Average Error: 52.7 → 0.1
Time: 7.5s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[-2 \cdot \frac{c}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
-2 \cdot \frac{c}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (* -2.0 (/ c (+ b (sqrt (- (* b b) (* c (* a 4.0))))))))
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 -2.0 * (c / (b + sqrt((b * b) - (c * (a * 4.0)))));
}

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 52.7

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

  2. Simplified52.7

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

  4. Applied flip--_binary64_175852.7

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

  5. Simplified0.4

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

  6. Simplified0.4

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

  8. Applied *-un-lft-identity_binary64_17830.4

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

  9. Applied times-frac_binary64_17890.2

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

  10. Applied associate-/l*_binary64_17280.2

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

  11. Simplified0.2

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

  13. Applied add-cube-cbrt_binary64_18181.4

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

  14. Applied times-frac_binary64_17891.4

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

  15. Applied add-cube-cbrt_binary64_18180.4

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

  16. Applied times-frac_binary64_17890.2

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

  17. Simplified0.2

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

  18. Simplified0.1

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

  19. Final simplification0.1

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

Reproduce

herbie shell --seed 2020356 
(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)))