Average Error: 52.3 → 51.5
Time: 36.2s
Precision: 64
\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{(\left(\sqrt{\sqrt{\sqrt[3]{\left((-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_* \cdot (-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*\right) \cdot (-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{a \cdot 3}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{(\left(\sqrt{\sqrt{\sqrt[3]{\left((-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_* \cdot (-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*\right) \cdot (-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{a \cdot 3}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r9553300 = b;
        double r9553301 = -r9553300;
        double r9553302 = r9553300 * r9553300;
        double r9553303 = 3.0;
        double r9553304 = a;
        double r9553305 = r9553303 * r9553304;
        double r9553306 = c;
        double r9553307 = r9553305 * r9553306;
        double r9553308 = r9553302 - r9553307;
        double r9553309 = sqrt(r9553308);
        double r9553310 = r9553301 + r9553309;
        double r9553311 = r9553310 / r9553305;
        return r9553311;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r9553312 = -3.0;
        double r9553313 = a;
        double r9553314 = c;
        double r9553315 = r9553313 * r9553314;
        double r9553316 = b;
        double r9553317 = r9553316 * r9553316;
        double r9553318 = fma(r9553312, r9553315, r9553317);
        double r9553319 = r9553318 * r9553318;
        double r9553320 = r9553319 * r9553318;
        double r9553321 = cbrt(r9553320);
        double r9553322 = sqrt(r9553321);
        double r9553323 = sqrt(r9553322);
        double r9553324 = sqrt(r9553318);
        double r9553325 = sqrt(r9553324);
        double r9553326 = -r9553316;
        double r9553327 = fma(r9553323, r9553325, r9553326);
        double r9553328 = 3.0;
        double r9553329 = r9553313 * r9553328;
        double r9553330 = r9553327 / r9553329;
        return r9553330;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 52.3

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

  2. Simplified52.2

    \[\leadsto \color{blue}{\frac{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}{3 \cdot a}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt52.0

    \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}} - b}{3 \cdot a}\]

  5. Applied fma-neg51.5

    \[\leadsto \frac{\color{blue}{(\left(\sqrt{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}}{3 \cdot a}\]
  6. Using strategy rm

  7. Applied add-cbrt-cube51.5

    \[\leadsto \frac{(\left(\sqrt{\sqrt{\color{blue}{\sqrt[3]{\left((-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_* \cdot (-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*\right) \cdot (-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{3 \cdot a}\]

  8. Final simplification51.5

    \[\leadsto \frac{(\left(\sqrt{\sqrt{\sqrt[3]{\left((-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_* \cdot (-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*\right) \cdot (-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}}\right) \cdot \left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{a \cdot 3}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))