Average Error: 43.7 → 0.3
Time: 5.8s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1}{\frac{2}{4} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1}{\frac{2}{4} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c}}
double f(double a, double b, double c) {
        double r41354 = b;
        double r41355 = -r41354;
        double r41356 = r41354 * r41354;
        double r41357 = 4.0;
        double r41358 = a;
        double r41359 = r41357 * r41358;
        double r41360 = c;
        double r41361 = r41359 * r41360;
        double r41362 = r41356 - r41361;
        double r41363 = sqrt(r41362);
        double r41364 = r41355 + r41363;
        double r41365 = 2.0;
        double r41366 = r41365 * r41358;
        double r41367 = r41364 / r41366;
        return r41367;
}
↓
double f(double a, double b, double c) {
        double r41368 = 1.0;
        double r41369 = 2.0;
        double r41370 = 4.0;
        double r41371 = r41369 / r41370;
        double r41372 = b;
        double r41373 = -r41372;
        double r41374 = r41372 * r41372;
        double r41375 = a;
        double r41376 = r41370 * r41375;
        double r41377 = c;
        double r41378 = r41376 * r41377;
        double r41379 = r41374 - r41378;
        double r41380 = sqrt(r41379);
        double r41381 = r41373 - r41380;
        double r41382 = r41381 / r41377;
        double r41383 = r41371 * r41382;
        double r41384 = r41368 / r41383;
        return r41384;
}

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 43.7

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

  3. Applied flip-+43.7

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

  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 clear-num0.5

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

  7. Simplified0.5

    \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]
  8. Using strategy rm

  9. Applied times-frac0.5

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

  10. Applied associate-*l*0.5

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2020036 
(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)))