Cubic critical, wide range

Average Error: 52.7 → 1.5

Time: 17.2s

Precision: binary64

Cost: 47360

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

↓

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

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))
   (+
    (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
    (* -0.5625 (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0)))))
  (*
   -0.16666666666666666
   (/ (* (pow (* c a) 4.0) 6.328125) (* 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)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.5625 * ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0))))) + (-0.16666666666666666 * ((pow((c * a), 4.0) * 6.328125) / (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)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-0.5625d0) * (((c ** 3.0d0) * (a ** 2.0d0)) / (b ** 5.0d0))))) + ((-0.16666666666666666d0) * ((((c * a) ** 4.0d0) * 6.328125d0) / (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)) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-0.5625 * ((Math.pow(c, 3.0) * Math.pow(a, 2.0)) / Math.pow(b, 5.0))))) + (-0.16666666666666666 * ((Math.pow((c * a), 4.0) * 6.328125) / (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)) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-0.5625 * ((math.pow(c, 3.0) * math.pow(a, 2.0)) / math.pow(b, 5.0))))) + (-0.16666666666666666 * ((math.pow((c * a), 4.0) * 6.328125) / (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(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.5625 * Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(c * a) ^ 4.0) * 6.328125) / 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)) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (-0.5625 * (((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))))) + (-0.16666666666666666 * ((((c * a) ^ 4.0) * 6.328125) / (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[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7

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

2. Taylor expanded in b around inf 1.5

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

3. Simplified 1.5

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

   [Start] 1.5	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
   rational_best-simplify-1 [=>] 1.5	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \color{blue}{\left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)} \]
   rational_best-simplify-43 [=>] 1.5	\[ \color{blue}{-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)} \]
   rational_best-simplify-1 [<=] 1.5	\[ -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \color{blue}{\left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]
   rational_best-simplify-1 [=>] 1.5	\[ \color{blue}{\left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) + -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}} \]

4. Taylor expanded in c around 0 1.5

   \[\leadsto \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + \color{blue}{-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}} \]

5. Simplified 1.5

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

   [Start] 1.5	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} \]
   rational_best-simplify-47 [<=] 1.5	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.16666666666666666 \cdot \frac{\color{blue}{{c}^{4} \cdot \left(5.0625 \cdot {a}^{4}\right) + {c}^{4} \cdot \left(1.265625 \cdot {a}^{4}\right)}}{a \cdot {b}^{7}} \]
   rational_best-simplify-44 [<=] 1.5	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.16666666666666666 \cdot \frac{\color{blue}{5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)} + {c}^{4} \cdot \left(1.265625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} \]
   rational_best-simplify-44 [<=] 1.5	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right) + \color{blue}{1.265625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}}{a \cdot {b}^{7}} \]
   rational_best-simplify-2 [=>] 1.5	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.16666666666666666 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot 5.0625} + 1.265625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} \]
   exponential-simplify-27 [=>] 1.5	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.16666666666666666 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot 5.0625 + 1.265625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} \]
   rational_best-simplify-2 [<=] 1.5	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.16666666666666666 \cdot \frac{{\color{blue}{\left(c \cdot a\right)}}^{4} \cdot 5.0625 + 1.265625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} \]
   rational_best-simplify-2 [=>] 1.5	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.16666666666666666 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 5.0625 + \color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot 1.265625}}{a \cdot {b}^{7}} \]
   exponential-simplify-27 [=>] 1.5	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.16666666666666666 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 5.0625 + \color{blue}{{\left(a \cdot c\right)}^{4}} \cdot 1.265625}{a \cdot {b}^{7}} \]
   rational_best-simplify-2 [<=] 1.5	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.16666666666666666 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 5.0625 + {\color{blue}{\left(c \cdot a\right)}}^{4} \cdot 1.265625}{a \cdot {b}^{7}} \]
   rational_best-simplify-47 [=>] 1.5	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.16666666666666666 \cdot \frac{\color{blue}{{\left(c \cdot a\right)}^{4} \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}} \]
   metadata-eval [=>] 1.5	\[ \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.16666666666666666 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot \color{blue}{6.328125}}{a \cdot {b}^{7}} \]

6. Final simplification 1.5

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

Alternatives

Alternative 1
Error	1.9
Cost	41152

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

Alternative 2
Error	1.9
Cost	41152

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

Alternative 3
Error	2.0
Cost	33664

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

Alternative 4
Error	2.4
Cost	27584

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

Alternative 5
Error	5.7
Cost	14916

\[\begin{array}{l} t_0 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6
Error	3.0
Cost	13760

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

Alternative 7
Error	6.0
Cost	320

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

Reproduce

herbie shell --seed 2023092 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))