Cubic critical, narrow range

Average Error: 28.3 → 6.0

Time: 43.1s

Precision: binary64

Cost: 47360

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0)))
  (+
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))
   (*
    -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 (-0.5625 * ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0))) + (((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)))) + (-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 = ((-0.5625d0) * (((c ** 3.0d0) * (a ** 2.0d0)) / (b ** 5.0d0))) + ((((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))) + ((-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 (-0.5625 * ((Math.pow(c, 3.0) * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) + (((-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)))) + (-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 (-0.5625 * ((math.pow(c, 3.0) * math.pow(a, 2.0)) / math.pow(b, 5.0))) + (((-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))) + (-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(-0.5625 * Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))) + Float64(Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))) + 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 = (-0.5625 * (((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))) + (((-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)))) + (-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[(-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[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $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]), $MachinePrecision]

Derivation

1. Initial program 28.3

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

2. Taylor expanded in b around inf 6.0

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

3. Simplified 6.0

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

   [Start] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
   rational_best.json-simplify-1 [=>] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \color{blue}{\left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)} \]
   rational_best.json-simplify-2 [=>] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) \]

4. Taylor expanded in c around 0 6.0

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

5. Simplified 6.0

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

   [Start] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{1.265625 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 5.0625 \cdot {\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}\right) \]
   exponential.json-simplify-27 [=>] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{1.265625 \cdot \color{blue}{{\left(a \cdot c\right)}^{4}} + 5.0625 \cdot {\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}\right) \]
   rational_best.json-simplify-2 [<=] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{1.265625 \cdot {\color{blue}{\left(c \cdot a\right)}}^{4} + 5.0625 \cdot {\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}\right) \]

6. Taylor expanded in c around 0 6.0

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

7. Simplified 6.0

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

   [Start] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) \]
   rational_best.json-simplify-47 [<=] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{\color{blue}{{c}^{4} \cdot \left(5.0625 \cdot {a}^{4}\right) + {c}^{4} \cdot \left(1.265625 \cdot {a}^{4}\right)}}{a \cdot {b}^{7}}\right) \]
   rational_best.json-simplify-44 [<=] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{\color{blue}{5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)} + {c}^{4} \cdot \left(1.265625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) \]
   rational_best.json-simplify-44 [<=] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right) + \color{blue}{1.265625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}}{a \cdot {b}^{7}}\right) \]
   rational_best.json-simplify-2 [=>] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot 5.0625} + 1.265625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) \]
   rational_best.json-simplify-2 [=>] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{\left({c}^{4} \cdot {a}^{4}\right) \cdot 5.0625 + \color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot 1.265625}}{a \cdot {b}^{7}}\right) \]
   rational_best.json-simplify-47 [=>] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}}\right) \]
   exponential.json-simplify-27 [=>] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot \left(1.265625 + 5.0625\right)}{a \cdot {b}^{7}}\right) \]
   rational_best.json-simplify-2 [<=] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{{\color{blue}{\left(c \cdot a\right)}}^{4} \cdot \left(1.265625 + 5.0625\right)}{a \cdot {b}^{7}}\right) \]
   metadata-eval [=>] 6.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot \color{blue}{6.328125}}{a \cdot {b}^{7}}\right) \]

8. Final simplification 6.0

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

Alternatives

Alternative 1
Error	7.1
Cost	41156

\[\begin{array}{l} t_0 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -0.032:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\\ \end{array} \]

Alternative 2
Error	7.1
Cost	41156

\[\begin{array}{l} t_0 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -0.032:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \left(-0.5 \cdot \frac{c}{b} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\\ \end{array} \]

Alternative 3
Error	7.3
Cost	35076

\[\begin{array}{l} t_0 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -0.032:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{c \cdot a}{b} + \left(-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + -1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)}{3 \cdot a}\\ \end{array} \]

Alternative 4
Error	7.3
Cost	35076

\[\begin{array}{l} t_0 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -0.032:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + \left(-1.5 \cdot \frac{c \cdot a}{b} + -1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)}{3 \cdot a}\\ \end{array} \]

Alternative 5
Error	9.6
Cost	21252

\[\begin{array}{l} t_0 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -0.0036:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\\ \end{array} \]

Alternative 6
Error	15.1
Cost	14916

\[\begin{array}{l} t_0 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7
Error	23.0
Cost	320

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

Reproduce

herbie shell --seed 2023088 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))