Average Error: 52.4 → 6.3
Time: 2.2min
Precision: binary64
Cost: 320
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[-0.5 \cdot \frac{c}{b}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
-0.5 \cdot \frac{c}{b}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c) :precision binary64 (* -0.5 (/ c b)))
double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - ((3.0 * a) * c))) / (3.0 * a);
}
↓
double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

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
Alternative 1
Accuracy6.3
Cost896
\[\frac{\frac{-1.5}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\frac{\sqrt[3]{3}}{\frac{c}{b}}}\]
Alternative 2
Accuracy6.3
Cost512
\[\frac{-1.5}{\frac{-3}{-\frac{c}{b}}}\]
Alternative 3
Accuracy6.4
Cost448
\[\frac{-0.5}{\frac{1}{\frac{c}{b}}}\]
Alternative 4
Accuracy6.5
Cost448
\[\frac{1}{-2 \cdot \frac{b}{c}}\]
Alternative 5
Accuracy6.5
Cost1024
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{-1.5}{\frac{\sqrt[3]{3}}{\frac{c}{b}}}\]
Alternative 6
Accuracy6.6
Cost704
\[\frac{\frac{-1.5 \cdot \left(a \cdot c\right)}{b}}{a \cdot 3}\]
Alternative 7
Accuracy6.6
Cost1024
\[\frac{c \cdot \frac{-1.5}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\frac{\sqrt[3]{3}}{\frac{1}{b}}}\]
Alternative 8
Accuracy7.3
Cost1152
\[\frac{-1.5}{\frac{3 \cdot \frac{\sqrt[3]{b} \cdot \sqrt[3]{b}}{\sqrt{c}}}{\frac{\sqrt{c}}{\sqrt[3]{b}}}}\]

Derivation

  1. Initial program 52.4

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

  2. Simplified52.4

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

  3. Taylor expanded around inf 6.3

    \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]

  4. Final simplification6.3

    \[\leadsto -0.5 \cdot \frac{c}{b}\]

Reproduce

herbie shell --seed 2020322 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))