Cubic critical

Average Error: 34.1 → 10.2

Time: 10.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -7.863505383668454 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{3}}{a}\\ \mathbf{elif}\;b \leq 8.84535440633062 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.863505383668454e+146)
   (/ (/ (* b -2.0) 3.0) a)
   (if (<= b 8.84535440633062e-68)
     (/ (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) 3.0) a)
     (* -0.5 (/ c 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 tmp;
	if (b <= -7.863505383668454e+146) {
		tmp = ((b * -2.0) / 3.0) / a;
	} else if (b <= 8.84535440633062e-68) {
		tmp = ((sqrt(((b * b) - ((3.0 * a) * c))) - b) / 3.0) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

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) :: tmp
    if (b <= (-7.863505383668454d+146)) then
        tmp = ((b * (-2.0d0)) / 3.0d0) / a
    else if (b <= 8.84535440633062d-68) then
        tmp = ((sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / 3.0d0) / a
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
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 tmp;
	if (b <= -7.863505383668454e+146) {
		tmp = ((b * -2.0) / 3.0) / a;
	} else if (b <= 8.84535440633062e-68) {
		tmp = ((Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / 3.0) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

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

↓

def code(a, b, c):
	tmp = 0
	if b <= -7.863505383668454e+146:
		tmp = ((b * -2.0) / 3.0) / a
	elif b <= 8.84535440633062e-68:
		tmp = ((math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / 3.0) / a
	else:
		tmp = -0.5 * (c / b)
	return tmp

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)
	tmp = 0.0
	if (b <= -7.863505383668454e+146)
		tmp = Float64(Float64(Float64(b * -2.0) / 3.0) / a);
	elseif (b <= 8.84535440633062e-68)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / 3.0) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

↓

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.863505383668454e+146)
		tmp = ((b * -2.0) / 3.0) / a;
	elseif (b <= 8.84535440633062e-68)
		tmp = ((sqrt(((b * b) - ((3.0 * a) * c))) - b) / 3.0) / a;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
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_] := If[LessEqual[b, -7.863505383668454e+146], N[(N[(N[(b * -2.0), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 8.84535440633062e-68], N[(N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -7.8635053836684542e146

   1. Initial program 61.2

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

   2. Applied associate-/r*_binary64 61.2

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

   3. Simplified 37.2

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

   4. Taylor expanded in b around -inf 2.9

      \[\leadsto \frac{\frac{\color{blue}{-2 \cdot b}}{3}}{a} \]

   if -7.8635053836684542e146 < b < 8.8453544063306207e-68

   1. Initial program 13.1

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

   2. Applied associate-/r*_binary64 13.2

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

   if 8.8453544063306207e-68 < b

   1. Initial program 53.2

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

   2. Taylor expanded in b around inf 8.5

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.863505383668454 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{3}}{a}\\ \mathbf{elif}\;b \leq 8.84535440633062 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022138 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))