quadp (p42, positive)

Average Error: 32.9 → 29.0

Time: 20.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 1.017935821261199 \cdot 10^{+105}:\\ \;\;\;\;\left(\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b} \cdot \frac{\frac{1}{2}}{a}\right) \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1320226 = b;
        double r1320227 = -r1320226;
        double r1320228 = r1320226 * r1320226;
        double r1320229 = 4.0;
        double r1320230 = a;
        double r1320231 = c;
        double r1320232 = r1320230 * r1320231;
        double r1320233 = r1320229 * r1320232;
        double r1320234 = r1320228 - r1320233;
        double r1320235 = sqrt(r1320234);
        double r1320236 = r1320227 + r1320235;
        double r1320237 = 2.0;
        double r1320238 = r1320237 * r1320230;
        double r1320239 = r1320236 / r1320238;
        return r1320239;
}

↓

double f(double a, double b, double c) {
        double r1320240 = b;
        double r1320241 = 1.017935821261199e+105;
        bool r1320242 = r1320240 <= r1320241;
        double r1320243 = -4.0;
        double r1320244 = a;
        double r1320245 = c;
        double r1320246 = r1320244 * r1320245;
        double r1320247 = r1320240 * r1320240;
        double r1320248 = fma(r1320243, r1320246, r1320247);
        double r1320249 = sqrt(r1320248);
        double r1320250 = r1320249 - r1320240;
        double r1320251 = sqrt(r1320250);
        double r1320252 = 0.5;
        double r1320253 = r1320252 / r1320244;
        double r1320254 = r1320251 * r1320253;
        double r1320255 = r1320254 * r1320251;
        double r1320256 = 0.0;
        double r1320257 = r1320242 ? r1320255 : r1320256;
        return r1320257;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	32.9
Target	20.3
Herbie	29.0

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

Derivation

1. Split input into 2 regimes

2. if b < 1.017935821261199e+105

   1. Initial program 25.4

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

   2. Simplified 25.4

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

   4. Applied *-un-lft-identity 25.4

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

   5. Applied div-inv 25.5

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

   6. Applied times-frac 25.5

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

   7. Simplified 25.5

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

   8. Simplified 25.5

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

   10. Applied add-sqr-sqrt 25.8

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

   11. Applied associate-*l* 25.8

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

   if 1.017935821261199e+105 < b

   1. Initial program 58.4

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

   2. Simplified 58.4

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

   4. Applied *-un-lft-identity 58.4

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

   5. Applied div-inv 58.4

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

   6. Applied times-frac 58.4

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

   7. Simplified 58.4

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

   8. Simplified 58.4

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

   9. Taylor expanded around 0 39.7

      \[\leadsto \color{blue}{0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 29.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.017935821261199 \cdot 10^{+105}:\\ \;\;\;\;\left(\sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b} \cdot \frac{\frac{1}{2}}{a}\right) \cdot \sqrt{\sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)} - b}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +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)))