Average Error: 52.3 → 51.2
Time: 13.2s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\sqrt{\frac{{\left({\left(\sqrt[3]{{\left(b \cdot b\right)}^{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)}} \cdot \left(\sqrt[3]{{\left(b \cdot b\right)}^{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)}} \cdot \sqrt[3]{{\left(b \cdot b\right)}^{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)}}\right)\right)}^{\left(\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)}\right)}^{\left(\sqrt[3]{\sqrt[3]{3}}\right)} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{b \cdot {b}^{3} + \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\sqrt{\frac{{\left({\left(\sqrt[3]{{\left(b \cdot b\right)}^{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)}} \cdot \left(\sqrt[3]{{\left(b \cdot b\right)}^{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)}} \cdot \sqrt[3]{{\left(b \cdot b\right)}^{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)}}\right)\right)}^{\left(\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)}\right)}^{\left(\sqrt[3]{\sqrt[3]{3}}\right)} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{b \cdot {b}^{3} + \left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right)}} - b}{a \cdot 2}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (/
  (-
   (sqrt
    (/
     (-
      (pow
       (pow
        (*
         (cbrt (pow (* b b) (* (cbrt 3.0) (cbrt 3.0))))
         (*
          (cbrt (pow (* b b) (* (cbrt 3.0) (cbrt 3.0))))
          (cbrt (pow (* b b) (* (cbrt 3.0) (cbrt 3.0))))))
        (cbrt (* (cbrt 3.0) (cbrt 3.0))))
       (cbrt (cbrt 3.0)))
      (pow (* (* 4.0 a) c) 3.0))
     (+ (* b (pow b 3.0)) (* (* (* 4.0 a) c) (+ (* b b) (* (* 4.0 a) c))))))
   b)
  (* a 2.0)))
double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - ((4.0 * a) * c))) / (2.0 * a);
}
↓
double code(double a, double b, double c) {
	return (sqrt((pow(pow((cbrt(pow((b * b), (cbrt(3.0) * cbrt(3.0)))) * (cbrt(pow((b * b), (cbrt(3.0) * cbrt(3.0)))) * cbrt(pow((b * b), (cbrt(3.0) * cbrt(3.0)))))), cbrt(cbrt(3.0) * cbrt(3.0))), cbrt(cbrt(3.0))) - pow(((4.0 * a) * c), 3.0)) / ((b * pow(b, 3.0)) + (((4.0 * a) * c) * ((b * b) + ((4.0 * a) * c))))) - b) / (a * 2.0);
}

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 52.3

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

  2. Simplified52.3

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

  4. Applied flip3--_binary6452.3

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

  5. Simplified52.0

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

  7. Applied add-cube-cbrt_binary6452.0

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

  8. Applied pow-unpow_binary6451.8

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

  10. Applied add-cube-cbrt_binary6451.8

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

  11. Applied cbrt-prod_binary6451.8

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

  12. Applied pow-unpow_binary6451.2

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

  14. Applied add-cube-cbrt_binary6451.2

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

  15. Final simplification51.2

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

Reproduce

herbie shell --seed 2020268 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :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) (* (* 4.0 a) c)))) (* 2.0 a)))