Average Error: 52.8 → 0.1
Time: 2.8m
Precision: 64
\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{\left(-b\right) - \sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r14090727 = b;
        double r14090728 = -r14090727;
        double r14090729 = r14090727 * r14090727;
        double r14090730 = 3.0;
        double r14090731 = a;
        double r14090732 = r14090730 * r14090731;
        double r14090733 = c;
        double r14090734 = r14090732 * r14090733;
        double r14090735 = r14090729 - r14090734;
        double r14090736 = sqrt(r14090735);
        double r14090737 = r14090728 + r14090736;
        double r14090738 = r14090737 / r14090732;
        return r14090738;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r14090739 = c;
        double r14090740 = b;
        double r14090741 = -r14090740;
        double r14090742 = -3.0;
        double r14090743 = a;
        double r14090744 = r14090743 * r14090739;
        double r14090745 = r14090740 * r14090740;
        double r14090746 = fma(r14090742, r14090744, r14090745);
        double r14090747 = sqrt(r14090746);
        double r14090748 = r14090741 - r14090747;
        double r14090749 = r14090739 / r14090748;
        return r14090749;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 52.8

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
  2. Using strategy rm

  3. Applied flip-+52.8

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

  4. Applied associate-/l/52.8

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

  5. Simplified0.5

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

  7. Applied associate-*r*0.5

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

  9. Applied times-frac0.5

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

  10. Simplified0.4

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

  11. Simplified0.4

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

  13. Applied pow10.4

    \[\leadsto \frac{c}{a} \cdot \color{blue}{{\left(\frac{a}{\left(-b\right) - \sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right)}^{1}}\]

  14. Applied pow10.4

    \[\leadsto \color{blue}{{\left(\frac{c}{a}\right)}^{1}} \cdot {\left(\frac{a}{\left(-b\right) - \sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right)}^{1}\]

  15. Applied pow-prod-down0.4

    \[\leadsto \color{blue}{{\left(\frac{c}{a} \cdot \frac{a}{\left(-b\right) - \sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right)}^{1}}\]

  16. Simplified0.1

    \[\leadsto {\color{blue}{\left(\frac{c}{\left(-b\right) - \sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right)}}^{1}\]

  17. Final simplification0.1

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

Reproduce

herbie shell --seed 2019112 +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)))