Average Error: 44.0 → 43.3
Time: 32.1s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(a \cdot -3\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -3\right) \cdot c\right)\right)}}\right), \left(-b\right)\right) \cdot \left(\frac{\sqrt{\frac{1}{3}}}{\sqrt{a}} \cdot \frac{\sqrt{\frac{1}{3}}}{\sqrt{a}}\right)\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(a \cdot -3\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -3\right) \cdot c\right)\right)}}\right), \left(-b\right)\right) \cdot \left(\frac{\sqrt{\frac{1}{3}}}{\sqrt{a}} \cdot \frac{\sqrt{\frac{1}{3}}}{\sqrt{a}}\right)
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r3128942 = b;
        double r3128943 = -r3128942;
        double r3128944 = r3128942 * r3128942;
        double r3128945 = 3.0;
        double r3128946 = a;
        double r3128947 = r3128945 * r3128946;
        double r3128948 = c;
        double r3128949 = r3128947 * r3128948;
        double r3128950 = r3128944 - r3128949;
        double r3128951 = sqrt(r3128950);
        double r3128952 = r3128943 + r3128951;
        double r3128953 = r3128952 / r3128947;
        return r3128953;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r3128954 = c;
        double r3128955 = a;
        double r3128956 = -3.0;
        double r3128957 = r3128955 * r3128956;
        double r3128958 = b;
        double r3128959 = r3128958 * r3128958;
        double r3128960 = fma(r3128954, r3128957, r3128959);
        double r3128961 = sqrt(r3128960);
        double r3128962 = sqrt(r3128961);
        double r3128963 = r3128957 * r3128954;
        double r3128964 = fma(r3128958, r3128958, r3128963);
        double r3128965 = sqrt(r3128964);
        double r3128966 = sqrt(r3128965);
        double r3128967 = -r3128958;
        double r3128968 = fma(r3128962, r3128966, r3128967);
        double r3128969 = 0.3333333333333333;
        double r3128970 = sqrt(r3128969);
        double r3128971 = sqrt(r3128955);
        double r3128972 = r3128970 / r3128971;
        double r3128973 = r3128972 * r3128972;
        double r3128974 = r3128968 * r3128973;
        return r3128974;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 44.0

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

  2. Simplified44.0

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

  4. Applied add-sqr-sqrt44.0

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

  5. Applied fma-neg43.4

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

  6. Taylor expanded around inf 43.4

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

  7. Simplified43.3

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

  9. Applied div-inv43.3

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

  10. Simplified43.3

    \[\leadsto \mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-3 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -3\right) \cdot c\right)\right)}}\right), \left(-b\right)\right) \cdot \color{blue}{\frac{\frac{1}{3}}{a}}\]
  11. Using strategy rm

  12. Applied add-sqr-sqrt43.3

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

  13. Applied add-sqr-sqrt43.3

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

  14. Applied times-frac43.3

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

  15. Final simplification43.3

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))