Cubic critical, medium range

Average Error: 43.9 → 0.4

Time: 16.9s

Precision: binary64

Cost: 7680

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

↓

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (/ (/ (- c) a) (+ (/ b a) (/ (sqrt (+ (* b b) (* a (* c -3.0)))) a))))

C

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 (-c / a) / ((b / a) + (sqrt(((b * b) + (a * (c * -3.0)))) / a));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-c / a) / ((b / a) + (sqrt(((b * b) + (a * (c * (-3.0d0))))) / a))
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

public static double code(double a, double b, double c) {
	return (-c / a) / ((b / a) + (Math.sqrt(((b * b) + (a * (c * -3.0)))) / a));
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

↓

def code(a, b, c):
	return (-c / a) / ((b / a) + (math.sqrt(((b * b) + (a * (c * -3.0)))) / a))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

↓

function code(a, b, c)
	return Float64(Float64(Float64(-c) / a) / Float64(Float64(b / a) + Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -3.0)))) / a)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

↓

function tmp = code(a, b, c)
	tmp = (-c / a) / ((b / a) + (sqrt(((b * b) + (a * (c * -3.0)))) / a));
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := N[(N[((-c) / a), $MachinePrecision] / N[(N[(b / a), $MachinePrecision] + N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.9

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

2. Simplified 43.9

   \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot -0.3333333333333333} \]
   Proof

   [Start] 43.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   *-lft-identity [<=] 43.9	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   metadata-eval [<=] 43.9	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   times-frac [<=] 43.9	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
   neg-mul-1 [<=] 43.9	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
   distribute-rgt-neg-in [=>] 43.9	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
   times-frac [=>] 43.9	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
   *-commutative [=>] 43.9	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \cdot \frac{-1}{3}} \]
   neg-mul-1 [=>] 43.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \cdot \frac{-1}{3} \]
   neg-sub0 [=>] 43.9	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1 \cdot a} \cdot \frac{-1}{3} \]
   associate-+l- [=>] 43.9	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{-1 \cdot a} \cdot \frac{-1}{3} \]
   sub0-neg [=>] 43.9	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{-1 \cdot a} \cdot \frac{-1}{3} \]
   neg-mul-1 [=>] 43.9	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{-1 \cdot a} \cdot \frac{-1}{3} \]
   times-frac [=>] 43.9	\[ \color{blue}{\left(\frac{-1}{-1} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}\right)} \cdot \frac{-1}{3} \]
   metadata-eval [=>] 43.9	\[ \left(\color{blue}{1} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}\right) \cdot \frac{-1}{3} \]
   *-lft-identity [=>] 43.9	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{a}} \cdot \frac{-1}{3} \]
   cancel-sign-sub-inv [=>] 43.9	\[ \frac{b - \sqrt{\color{blue}{b \cdot b + \left(-3 \cdot a\right) \cdot c}}}{a} \cdot \frac{-1}{3} \]
   +-commutative [=>] 43.9	\[ \frac{b - \sqrt{\color{blue}{\left(-3 \cdot a\right) \cdot c + b \cdot b}}}{a} \cdot \frac{-1}{3} \]
   *-commutative [=>] 43.9	\[ \frac{b - \sqrt{\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b}}{a} \cdot \frac{-1}{3} \]
   distribute-lft-neg-in [=>] 43.9	\[ \frac{b - \sqrt{c \cdot \color{blue}{\left(\left(-3\right) \cdot a\right)} + b \cdot b}}{a} \cdot \frac{-1}{3} \]
   associate-*r* [=>] 43.9	\[ \frac{b - \sqrt{\color{blue}{\left(c \cdot \left(-3\right)\right) \cdot a} + b \cdot b}}{a} \cdot \frac{-1}{3} \]
   *-commutative [=>] 43.9	\[ \frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot \left(-3\right)\right)} + b \cdot b}}{a} \cdot \frac{-1}{3} \]
   fma-def [=>] 43.9	\[ \frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot \left(-3\right), b \cdot b\right)}}}{a} \cdot \frac{-1}{3} \]
   metadata-eval [=>] 43.9	\[ \frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot \color{blue}{-3}, b \cdot b\right)}}{a} \cdot \frac{-1}{3} \]
   metadata-eval [=>] 43.9	\[ \frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \color{blue}{-0.3333333333333333} \]

3. Applied egg-rr 44.3

   \[\leadsto \color{blue}{\frac{\left(\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right) \cdot -0.3333333333333333}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}} \]

4. Simplified 43.5

   \[\leadsto \color{blue}{\frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}{a}}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot -0.3333333333333333} \]
   Proof

   [Start] 44.3	\[ \frac{\left(\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right) \cdot -0.3333333333333333}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-/l* [=>] 44.3	\[ \color{blue}{\frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{\frac{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{-0.3333333333333333}}} \]
   associate-/r/ [=>] 44.3	\[ \color{blue}{\frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot -0.3333333333333333} \]
   associate-*l/ [=>] 43.9	\[ \frac{\frac{b}{a} \cdot \frac{b}{a} - \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{a}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot -0.3333333333333333 \]
   associate-*r/ [=>] 43.6	\[ \frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot -0.3333333333333333 \]
   rem-square-sqrt [=>] 43.5	\[ \frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\frac{\color{blue}{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot -0.3333333333333333 \]

5. Taylor expanded in b around 0 0.6

   \[\leadsto \frac{\color{blue}{3 \cdot \frac{c}{a}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot -0.3333333333333333 \]

6. Applied egg-rr 0.4

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

7. Applied egg-rr 0.4

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

8. Final simplification 0.4

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

Alternatives

Alternative 1
Error	5.8
Cost	1152

\[\frac{\frac{-c}{a}}{\frac{b}{a} + \left(\frac{b}{a} + -1.5 \cdot \frac{c}{b}\right)} \]

Alternative 2
Error	11.9
Cost	320

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

Reproduce

herbie shell --seed 2022356 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))