Average Error: 52.6 → 0.1
Time: 2.4min
Precision: binary64
Cost: 7424
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[2 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
2 \cdot \frac{c}{\left(-b\right) - \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

Alternatives

Alternative 1
Error2.9
Cost1088
\[\frac{2 \cdot a}{\frac{a}{\frac{c}{2 \cdot \left(\frac{c}{\frac{b}{a}} - b\right)}}}\]
Alternative 2
Error6.1
Cost256
\[-\frac{c}{b}\]
Alternative 3
Error57.6
Cost64
\[-1\]
Alternative 4
Error61.9
Cost64
\[0\]
Alternative 5
Error63.0
Cost64
\[1\]

Error

Time

Derivation

  1. Initial program 52.6

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

  3. Applied flip-+_binary64_107552.6

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

  4. Simplified0.4

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

  6. Applied associate-/r*_binary64_10450.4

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

  7. Simplified0.2

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

  9. Applied associate-/l*_binary64_10460.2

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

  10. Simplified0.2

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

  12. Applied associate-/r/_binary64_10470.1

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

  13. Simplified0.1

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2021040 
(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)))