Average Error: 43.9 → 42.2
Time: 46.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{(\left(\sqrt{\sqrt{(-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{\sqrt[3]{(\left(-3 \cdot c\right) \cdot a + \left(b \cdot b\right))_* \cdot (\left(-3 \cdot c\right) \cdot a + \left(b \cdot b\right))_*} \cdot {\left((-3 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}}}}\right) + \left(-b\right))_*}{a \cdot 3}\]
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r18512119 = b;
        double r18512120 = -r18512119;
        double r18512121 = r18512119 * r18512119;
        double r18512122 = 3.0;
        double r18512123 = a;
        double r18512124 = r18512122 * r18512123;
        double r18512125 = c;
        double r18512126 = r18512124 * r18512125;
        double r18512127 = r18512121 - r18512126;
        double r18512128 = sqrt(r18512127);
        double r18512129 = r18512120 + r18512128;
        double r18512130 = r18512129 / r18512124;
        return r18512130;
}
↓
double f(double a, double b, double c, double __attribute__((unused)) d) {
        double r18512131 = -3.0;
        double r18512132 = a;
        double r18512133 = c;
        double r18512134 = r18512132 * r18512133;
        double r18512135 = b;
        double r18512136 = r18512135 * r18512135;
        double r18512137 = fma(r18512131, r18512134, r18512136);
        double r18512138 = sqrt(r18512137);
        double r18512139 = sqrt(r18512138);
        double r18512140 = r18512131 * r18512133;
        double r18512141 = fma(r18512140, r18512132, r18512136);
        double r18512142 = r18512141 * r18512141;
        double r18512143 = cbrt(r18512142);
        double r18512144 = 0.3333333333333333;
        double r18512145 = pow(r18512137, r18512144);
        double r18512146 = r18512143 * r18512145;
        double r18512147 = sqrt(r18512146);
        double r18512148 = sqrt(r18512147);
        double r18512149 = -r18512135;
        double r18512150 = fma(r18512139, r18512148, r18512149);
        double r18512151 = 3.0;
        double r18512152 = r18512132 * r18512151;
        double r18512153 = r18512150 / r18512152;
        return r18512153;
}


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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 43.9

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

  2. Simplified43.9

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

  4. Applied add-sqr-sqrt43.9

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

  5. Applied sqrt-prod43.9

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

  6. Applied fma-neg43.2

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

  8. Applied add-cbrt-cube43.3

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

  10. Applied pow1/342.9

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

  12. Applied unpow-prod-down42.9

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

  13. Simplified42.2

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

  14. Final simplification42.2

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

Reproduce

herbie shell --seed 2019101 +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)))