Cubic critical, medium range

Average Error: 43.8 → 3.0

Time: 17.1s

Precision: binary64

Cost: 47296

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

Math

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

↓

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

FPCore

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

↓

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

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

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

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

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

Derivation

1. Initial program 43.8

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

2. Applied egg-rr 43.8

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

3. Taylor expanded in b around inf 3.0

   \[\leadsto -\color{blue}{\left(0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(0.5 \cdot \frac{c}{b} + \left(0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + 0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)\right)} \]

4. Simplified 3.0

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

   [Start] 3.0	\[ -\left(0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(0.5 \cdot \frac{c}{b} + \left(0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + 0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)\right) \]
   rational.json-simplify-41 [=>] 3.0	\[ -\color{blue}{\left(0.5 \cdot \frac{c}{b} + \left(\left(0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + 0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) + 0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]
   rational.json-simplify-1 [=>] 3.0	\[ -\left(0.5 \cdot \frac{c}{b} + \color{blue}{\left(0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + 0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)}\right) \]
   rational.json-simplify-1 [=>] 3.0	\[ -\left(0.5 \cdot \frac{c}{b} + \left(0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \color{blue}{\left(0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + 0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)}\right)\right) \]
   rational.json-simplify-41 [=>] 3.0	\[ -\left(0.5 \cdot \frac{c}{b} + \color{blue}{\left(0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + 0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)}\right) \]

5. Applied egg-rr 3.0

   \[\leadsto -\left(0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot 0.375\right) + \left({a}^{2} \cdot \left(0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + \color{blue}{\left(\frac{{\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}} \cdot 1.0546875 + 0\right)}\right)\right)\right) \]

6. Simplified 3.0

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

   [Start] 3.0	\[ -\left(0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot 0.375\right) + \left({a}^{2} \cdot \left(0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + \left(\frac{{\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}} \cdot 1.0546875 + 0\right)\right)\right)\right) \]
   rational.json-simplify-4 [=>] 3.0	\[ -\left(0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot 0.375\right) + \left({a}^{2} \cdot \left(0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + \color{blue}{\frac{{\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}} \cdot 1.0546875}\right)\right)\right) \]
   rational.json-simplify-2 [=>] 3.0	\[ -\left(0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot 0.375\right) + \left({a}^{2} \cdot \left(0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + \color{blue}{1.0546875 \cdot \frac{{\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}}\right)\right)\right) \]

7. Final simplification 3.0

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

Alternatives

Alternative 1
Error	3.1
Cost	41088

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

Alternative 2
Error	3.3
Cost	41024

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

Alternative 3
Error	3.3
Cost	41024

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

Alternative 4
Error	4.1
Cost	33796

\[\begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{3 \cdot a} \cdot \left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\\ \end{array} \]

Alternative 5
Error	4.1
Cost	33796

\[\begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{3 \cdot a} \cdot \left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + {a}^{2} \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\\ \end{array} \]

Alternative 6
Error	4.2
Cost	27652

\[\begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{3 \cdot a} \cdot \left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{a \cdot \left(0.5 \cdot \frac{c}{b}\right) + \left(\frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} \cdot 0.375 + \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}} \cdot 0.5625\right)}{a}\\ \end{array} \]

Alternative 7
Error	4.3
Cost	27588

\[\begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{3 \cdot a} \cdot \left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \frac{c \cdot a}{b} + \left(-0.375 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)}{a}\\ \end{array} \]

Alternative 8
Error	4.3
Cost	27588

\[\begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{3 \cdot a} \cdot \left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(\frac{a}{b} \cdot -0.5\right) + \left(-0.375 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)}{a}\\ \end{array} \]

Alternative 9
Error	10.6
Cost	15044

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -7.5 \cdot 10^{-10}:\\ \;\;\;\;\left(\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{2}{a}\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10
Error	10.6
Cost	15044

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -7.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{\frac{a}{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11
Error	10.6
Cost	14916

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -7.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12
Error	5.8
Cost	13892

\[\begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{3 \cdot a} \cdot \left(-\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right)\\ \end{array} \]

Alternative 13
Error	11.2
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \leq 0.000425:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 14
Error	12.0
Cost	320

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

Reproduce

herbie shell --seed 2023068 
(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)))