Cubic critical, medium range

Average Error: 43.8 → 3.1

Time: 4.8s

Precision: binary64

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

Math

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

↓

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

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 3.0) (pow a 2.0)) (pow b 5.0)) -0.5625)
  (+
   (* (/ (* (pow c 4.0) (pow a 3.0)) (pow b 7.0)) -1.0546875)
   (- (* (/ c b) -0.5) (* 0.375 (/ (* a (pow c 2.0)) (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, 3.0) * pow(a, 2.0)) / pow(b, 5.0)) * -0.5625) + ((((pow(c, 4.0) * pow(a, 3.0)) / pow(b, 7.0)) * -1.0546875) + (((c / b) * -0.5) - (0.375 * ((a * pow(c, 2.0)) / 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 ** 3.0d0) * (a ** 2.0d0)) / (b ** 5.0d0)) * (-0.5625d0)) + (((((c ** 4.0d0) * (a ** 3.0d0)) / (b ** 7.0d0)) * (-1.0546875d0)) + (((c / b) * (-0.5d0)) - (0.375d0 * ((a * (c ** 2.0d0)) / (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, 3.0) * Math.pow(a, 2.0)) / Math.pow(b, 5.0)) * -0.5625) + ((((Math.pow(c, 4.0) * Math.pow(a, 3.0)) / Math.pow(b, 7.0)) * -1.0546875) + (((c / b) * -0.5) - (0.375 * ((a * Math.pow(c, 2.0)) / 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, 3.0) * math.pow(a, 2.0)) / math.pow(b, 5.0)) * -0.5625) + ((((math.pow(c, 4.0) * math.pow(a, 3.0)) / math.pow(b, 7.0)) * -1.0546875) + (((c / b) * -0.5) - (0.375 * ((a * math.pow(c, 2.0)) / 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(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0)) * -0.5625) + Float64(Float64(Float64(Float64((c ^ 4.0) * (a ^ 3.0)) / (b ^ 7.0)) * -1.0546875) + Float64(Float64(Float64(c / b) * -0.5) - Float64(0.375 * Float64(Float64(a * (c ^ 2.0)) / (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 ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0)) * -0.5625) + (((((c ^ 4.0) * (a ^ 3.0)) / (b ^ 7.0)) * -1.0546875) + (((c / b) * -0.5) - (0.375 * ((a * (c ^ 2.0)) / (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[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * -0.5625), $MachinePrecision] + N[(N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * -1.0546875), $MachinePrecision] + N[(N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision] - N[(0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus a

Bits error versus b

Bits error versus c

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. Simplified 43.8

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

3. Taylor expanded in b around inf 3.1

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

4. Final simplification 3.1

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

Reproduce

herbie shell --seed 2022148 
(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)))