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

   3. Simplified 53.4

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

   4. Taylor expanded around -inf 8.1

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

   5. Simplified 8.1

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

   if -1.5961406266953245e-58 < b < 3.1115579814291686e+29

   1. Initial program 14.7

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

   2. Taylor expanded around -inf 14.7

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

   3. Simplified 14.7

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

   5. Applied *-un-lft-identity 14.7

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

   6. Applied associate-/l* 14.9

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

   if 3.1115579814291686e+29 < b

   1. Initial program 34.4

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

   2. Taylor expanded around -inf 34.4

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

   3. Simplified 34.4

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

   5. Applied *-un-lft-identity 34.4

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

   6. Applied associate-/l* 34.5

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

   8. Applied associate-/r/ 34.5

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

   9. Taylor expanded around 0 6.5

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

   10. Simplified 6.5

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

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.5961406266953245 \cdot 10^{-58}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.1115579814291686 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 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)))