The quadratic formula (r1)

Average Error: 33.0 → 8.7

Time: 15.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.6519381339788066 \cdot 10^{+37}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le -1.3761661522305357 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}{\frac{a}{\sqrt{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}}}{2}\\ \mathbf{elif}\;b \le 6.555431533807236 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{a \cdot \frac{-4 \cdot c}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r4930765 = b;
        double r4930766 = -r4930765;
        double r4930767 = r4930765 * r4930765;
        double r4930768 = 4.0;
        double r4930769 = a;
        double r4930770 = r4930768 * r4930769;
        double r4930771 = c;
        double r4930772 = r4930770 * r4930771;
        double r4930773 = r4930767 - r4930772;
        double r4930774 = sqrt(r4930773);
        double r4930775 = r4930766 + r4930774;
        double r4930776 = 2.0;
        double r4930777 = r4930776 * r4930769;
        double r4930778 = r4930775 / r4930777;
        return r4930778;
}

↓

double f(double a, double b, double c) {
        double r4930779 = b;
        double r4930780 = -1.6519381339788066e+37;
        bool r4930781 = r4930779 <= r4930780;
        double r4930782 = c;
        double r4930783 = r4930782 / r4930779;
        double r4930784 = a;
        double r4930785 = r4930779 / r4930784;
        double r4930786 = r4930783 - r4930785;
        double r4930787 = 2.0;
        double r4930788 = r4930786 * r4930787;
        double r4930789 = r4930788 / r4930787;
        double r4930790 = -1.3761661522305357e-153;
        bool r4930791 = r4930779 <= r4930790;
        double r4930792 = -4.0;
        double r4930793 = r4930792 * r4930782;
        double r4930794 = r4930779 * r4930779;
        double r4930795 = fma(r4930793, r4930784, r4930794);
        double r4930796 = sqrt(r4930795);
        double r4930797 = r4930796 - r4930779;
        double r4930798 = sqrt(r4930797);
        double r4930799 = r4930784 / r4930798;
        double r4930800 = r4930798 / r4930799;
        double r4930801 = r4930800 / r4930787;
        double r4930802 = 6.555431533807236e+28;
        bool r4930803 = r4930779 <= r4930802;
        double r4930804 = r4930796 + r4930779;
        double r4930805 = r4930793 / r4930804;
        double r4930806 = r4930784 * r4930805;
        double r4930807 = r4930806 / r4930784;
        double r4930808 = r4930807 / r4930787;
        double r4930809 = -2.0;
        double r4930810 = r4930809 * r4930783;
        double r4930811 = r4930810 / r4930787;
        double r4930812 = r4930803 ? r4930808 : r4930811;
        double r4930813 = r4930791 ? r4930801 : r4930812;
        double r4930814 = r4930781 ? r4930789 : r4930813;
        return r4930814;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.0
Target	20.1
Herbie	8.7

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

Derivation

1. Split input into 4 regimes

2. if b < -1.6519381339788066e+37

   1. Initial program 33.6

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

   2. Simplified 33.6

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

   3. Taylor expanded around -inf 6.5

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

   4. Simplified 6.5

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

   if -1.6519381339788066e+37 < b < -1.3761661522305357e-153

   1. Initial program 5.2

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

   2. Simplified 5.2

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

   4. Applied clear-num 5.3

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

   6. Applied *-un-lft-identity 5.3

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

   7. Applied *-un-lft-identity 5.3

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

   8. Applied times-frac 5.3

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

   9. Applied add-cube-cbrt 5.3

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

   10. Applied times-frac 5.3

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

   11. Simplified 5.3

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

   12. Simplified 5.2

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

   14. Applied add-sqr-sqrt 5.6

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

   15. Applied associate-/l* 5.6

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

   if -1.3761661522305357e-153 < b < 6.555431533807236e+28

   1. Initial program 24.5

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

   2. Simplified 24.5

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

   4. Applied clear-num 24.5

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

   6. Applied *-un-lft-identity 24.5

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

   7. Applied *-un-lft-identity 24.5

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

   8. Applied times-frac 24.5

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

   9. Applied add-cube-cbrt 24.5

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

   10. Applied times-frac 24.5

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

   11. Simplified 24.5

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

   12. Simplified 24.5

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

   14. Applied flip-- 25.0

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

   15. Simplified 17.1

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

   17. Applied *-un-lft-identity 17.1

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

   18. Applied times-frac 14.9

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

   19. Simplified 14.9

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

   if 6.555431533807236e+28 < b

   1. Initial program 56.3

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

   2. Simplified 56.3

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

   3. Taylor expanded around inf 4.5

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
3. Recombined 4 regimes into one program.

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6519381339788066 \cdot 10^{+37}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le -1.3761661522305357 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}{\frac{a}{\sqrt{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}}}{2}\\ \mathbf{elif}\;b \le 6.555431533807236 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{a \cdot \frac{-4 \cdot c}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

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