Average Error: 43.4 → 0.4
Time: 6.2s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\mathsf{fma}\left(4 \cdot \left(a \cdot c\right), \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}}\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\mathsf{fma}\left(4 \cdot \left(a \cdot c\right), \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}}\right)}
double f(double a, double b, double c) {
        double r35385 = b;
        double r35386 = -r35385;
        double r35387 = r35385 * r35385;
        double r35388 = 4.0;
        double r35389 = a;
        double r35390 = r35388 * r35389;
        double r35391 = c;
        double r35392 = r35390 * r35391;
        double r35393 = r35387 - r35392;
        double r35394 = sqrt(r35393);
        double r35395 = r35386 + r35394;
        double r35396 = 2.0;
        double r35397 = r35396 * r35389;
        double r35398 = r35395 / r35397;
        return r35398;
}
↓
double f(double a, double b, double c) {
        double r35399 = 0.0;
        double r35400 = 4.0;
        double r35401 = a;
        double r35402 = c;
        double r35403 = r35401 * r35402;
        double r35404 = r35400 * r35403;
        double r35405 = r35399 + r35404;
        double r35406 = 2.0;
        double r35407 = r35406 * r35401;
        double r35408 = b;
        double r35409 = -r35408;
        double r35410 = r35408 * r35408;
        double r35411 = 3.0;
        double r35412 = pow(r35410, r35411);
        double r35413 = r35400 * r35401;
        double r35414 = r35413 * r35402;
        double r35415 = pow(r35414, r35411);
        double r35416 = r35412 - r35415;
        double r35417 = fma(r35408, r35408, r35414);
        double r35418 = r35410 * r35410;
        double r35419 = fma(r35404, r35417, r35418);
        double r35420 = r35416 / r35419;
        double r35421 = sqrt(r35420);
        double r35422 = r35409 - r35421;
        double r35423 = r35407 * r35422;
        double r35424 = r35405 / r35423;
        return r35424;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.4

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

  3. Applied flip-+43.4

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

  4. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
  5. Using strategy rm

  6. Applied div-inv0.5

    \[\leadsto \frac{\color{blue}{\left(0 + 4 \cdot \left(a \cdot c\right)\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

  7. Applied associate-/l*0.5

    \[\leadsto \color{blue}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\frac{2 \cdot a}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

  8. Simplified0.4

    \[\leadsto \frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]
  9. Using strategy rm

  10. Applied flip3--0.4

    \[\leadsto \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}\right)}\]

  11. Simplified0.4

    \[\leadsto \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\color{blue}{\mathsf{fma}\left(4 \cdot \left(a \cdot c\right), \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}}}\right)}\]

  12. Final simplification0.4

    \[\leadsto \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\mathsf{fma}\left(4 \cdot \left(a \cdot c\right), \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), \left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}}\right)}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))