quadm (p42, negative)

Average Error: 34.4 → 10.3

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.0674124610604968 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le 5.96876625840091586 \cdot 10^{107}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right)\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r88743 = b;
        double r88744 = -r88743;
        double r88745 = r88743 * r88743;
        double r88746 = 4.0;
        double r88747 = a;
        double r88748 = c;
        double r88749 = r88747 * r88748;
        double r88750 = r88746 * r88749;
        double r88751 = r88745 - r88750;
        double r88752 = sqrt(r88751);
        double r88753 = r88744 - r88752;
        double r88754 = 2.0;
        double r88755 = r88754 * r88747;
        double r88756 = r88753 / r88755;
        return r88756;
}

↓

double f(double a, double b, double c) {
        double r88757 = b;
        double r88758 = -1.0674124610604968e-82;
        bool r88759 = r88757 <= r88758;
        double r88760 = 1.0;
        double r88761 = 2.0;
        double r88762 = r88760 / r88761;
        double r88763 = -2.0;
        double r88764 = c;
        double r88765 = r88764 / r88757;
        double r88766 = r88763 * r88765;
        double r88767 = r88762 * r88766;
        double r88768 = 5.968766258400916e+107;
        bool r88769 = r88757 <= r88768;
        double r88770 = -r88757;
        double r88771 = r88757 * r88757;
        double r88772 = 4.0;
        double r88773 = a;
        double r88774 = r88773 * r88764;
        double r88775 = r88772 * r88774;
        double r88776 = r88771 - r88775;
        double r88777 = sqrt(r88776);
        double r88778 = r88770 - r88777;
        double r88779 = r88778 / r88773;
        double r88780 = r88762 * r88779;
        double r88781 = r88761 * r88765;
        double r88782 = 2.0;
        double r88783 = r88757 / r88773;
        double r88784 = r88782 * r88783;
        double r88785 = r88781 - r88784;
        double r88786 = r88762 * r88785;
        double r88787 = r88769 ? r88780 : r88786;
        double r88788 = r88759 ? r88767 : r88787;
        return r88788;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.4
Target	21.5
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 < -1.0674124610604968e-82

   1. Initial program 52.3

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

   3. Applied clear-num 52.4

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

   5. Applied *-un-lft-identity 52.4

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

   6. Applied times-frac 52.4

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

   7. Applied add-cube-cbrt 52.4

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

   8. Applied times-frac 52.4

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

   9. Simplified 52.4

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

   10. Simplified 52.3

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

   11. Taylor expanded around -inf 8.9

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

   if -1.0674124610604968e-82 < b < 5.968766258400916e+107

   1. Initial program 13.7

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

   3. Applied clear-num 13.9

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

   5. Applied *-un-lft-identity 13.9

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

   6. Applied times-frac 13.8

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

   7. Applied add-cube-cbrt 13.8

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

   8. Applied times-frac 13.8

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

   9. Simplified 13.8

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

   10. Simplified 13.7

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

   if 5.968766258400916e+107 < b

   1. Initial program 50.0

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

   3. Applied clear-num 50.0

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

   5. Applied *-un-lft-identity 50.0

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

   6. Applied times-frac 50.0

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

   7. Applied add-cube-cbrt 50.0

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

   8. Applied times-frac 50.0

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

   9. Simplified 50.0

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

   10. Simplified 50.0

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

   11. Taylor expanded around inf 3.8

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.0674124610604968 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \le 5.96876625840091586 \cdot 10^{107}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(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)))