quadp (p42, positive)

Average Error: 33.2 → 12.6

Time: 23.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 2.1712981559507382 \cdot 10^{-281}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 3.4192303804034762 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{a \cdot -4}{a} \cdot \frac{c}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + b}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot -4}{a} \cdot \frac{c}{2 \cdot b}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r2491094 = b;
        double r2491095 = -r2491094;
        double r2491096 = r2491094 * r2491094;
        double r2491097 = 4.0;
        double r2491098 = a;
        double r2491099 = c;
        double r2491100 = r2491098 * r2491099;
        double r2491101 = r2491097 * r2491100;
        double r2491102 = r2491096 - r2491101;
        double r2491103 = sqrt(r2491102);
        double r2491104 = r2491095 + r2491103;
        double r2491105 = 2.0;
        double r2491106 = r2491105 * r2491098;
        double r2491107 = r2491104 / r2491106;
        return r2491107;
}

↓

double f(double a, double b, double c) {
        double r2491108 = b;
        double r2491109 = 2.1712981559507382e-281;
        bool r2491110 = r2491108 <= r2491109;
        double r2491111 = a;
        double r2491112 = c;
        double r2491113 = r2491111 * r2491112;
        double r2491114 = -4.0;
        double r2491115 = r2491108 * r2491108;
        double r2491116 = fma(r2491113, r2491114, r2491115);
        double r2491117 = sqrt(r2491116);
        double r2491118 = r2491117 / r2491111;
        double r2491119 = r2491108 / r2491111;
        double r2491120 = r2491118 - r2491119;
        double r2491121 = 2.0;
        double r2491122 = r2491120 / r2491121;
        double r2491123 = 3.4192303804034762e+140;
        bool r2491124 = r2491108 <= r2491123;
        double r2491125 = r2491111 * r2491114;
        double r2491126 = r2491125 / r2491111;
        double r2491127 = fma(r2491112, r2491125, r2491115);
        double r2491128 = sqrt(r2491127);
        double r2491129 = r2491128 + r2491108;
        double r2491130 = r2491112 / r2491129;
        double r2491131 = r2491126 * r2491130;
        double r2491132 = r2491131 / r2491121;
        double r2491133 = r2491121 * r2491108;
        double r2491134 = r2491112 / r2491133;
        double r2491135 = r2491126 * r2491134;
        double r2491136 = r2491135 / r2491121;
        double r2491137 = r2491124 ? r2491132 : r2491136;
        double r2491138 = r2491110 ? r2491122 : r2491137;
        return r2491138;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.2
Target	20.2
Herbie	12.6

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

Derivation

1. Split input into 3 regimes

2. if b < 2.1712981559507382e-281

   1. Initial program 20.3

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

   2. Simplified 20.3

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

   4. Applied div-sub 20.3

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

   if 2.1712981559507382e-281 < b < 3.4192303804034762e+140

   1. Initial program 35.0

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

   2. Simplified 35.0

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

   4. Applied flip-- 35.1

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

   5. Simplified 16.3

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, 0\right)}}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}{2}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 16.3

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

   8. Applied *-un-lft-identity 16.3

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

   9. Applied times-frac 16.3

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

   10. Simplified 16.3

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

   11. Simplified 8.0

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

   if 3.4192303804034762e+140 < b

   1. Initial program 61.5

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

   2. Simplified 61.5

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

   4. Applied flip-- 61.6

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

   5. Simplified 35.5

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, 0\right)}}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}{2}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 35.5

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

   8. Applied *-un-lft-identity 35.5

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

   9. Applied times-frac 35.5

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

   10. Simplified 35.5

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

   11. Simplified 34.6

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

   13. Applied add-cube-cbrt 34.7

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

   14. Applied fma-def 34.7

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

   15. Taylor expanded around 0 1.8

       \[\leadsto \frac{1 \cdot \left(\frac{c}{\color{blue}{2 \cdot b}} \cdot \frac{-4 \cdot a}{a}\right)}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.1712981559507382 \cdot 10^{-281}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 3.4192303804034762 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{a \cdot -4}{a} \cdot \frac{c}{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} + b}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot -4}{a} \cdot \frac{c}{2 \cdot b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :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)))