Average Error: 52.7 → 0.1
Time: 21.0s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{1 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}}
double f(double a, double b, double c) {
        double r3705605 = b;
        double r3705606 = -r3705605;
        double r3705607 = r3705605 * r3705605;
        double r3705608 = 3.0;
        double r3705609 = a;
        double r3705610 = r3705608 * r3705609;
        double r3705611 = c;
        double r3705612 = r3705610 * r3705611;
        double r3705613 = r3705607 - r3705612;
        double r3705614 = sqrt(r3705613);
        double r3705615 = r3705606 + r3705614;
        double r3705616 = r3705615 / r3705610;
        return r3705616;
}
↓
double f(double a, double b, double c) {
        double r3705617 = 1.0;
        double r3705618 = c;
        double r3705619 = r3705617 * r3705618;
        double r3705620 = b;
        double r3705621 = -r3705620;
        double r3705622 = r3705620 * r3705620;
        double r3705623 = a;
        double r3705624 = 3.0;
        double r3705625 = r3705623 * r3705624;
        double r3705626 = r3705625 * r3705618;
        double r3705627 = r3705622 - r3705626;
        double r3705628 = sqrt(r3705627);
        double r3705629 = r3705621 - r3705628;
        double r3705630 = r3705619 / r3705629;
        return r3705630;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 52.7

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

  3. Applied flip-+52.7

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

  4. Simplified0.4

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

  6. Applied div-inv0.5

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

  7. Applied associate-/l*0.5

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

  8. Simplified0.4

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

  10. Applied associate-/r*0.2

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

  11. Taylor expanded around 0 0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b c)
  :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) (* (* 3.0 a) c)))) (* 3.0 a)))