Average Error: 52.8 → 50.9
Time: 1.1m
Precision: 64
\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{(\left(\sqrt{\sqrt{{\left((c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}} \cdot \sqrt[3]{(\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_* \cdot (\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_*}}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{(\left(\sqrt{\sqrt{{\left((c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}} \cdot \sqrt[3]{(\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_* \cdot (\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_*}}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}
double f(double a, double b, double c) {
        double r2436124 = b;
        double r2436125 = -r2436124;
        double r2436126 = r2436124 * r2436124;
        double r2436127 = 4.0;
        double r2436128 = a;
        double r2436129 = r2436127 * r2436128;
        double r2436130 = c;
        double r2436131 = r2436129 * r2436130;
        double r2436132 = r2436126 - r2436131;
        double r2436133 = sqrt(r2436132);
        double r2436134 = r2436125 + r2436133;
        double r2436135 = 2.0;
        double r2436136 = r2436135 * r2436128;
        double r2436137 = r2436134 / r2436136;
        return r2436137;
}
↓
double f(double a, double b, double c) {
        double r2436138 = c;
        double r2436139 = a;
        double r2436140 = -4.0;
        double r2436141 = r2436139 * r2436140;
        double r2436142 = b;
        double r2436143 = r2436142 * r2436142;
        double r2436144 = fma(r2436138, r2436141, r2436143);
        double r2436145 = 0.3333333333333333;
        double r2436146 = pow(r2436144, r2436145);
        double r2436147 = r2436138 * r2436140;
        double r2436148 = fma(r2436147, r2436139, r2436143);
        double r2436149 = r2436148 * r2436148;
        double r2436150 = cbrt(r2436149);
        double r2436151 = r2436146 * r2436150;
        double r2436152 = sqrt(r2436151);
        double r2436153 = sqrt(r2436152);
        double r2436154 = sqrt(r2436144);
        double r2436155 = sqrt(r2436154);
        double r2436156 = -r2436142;
        double r2436157 = fma(r2436153, r2436155, r2436156);
        double r2436158 = 2.0;
        double r2436159 = r2436157 / r2436158;
        double r2436160 = r2436159 / r2436139;
        return r2436160;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.8

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

  2. Simplified52.8

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*} - b}{2}}{a}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt52.8

    \[\leadsto \frac{\frac{\sqrt{\color{blue}{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*} \cdot \sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}} - b}{2}}{a}\]

  5. Applied sqrt-prod52.5

    \[\leadsto \frac{\frac{\color{blue}{\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}} - b}{2}}{a}\]

  6. Applied fma-neg52.0

    \[\leadsto \frac{\frac{\color{blue}{(\left(\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}}{2}}{a}\]
  7. Using strategy rm

  8. Applied add-cbrt-cube52.0

    \[\leadsto \frac{\frac{(\left(\sqrt{\sqrt{\color{blue}{\sqrt[3]{\left((c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_* \cdot (c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*\right) \cdot (c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}\]
  9. Using strategy rm

  10. Applied pow1/351.3

    \[\leadsto \frac{\frac{(\left(\sqrt{\sqrt{\color{blue}{{\left(\left((c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_* \cdot (c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*\right) \cdot (c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}}}}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}\]
  11. Using strategy rm

  12. Applied unpow-prod-down51.3

    \[\leadsto \frac{\frac{(\left(\sqrt{\sqrt{\color{blue}{{\left((c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_* \cdot (c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}} \cdot {\left((c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}}}}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}\]

  13. Simplified50.9

    \[\leadsto \frac{\frac{(\left(\sqrt{\sqrt{\color{blue}{\sqrt[3]{(\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_* \cdot (\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_*}} \cdot {\left((c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}}}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(-4 \cdot a\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}\]

  14. Final simplification50.9

    \[\leadsto \frac{\frac{(\left(\sqrt{\sqrt{{\left((c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*\right)}^{\frac{1}{3}} \cdot \sqrt[3]{(\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_* \cdot (\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_*}}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))