Quadratic roots, wide range

Average Error: 52.7 → 50.9

Time: 1.1m

Precision: 64

\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r2439223 = b;
        double r2439224 = -r2439223;
        double r2439225 = r2439223 * r2439223;
        double r2439226 = 4.0;
        double r2439227 = a;
        double r2439228 = r2439226 * r2439227;
        double r2439229 = c;
        double r2439230 = r2439228 * r2439229;
        double r2439231 = r2439225 - r2439230;
        double r2439232 = sqrt(r2439231);
        double r2439233 = r2439224 + r2439232;
        double r2439234 = 2.0;
        double r2439235 = r2439234 * r2439227;
        double r2439236 = r2439233 / r2439235;
        return r2439236;
}

↓

double f(double a, double b, double c) {
        double r2439237 = c;
        double r2439238 = a;
        double r2439239 = -4.0;
        double r2439240 = r2439238 * r2439239;
        double r2439241 = b;
        double r2439242 = r2439241 * r2439241;
        double r2439243 = fma(r2439237, r2439240, r2439242);
        double r2439244 = 0.3333333333333333;
        double r2439245 = pow(r2439243, r2439244);
        double r2439246 = r2439237 * r2439239;
        double r2439247 = fma(r2439246, r2439238, r2439242);
        double r2439248 = r2439247 * r2439247;
        double r2439249 = cbrt(r2439248);
        double r2439250 = r2439245 * r2439249;
        double r2439251 = sqrt(r2439250);
        double r2439252 = sqrt(r2439251);
        double r2439253 = sqrt(r2439243);
        double r2439254 = sqrt(r2439253);
        double r2439255 = -r2439241;
        double r2439256 = fma(r2439252, r2439254, r2439255);
        double r2439257 = 2.0;
        double r2439258 = r2439256 / r2439257;
        double r2439259 = r2439258 / r2439238;
        return r2439259;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.7

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

2. Simplified 52.7

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

4. Applied add-sqr-sqrt 52.4

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

5. Applied fma-neg 51.8

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

7. Applied add-cbrt-cube 51.9

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

9. Applied pow1/3 51.3

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

11. Applied unpow-prod-down 51.3

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

12. Simplified 50.9

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

13. Final simplification 50.9

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

Reproduce

herbie shell --seed 2019119 +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)))