Average Error: 28.5 → 0.3
Time: 7.8s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{c \cdot -2}{b + \sqrt{\sqrt[3]{{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{3}}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{c \cdot -2}{b + \sqrt{\sqrt[3]{{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right)}^{3}}}}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (/ (* c -2.0) (+ b (sqrt (cbrt (pow (- (* b b) (* c (* a 4.0))) 3.0))))))
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 (c * -2.0) / (b + sqrt(cbrt(pow(((b * b) - (c * (a * 4.0))), 3.0))));
}

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 28.5

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

  2. Simplified28.5

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

  4. Applied flip--_binary64_209928.6

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

  5. Simplified0.5

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

  6. Simplified0.5

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

  8. Applied clear-num_binary64_21230.5

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

  9. Simplified0.4

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

  11. Applied add-cbrt-cube_binary64_21600.4

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

  12. Simplified0.4

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

  14. Applied associate-/r/_binary64_20700.5

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

  15. Applied associate-/r*_binary64_20680.4

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

  16. Simplified0.3

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

  17. Final simplification0.3

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

Reproduce

herbie shell --seed 2020346 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))