Average Error: 52.4 → 0.5
Time: 18.7s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{3 \cdot \left(a \cdot c\right)}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{3 \cdot \left(a \cdot c\right)}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}
double f(double a, double b, double c) {
        double r74156 = b;
        double r74157 = -r74156;
        double r74158 = r74156 * r74156;
        double r74159 = 3.0;
        double r74160 = a;
        double r74161 = r74159 * r74160;
        double r74162 = c;
        double r74163 = r74161 * r74162;
        double r74164 = r74158 - r74163;
        double r74165 = sqrt(r74164);
        double r74166 = r74157 + r74165;
        double r74167 = r74166 / r74161;
        return r74167;
}
↓
double f(double a, double b, double c) {
        double r74168 = 3.0;
        double r74169 = a;
        double r74170 = c;
        double r74171 = r74169 * r74170;
        double r74172 = r74168 * r74171;
        double r74173 = b;
        double r74174 = -r74173;
        double r74175 = r74173 * r74173;
        double r74176 = r74168 * r74169;
        double r74177 = r74176 * r74170;
        double r74178 = r74175 - r74177;
        double r74179 = sqrt(r74178);
        double r74180 = r74174 - r74179;
        double r74181 = r74180 * r74176;
        double r74182 = r74172 / r74181;
        return r74182;
}

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 52.4

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

  3. Applied flip-+52.4

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

  4. Simplified0.5

    \[\leadsto \frac{\frac{\color{blue}{0 + \left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
  5. Using strategy rm

  6. Applied div-inv0.6

    \[\leadsto \frac{\color{blue}{\left(0 + \left(a \cdot c\right) \cdot 3\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

  7. Applied associate-/l*0.6

    \[\leadsto \color{blue}{\frac{0 + \left(a \cdot c\right) \cdot 3}{\frac{3 \cdot a}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}\]

  8. Simplified0.5

    \[\leadsto \frac{0 + \left(a \cdot c\right) \cdot 3}{\color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}}\]

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e+31) (< 4.93038e-32 b 2.02824e+31) (< 4.93038e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))