Cubic critical, medium range

Average Error: 43.9 → 2.9

Time: 1.0min

Precision: binary64

Cost: 53568

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

Math

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

↓

\[\left({c}^{4} \cdot {a}^{3}\right) \cdot \frac{-1.0546875}{{b}^{7}} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (+
  (* (* (pow c 4.0) (pow a 3.0)) (/ -1.0546875 (pow b 7.0)))
  (+
   (* -0.5625 (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0)))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* (pow c 2.0) a) (pow b 3.0)))))))

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 ((pow(c, 4.0) * pow(a, 3.0)) * (-1.0546875 / pow(b, 7.0))) + ((-0.5625 * ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((pow(c, 2.0) * a) / pow(b, 3.0)))));
}

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 ** 4.0d0) * (a ** 3.0d0)) * ((-1.0546875d0) / (b ** 7.0d0))) + (((-0.5625d0) * (((c ** 3.0d0) * (a ** 2.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + ((-0.375d0) * (((c ** 2.0d0) * a) / (b ** 3.0d0)))))
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 ((Math.pow(c, 4.0) * Math.pow(a, 3.0)) * (-1.0546875 / Math.pow(b, 7.0))) + ((-0.5625 * ((Math.pow(c, 3.0) * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((Math.pow(c, 2.0) * a) / Math.pow(b, 3.0)))));
}

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 ((math.pow(c, 4.0) * math.pow(a, 3.0)) * (-1.0546875 / math.pow(b, 7.0))) + ((-0.5625 * ((math.pow(c, 3.0) * math.pow(a, 2.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((math.pow(c, 2.0) * a) / math.pow(b, 3.0)))))

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 ^ 4.0) * (a ^ 3.0)) * Float64(-1.0546875 / (b ^ 7.0))) + Float64(Float64(-0.5625 * Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64((c ^ 2.0) * a) / (b ^ 3.0))))))
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 ^ 4.0) * (a ^ 3.0)) * (-1.0546875 / (b ^ 7.0))) + ((-0.5625 * (((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + (-0.375 * (((c ^ 2.0) * a) / (b ^ 3.0)))));
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[(N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] * N[(-1.0546875 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5625 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(N[Power[c, 2.0], $MachinePrecision] * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

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

   [Start] 43.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   rational_best-simplify-3 [=>] 43.9	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}}{3 \cdot a} \]
   rational_best-simplify-9 [<=] 43.9	\[ \frac{\color{blue}{\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - 0\right)} + \left(-b\right)}{3 \cdot a} \]
   rational_best-simplify-56 [=>] 43.9	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - \left(0 + b\right)}}{3 \cdot a} \]
   rational_best-simplify-6 [=>] 43.9	\[ \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - \color{blue}{b}}{3 \cdot a} \]

3. Taylor expanded in a around 0 2.9

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

4. Taylor expanded in c around 0 2.9

   \[\leadsto \color{blue}{-1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]

5. Simplified 2.9

   \[\leadsto \color{blue}{\left({c}^{4} \cdot {a}^{3}\right) \cdot \frac{-1.0546875}{{b}^{7}}} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
   Proof

   [Start] 2.9	\[ -1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
   rational_best-simplify-55 [=>] 2.9	\[ \color{blue}{\left({c}^{4} \cdot {a}^{3}\right) \cdot \frac{-1.0546875}{{b}^{7}}} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]

6. Final simplification 2.9

   \[\leadsto \left({c}^{4} \cdot {a}^{3}\right) \cdot \frac{-1.0546875}{{b}^{7}} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]

Alternatives

Alternative 1
Error	3.1
Cost	47360

\[\left(c \cdot \frac{-0.5}{b} + \left(a \cdot {c}^{2}\right) \cdot \frac{-0.375}{{b}^{3}}\right) + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left({\left(c \cdot a\right)}^{4} \cdot 6.328125\right) \cdot \frac{\frac{-0.16666666666666666}{a}}{{b}^{7}}\right) \]

Alternative 2
Error	3.3
Cost	41152

\[\frac{0.3333333333333333}{a} \cdot \left(-1.5 \cdot \frac{c \cdot a}{b} + \left({\left(c \cdot a\right)}^{3} \cdot \frac{-1.6875}{{b}^{5}} + \left(-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + \left({\left(c \cdot a\right)}^{4} \cdot 6.328125\right) \cdot \frac{-0.5}{{b}^{7}}\right)\right)\right) \]

Alternative 3
Error	3.2
Cost	41152

\[\frac{\left(\left(c \cdot a\right) \cdot \frac{-1.5}{b} + \left(-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + {\left(c \cdot a\right)}^{3} \cdot \frac{-1.6875}{{b}^{5}}\right)\right) + \left({\left(c \cdot a\right)}^{4} \cdot 6.328125\right) \cdot \frac{-0.5}{{b}^{7}}}{3 \cdot a} \]

Alternative 4
Error	4.0
Cost	33664

\[\left(c \cdot \frac{-0.5}{b} + \left(a \cdot {c}^{2}\right) \cdot \frac{-0.375}{{b}^{3}}\right) + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} \]

Alternative 5
Error	4.3
Cost	27584

\[\frac{0.3333333333333333}{a} \cdot \left({\left(c \cdot a\right)}^{2} \cdot \frac{-1.125}{{b}^{3}} + \left(-1.5 \cdot \frac{c \cdot a}{b} + {\left(c \cdot a\right)}^{3} \cdot \frac{-1.6875}{{b}^{5}}\right)\right) \]

Alternative 6
Error	4.2
Cost	27584

\[\frac{-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + \left(\left(c \cdot a\right) \cdot \frac{-1.5}{b} + {\left(c \cdot a\right)}^{3} \cdot \frac{-1.6875}{{b}^{5}}\right)}{3 \cdot a} \]

Alternative 7
Error	10.8
Cost	15172

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{0.3333333333333333 \cdot \left(b - \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -3}\right)}{-a}}{1.5} \cdot 1.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8
Error	10.8
Cost	14852

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{b \cdot b + \frac{c \cdot a}{-0.3333333333333333}} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9
Error	5.9
Cost	14208

\[\left(\frac{-\frac{c}{b}}{4} + \frac{{c}^{2}}{{b}^{3}} \cdot \left(-0.375 \cdot a\right)\right) - \frac{\frac{c}{b}}{4} \]

Alternative 10
Error	5.9
Cost	13760

\[-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} \]

Alternative 11
Error	11.2
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 0.0017:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{b \cdot b + \frac{a \cdot c}{-0.3333333333333333}} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12
Error	11.2
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 0.00166:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \frac{a \cdot c}{-0.3333333333333333}} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 13
Error	11.9
Cost	320

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

Alternative 14
Error	62.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023099 
(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)))