Cubic critical

Average Error: 34.3 → 10.4

Time: 20.9s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+135}:\\ \;\;\;\;\frac{1}{-1.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

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 -1.02e+135)
   (/ 1.0 (* -1.5 (/ a b)))
   (if (<= b 8.5e-51)
     (/ (- (sqrt (+ (* b b) (* c (* a -3.0)))) b) (* a 3.0))
     (* -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 <= -1.02e+135) {
		tmp = 1.0 / (-1.5 * (a / b));
	} else if (b <= 8.5e-51) {
		tmp = (sqrt(((b * b) + (c * (a * -3.0)))) - b) / (a * 3.0);
	} 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 <= (-1.02d+135)) then
        tmp = 1.0d0 / ((-1.5d0) * (a / b))
    else if (b <= 8.5d-51) then
        tmp = (sqrt(((b * b) + (c * (a * (-3.0d0))))) - b) / (a * 3.0d0)
    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 <= -1.02e+135) {
		tmp = 1.0 / (-1.5 * (a / b));
	} else if (b <= 8.5e-51) {
		tmp = (Math.sqrt(((b * b) + (c * (a * -3.0)))) - b) / (a * 3.0);
	} 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 <= -1.02e+135:
		tmp = 1.0 / (-1.5 * (a / b))
	elif b <= 8.5e-51:
		tmp = (math.sqrt(((b * b) + (c * (a * -3.0)))) - b) / (a * 3.0)
	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 <= -1.02e+135)
		tmp = Float64(1.0 / Float64(-1.5 * Float64(a / b)));
	elseif (b <= 8.5e-51)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	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 <= -1.02e+135)
		tmp = 1.0 / (-1.5 * (a / b));
	elseif (b <= 8.5e-51)
		tmp = (sqrt(((b * b) + (c * (a * -3.0)))) - b) / (a * 3.0);
	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, -1.02e+135], N[(1.0 / N[(-1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.5e-51], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.01999999999999993e135

   1. Initial program 57.5

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

   2. Taylor expanded in b around -inf 4.1

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

   3. Applied egg-rr 4.1

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666 \cdot b}}} \]

   4. Taylor expanded in a around 0 4.1

      \[\leadsto \frac{1}{\color{blue}{-1.5 \cdot \frac{a}{b}}} \]

   if -1.01999999999999993e135 < b < 8.50000000000000036e-51

   1. Initial program 13.8

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

   if 8.50000000000000036e-51 < b

   1. Initial program 53.7

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

   2. Applied egg-rr 53.8

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(-1, a \cdot \left(3 \cdot c\right), b \cdot b\right)}}}{3 \cdot a} \]

   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 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+135}:\\ \;\;\;\;\frac{1}{-1.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.4
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{+135}:\\ \;\;\;\;\frac{1}{-1.5 \cdot \frac{a}{b}}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-51}:\\ \;\;\;\;\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 2
Error	13.9
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7.9 \cdot 10^{-22}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{0.5}{\frac{b}{c}}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3
Error	13.9
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7.9 \cdot 10^{-22}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{0.5}{\frac{b}{c}}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4
Error	23.0
Cost	836

\[\begin{array}{l} \mathbf{if}\;b \leq -2.02 \cdot 10^{-302}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{0.5}{\frac{b}{c}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5
Error	23.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-1.5}{b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6
Error	23.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{-1.5 \cdot \frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7
Error	40.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+45}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b}\\ \end{array} \]

Alternative 8
Error	23.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-235}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9
Error	59.3
Cost	320

\[\frac{b}{\frac{a}{-0.3333333333333333}} \]

Alternative 10
Error	45.7
Cost	320

\[b \cdot \frac{-0.6666666666666666}{a} \]

Reproduce

herbie shell --seed 2022294 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))