Cubic critical, medium range

Average Error: 44.2 → 2.8

Time: 18.5s

Precision: binary64

Cost: 73600

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{{c}^{2}}{{b}^{3}}\\ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \left(a \cdot t_0\right) + -0.5625 \cdot \left({a}^{2} \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right) + -0.16666666666666666 \cdot \left(\left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + 1.265625 \cdot {t_0}^{2}\right) \cdot \frac{{a}^{3}}{b}\right) \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (pow c 2.0) (pow b 3.0))))
   (+
    (+
     (* -0.5 (/ c b))
     (+
      (* -0.375 (* a t_0))
      (* -0.5625 (* (pow a 2.0) (/ (pow c 3.0) (pow b 5.0))))))
    (*
     -0.16666666666666666
     (*
      (+ (* 5.0625 (/ (pow c 4.0) (pow b 6.0))) (* 1.265625 (pow t_0 2.0)))
      (/ (pow a 3.0) b))))))

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) {
	double t_0 = pow(c, 2.0) / pow(b, 3.0);
	return ((-0.5 * (c / b)) + ((-0.375 * (a * t_0)) + (-0.5625 * (pow(a, 2.0) * (pow(c, 3.0) / pow(b, 5.0)))))) + (-0.16666666666666666 * (((5.0625 * (pow(c, 4.0) / pow(b, 6.0))) + (1.265625 * pow(t_0, 2.0))) * (pow(a, 3.0) / b)));
}

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
    real(8) :: t_0
    t_0 = (c ** 2.0d0) / (b ** 3.0d0)
    code = (((-0.5d0) * (c / b)) + (((-0.375d0) * (a * t_0)) + ((-0.5625d0) * ((a ** 2.0d0) * ((c ** 3.0d0) / (b ** 5.0d0)))))) + ((-0.16666666666666666d0) * (((5.0625d0 * ((c ** 4.0d0) / (b ** 6.0d0))) + (1.265625d0 * (t_0 ** 2.0d0))) * ((a ** 3.0d0) / b)))
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) {
	double t_0 = Math.pow(c, 2.0) / Math.pow(b, 3.0);
	return ((-0.5 * (c / b)) + ((-0.375 * (a * t_0)) + (-0.5625 * (Math.pow(a, 2.0) * (Math.pow(c, 3.0) / Math.pow(b, 5.0)))))) + (-0.16666666666666666 * (((5.0625 * (Math.pow(c, 4.0) / Math.pow(b, 6.0))) + (1.265625 * Math.pow(t_0, 2.0))) * (Math.pow(a, 3.0) / b)));
}

Python

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

↓

def code(a, b, c):
	t_0 = math.pow(c, 2.0) / math.pow(b, 3.0)
	return ((-0.5 * (c / b)) + ((-0.375 * (a * t_0)) + (-0.5625 * (math.pow(a, 2.0) * (math.pow(c, 3.0) / math.pow(b, 5.0)))))) + (-0.16666666666666666 * (((5.0625 * (math.pow(c, 4.0) / math.pow(b, 6.0))) + (1.265625 * math.pow(t_0, 2.0))) * (math.pow(a, 3.0) / b)))

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)
	t_0 = Float64((c ^ 2.0) / (b ^ 3.0))
	return Float64(Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(a * t_0)) + Float64(-0.5625 * Float64((a ^ 2.0) * Float64((c ^ 3.0) / (b ^ 5.0)))))) + Float64(-0.16666666666666666 * Float64(Float64(Float64(5.0625 * Float64((c ^ 4.0) / (b ^ 6.0))) + Float64(1.265625 * (t_0 ^ 2.0))) * Float64((a ^ 3.0) / b))))
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)
	t_0 = (c ^ 2.0) / (b ^ 3.0);
	tmp = ((-0.5 * (c / b)) + ((-0.375 * (a * t_0)) + (-0.5625 * ((a ^ 2.0) * ((c ^ 3.0) / (b ^ 5.0)))))) + (-0.16666666666666666 * (((5.0625 * ((c ^ 4.0) / (b ^ 6.0))) + (1.265625 * (t_0 ^ 2.0))) * ((a ^ 3.0) / b)));
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_] := Block[{t$95$0 = N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(5.0625 * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.265625 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[a, 3.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 44.2

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

2. Simplified 44.2

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

   [Start] 44.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   rational.json-simplify-2 [=>] 44.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{c \cdot \left(3 \cdot a\right)}}}{3 \cdot a} \]
   rational.json-simplify-2 [=>] 44.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 3\right)}}}{3 \cdot a} \]
   rational.json-simplify-43 [=>] 44.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{a \cdot \left(3 \cdot c\right)}}}{3 \cdot a} \]

3. Taylor expanded in a around 0 2.8

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

4. Simplified 2.8

   \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \left(a \cdot \frac{{c}^{2}}{{b}^{3}}\right) + -0.5625 \cdot \left({a}^{2} \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right) + -0.16666666666666666 \cdot \left(\left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + 1.265625 \cdot {\left(\frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right) \cdot \frac{{a}^{3}}{b}\right)} \]
   Proof

   [Start] 2.8	\[ -0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
   rational.json-simplify-1 [=>] 2.8	\[ \color{blue}{\left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) + -0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b}} \]
   rational.json-simplify-41 [=>] 2.8	\[ \color{blue}{\left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} + -0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} \]
   rational.json-simplify-49 [=>] 2.8	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{3}}\right)} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} \]
   rational.json-simplify-49 [=>] 2.8	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \left(a \cdot \frac{{c}^{2}}{{b}^{3}}\right) + -0.5625 \cdot \color{blue}{\left({a}^{2} \cdot \frac{{c}^{3}}{{b}^{5}}\right)}\right)\right) + -0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} \]
   rational.json-simplify-49 [=>] 2.8	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \left(a \cdot \frac{{c}^{2}}{{b}^{3}}\right) + -0.5625 \cdot \left({a}^{2} \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right) + -0.16666666666666666 \cdot \color{blue}{\left(\left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right) \cdot \frac{{a}^{3}}{b}\right)} \]

5. Final simplification 2.8

   \[\leadsto \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \left(a \cdot \frac{{c}^{2}}{{b}^{3}}\right) + -0.5625 \cdot \left({a}^{2} \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\right) + -0.16666666666666666 \cdot \left(\left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + 1.265625 \cdot {\left(\frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right) \cdot \frac{{a}^{3}}{b}\right) \]

Alternatives

Alternative 1
Error	2.8
Cost	47360

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

Alternative 2
Error	3.0
Cost	41088

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

Alternative 3
Error	2.9
Cost	41088

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

Alternative 4
Error	3.2
Cost	41024

\[\left(-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + \left(\frac{c}{b} \cdot \left(a \cdot -1.5\right) + \left(-1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}} + \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot -3.1640625\right)\right)\right) \cdot \frac{0.3333333333333333}{a} \]

Alternative 5
Error	3.1
Cost	41024

\[\frac{-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + \left(a \cdot \left(-1.5 \cdot \frac{c}{b}\right) + \left(-1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}} + \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot -3.1640625\right)\right)}{a} \cdot 0.3333333333333333 \]

Alternative 6
Error	3.8
Cost	33664

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

Alternative 7
Error	3.7
Cost	33664

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

Alternative 8
Error	4.0
Cost	27520

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

Alternative 9
Error	3.8
Cost	27520

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

Alternative 10
Error	10.7
Cost	15044

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -100:\\ \;\;\;\;\left(\left(\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)}\right) \cdot \frac{2}{a}\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11
Error	10.7
Cost	14916

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -100:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12
Error	5.7
Cost	13760

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

Alternative 13
Error	11.4
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \leq 0.52:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 14
Error	11.8
Cost	320

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

Reproduce

herbie shell --seed 2023074 
(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)))