The quadratic formula (r2)

Average Error: 33.6 → 9.9

Time: 22.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -8.1855168042470635 \cdot 10^{-53}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.634898599408338 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2}}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r2586623 = b;
        double r2586624 = -r2586623;
        double r2586625 = r2586623 * r2586623;
        double r2586626 = 4.0;
        double r2586627 = a;
        double r2586628 = c;
        double r2586629 = r2586627 * r2586628;
        double r2586630 = r2586626 * r2586629;
        double r2586631 = r2586625 - r2586630;
        double r2586632 = sqrt(r2586631);
        double r2586633 = r2586624 - r2586632;
        double r2586634 = 2.0;
        double r2586635 = r2586634 * r2586627;
        double r2586636 = r2586633 / r2586635;
        return r2586636;
}

↓

double f(double a, double b, double c) {
        double r2586637 = b;
        double r2586638 = -8.1855168042470635e-53;
        bool r2586639 = r2586637 <= r2586638;
        double r2586640 = c;
        double r2586641 = r2586640 / r2586637;
        double r2586642 = -r2586641;
        double r2586643 = 3.634898599408338e+146;
        bool r2586644 = r2586637 <= r2586643;
        double r2586645 = -0.5;
        double r2586646 = -4.0;
        double r2586647 = r2586646 * r2586640;
        double r2586648 = a;
        double r2586649 = r2586637 * r2586637;
        double r2586650 = fma(r2586647, r2586648, r2586649);
        double r2586651 = sqrt(r2586650);
        double r2586652 = r2586651 + r2586637;
        double r2586653 = r2586645 * r2586652;
        double r2586654 = r2586653 / r2586648;
        double r2586655 = 2.0;
        double r2586656 = r2586637 / r2586648;
        double r2586657 = r2586640 / r2586656;
        double r2586658 = r2586657 - r2586637;
        double r2586659 = r2586655 * r2586658;
        double r2586660 = r2586659 / r2586655;
        double r2586661 = r2586660 / r2586648;
        double r2586662 = r2586644 ? r2586654 : r2586661;
        double r2586663 = r2586639 ? r2586642 : r2586662;
        return r2586663;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.6
Target	20.6
Herbie	9.9

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

Derivation

1. Split input into 3 regimes

2. if b < -8.1855168042470635e-53

   1. Initial program 54.3

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

   2. Simplified 54.3

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

   4. Applied *-un-lft-identity 54.3

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

   5. Applied div-inv 54.3

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

   6. Applied times-frac 54.3

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

   7. Simplified 54.3

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

   8. Simplified 54.3

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

   9. Taylor expanded around -inf 7.5

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

   10. Simplified 7.5

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

   if -8.1855168042470635e-53 < b < 3.634898599408338e+146

   1. Initial program 13.3

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

   2. Simplified 13.3

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

   4. Applied *-un-lft-identity 13.3

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

   5. Applied div-inv 13.3

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

   6. Applied times-frac 13.4

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

   7. Simplified 13.4

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

   8. Simplified 13.4

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

   10. Applied associate-*r/ 13.3

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

   11. Simplified 13.3

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

   12. Taylor expanded around -inf 13.3

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

   13. Simplified 13.3

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

   if 3.634898599408338e+146 < b

   1. Initial program 58.5

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

   2. Simplified 58.5

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

   3. Taylor expanded around inf 10.9

      \[\leadsto \frac{\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2}}{a}\]

   4. Simplified 2.1

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

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.1855168042470635 \cdot 10^{-53}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.634898599408338 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{-1}{2} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +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)))