Cubic critical

Average Error: 34.0 → 10.0

Time: 21.4s

Precision: binary64

Cost: 7688

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{+87}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

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 -3.9e+87)
   (+ (* (/ b a) -0.6666666666666666) (* 0.5 (/ c b)))
   (if (<= b 2.1e-56)
     (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 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 <= -3.9e+87) {
		tmp = ((b / a) * -0.6666666666666666) + (0.5 * (c / b));
	} else if (b <= 2.1e-56) {
		tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (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 <= (-3.9d+87)) then
        tmp = ((b / a) * (-0.6666666666666666d0)) + (0.5d0 * (c / b))
    else if (b <= 2.1d-56) then
        tmp = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (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 <= -3.9e+87) {
		tmp = ((b / a) * -0.6666666666666666) + (0.5 * (c / b));
	} else if (b <= 2.1e-56) {
		tmp = (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (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 <= -3.9e+87:
		tmp = ((b / a) * -0.6666666666666666) + (0.5 * (c / b))
	elif b <= 2.1e-56:
		tmp = (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (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 <= -3.9e+87)
		tmp = Float64(Float64(Float64(b / a) * -0.6666666666666666) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 2.1e-56)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(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 <= -3.9e+87)
		tmp = ((b / a) * -0.6666666666666666) + (0.5 * (c / b));
	elseif (b <= 2.1e-56)
		tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (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, -3.9e+87], N[(N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.1e-56], 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], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.9000000000000002e87

   1. Initial program 45.2

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

   2. Simplified 45.2

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

      [Start] 45.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      rational.json-simplify-2 [=>] 45.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{c \cdot \left(3 \cdot a\right)}}}{3 \cdot a} \]
      rational.json-simplify-2 [=>] 45.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 3\right)}}}{3 \cdot a} \]
      rational.json-simplify-43 [=>] 45.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{a \cdot \left(3 \cdot c\right)}}}{3 \cdot a} \]

   3. Taylor expanded in b around -inf 4.3

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   4. Simplified 4.3

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}} \]
      Proof

      [Start] 4.3	\[ -0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b} \]
      rational.json-simplify-2 [=>] 4.3	\[ \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} + 0.5 \cdot \frac{c}{b} \]

   if -3.9000000000000002e87 < b < 2.10000000000000006e-56

   1. Initial program 13.7

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

   if 2.10000000000000006e-56 < b

   1. Initial program 54.4

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

   2. Simplified 54.4

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

      [Start] 54.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      rational.json-simplify-2 [=>] 54.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{c \cdot \left(3 \cdot a\right)}}}{3 \cdot a} \]
      rational.json-simplify-2 [=>] 54.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 3\right)}}}{3 \cdot a} \]
      rational.json-simplify-43 [=>] 54.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{a \cdot \left(3 \cdot c\right)}}}{3 \cdot a} \]

   3. Taylor expanded in b around inf 8.0

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

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.9 \cdot 10^{+87}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.1
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+89}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} + \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 2
Error	10.1
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} + \left(-b\right)}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3
Error	10.1
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-52}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4
Error	13.8
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-56}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{a \cdot \left(-3 \cdot c\right)} + \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5
Error	13.8
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(c \cdot a\right)} + \left(-b\right)}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6
Error	13.8
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{-3 \cdot \left(c \cdot a\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7
Error	13.7
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{-67}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-56}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{c \cdot \left(a \cdot -3\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8
Error	22.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-248}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(b \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9
Error	36.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 8.6 \cdot 10^{-249}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{-249}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11
Error	39.9
Cost	320

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

Reproduce

herbie shell --seed 2023064 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))