Average Error: 52.4 → 0.4
Time: 19.0s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{c \cdot \left(a \cdot 3\right) + \left(b \cdot b - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \left(\sqrt[3]{c} \cdot \left(a \cdot 3\right)\right)}}}{a \cdot 3}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\frac{c \cdot \left(a \cdot 3\right) + \left(b \cdot b - b \cdot b\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \left(\sqrt[3]{c} \cdot \left(a \cdot 3\right)\right)}}}{a \cdot 3}
double f(double a, double b, double c) {
        double r74099 = b;
        double r74100 = -r74099;
        double r74101 = r74099 * r74099;
        double r74102 = 3.0;
        double r74103 = a;
        double r74104 = r74102 * r74103;
        double r74105 = c;
        double r74106 = r74104 * r74105;
        double r74107 = r74101 - r74106;
        double r74108 = sqrt(r74107);
        double r74109 = r74100 + r74108;
        double r74110 = r74109 / r74104;
        return r74110;
}
↓
double f(double a, double b, double c) {
        double r74111 = c;
        double r74112 = a;
        double r74113 = 3.0;
        double r74114 = r74112 * r74113;
        double r74115 = r74111 * r74114;
        double r74116 = b;
        double r74117 = r74116 * r74116;
        double r74118 = r74117 - r74117;
        double r74119 = r74115 + r74118;
        double r74120 = -r74116;
        double r74121 = cbrt(r74111);
        double r74122 = r74121 * r74121;
        double r74123 = r74121 * r74114;
        double r74124 = r74122 * r74123;
        double r74125 = r74117 - r74124;
        double r74126 = sqrt(r74125);
        double r74127 = r74120 - r74126;
        double r74128 = r74119 / r74127;
        double r74129 = r74128 / r74114;
        return r74129;
}

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. Simplified0.4

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

  5. Simplified0.4

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

  7. Applied add-cube-cbrt0.4

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

  8. Applied associate-*l*0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019194 
(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)))