quadm (p42, negative)

Average Error: 33.8 → 6.7

Time: 34.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.3422084503380959 \cdot 10^{+126}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -2.9582179666484207 \cdot 10^{-229}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 9.112814637305151 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r2807758 = b;
        double r2807759 = -r2807758;
        double r2807760 = r2807758 * r2807758;
        double r2807761 = 4.0;
        double r2807762 = a;
        double r2807763 = c;
        double r2807764 = r2807762 * r2807763;
        double r2807765 = r2807761 * r2807764;
        double r2807766 = r2807760 - r2807765;
        double r2807767 = sqrt(r2807766);
        double r2807768 = r2807759 - r2807767;
        double r2807769 = 2.0;
        double r2807770 = r2807769 * r2807762;
        double r2807771 = r2807768 / r2807770;
        return r2807771;
}

↓

double f(double a, double b, double c) {
        double r2807772 = b;
        double r2807773 = -1.3422084503380959e+126;
        bool r2807774 = r2807772 <= r2807773;
        double r2807775 = c;
        double r2807776 = -r2807775;
        double r2807777 = r2807776 / r2807772;
        double r2807778 = -2.9582179666484207e-229;
        bool r2807779 = r2807772 <= r2807778;
        double r2807780 = 2.0;
        double r2807781 = r2807780 * r2807775;
        double r2807782 = r2807772 * r2807772;
        double r2807783 = 4.0;
        double r2807784 = a;
        double r2807785 = r2807784 * r2807775;
        double r2807786 = r2807783 * r2807785;
        double r2807787 = r2807782 - r2807786;
        double r2807788 = sqrt(r2807787);
        double r2807789 = r2807788 - r2807772;
        double r2807790 = sqrt(r2807789);
        double r2807791 = r2807781 / r2807790;
        double r2807792 = 1.0;
        double r2807793 = r2807792 / r2807790;
        double r2807794 = r2807791 * r2807793;
        double r2807795 = 9.112814637305151e+83;
        bool r2807796 = r2807772 <= r2807795;
        double r2807797 = 0.5;
        double r2807798 = r2807797 / r2807784;
        double r2807799 = -r2807772;
        double r2807800 = r2807799 - r2807788;
        double r2807801 = r2807798 * r2807800;
        double r2807802 = r2807775 / r2807772;
        double r2807803 = r2807772 / r2807784;
        double r2807804 = r2807802 - r2807803;
        double r2807805 = r2807796 ? r2807801 : r2807804;
        double r2807806 = r2807779 ? r2807794 : r2807805;
        double r2807807 = r2807774 ? r2807777 : r2807806;
        return r2807807;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.8
Target	20.8
Herbie	6.7

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

Derivation

1. Split input into 4 regimes

2. if b < -1.3422084503380959e+126

   1. Initial program 59.9

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

   2. Taylor expanded around -inf 2.0

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

   3. Simplified 2.0

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

   if -1.3422084503380959e+126 < b < -2.9582179666484207e-229

   1. Initial program 37.3

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

   3. Applied *-un-lft-identity 37.3

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

   4. Applied associate-/l* 37.3

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

   6. Applied flip-- 37.4

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

   7. Applied associate-/r/ 37.5

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

   8. Applied *-un-lft-identity 37.5

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

   9. Applied times-frac 37.5

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

   10. Simplified 15.9

       \[\leadsto \color{blue}{\left(\frac{\frac{1}{2}}{a} \cdot \left(0 - \left(-4 \cdot c\right) \cdot a\right)\right)} \cdot \frac{1}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

   11. Simplified 15.9

       \[\leadsto \left(\frac{\frac{1}{2}}{a} \cdot \left(0 - \left(-4 \cdot c\right) \cdot a\right)\right) \cdot \color{blue}{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\]

   12. Taylor expanded around inf 7.4

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

   14. Applied add-sqr-sqrt 7.6

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

   15. Applied add-cube-cbrt 7.6

       \[\leadsto \left(2 \cdot c\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\]

   16. Applied times-frac 7.7

       \[\leadsto \left(2 \cdot c\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\right)}\]

   17. Applied associate-*r* 7.7

       \[\leadsto \color{blue}{\left(\left(2 \cdot c\right) \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\]

   18. Simplified 7.6

       \[\leadsto \color{blue}{\frac{c \cdot 2}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\]

   if -2.9582179666484207e-229 < b < 9.112814637305151e+83

   1. Initial program 10.0

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

   3. Applied div-inv 10.1

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

   4. Simplified 10.1

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

   if 9.112814637305151e+83 < b

   1. Initial program 42.5

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

   2. Taylor expanded around inf 4.1

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3422084503380959 \cdot 10^{+126}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -2.9582179666484207 \cdot 10^{-229}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{1}{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 9.112814637305151 \cdot 10^{+83}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :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)))