Cubic critical

Average Accuracy: 46.0% → 84.4%

Time: 24.3s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

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 -1e+154)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 2.65e-68)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (/ (* c -0.5) 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 <= -1e+154) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2.65e-68) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / 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 <= (-1d+154)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 2.65d-68) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else
        tmp = (c * (-0.5d0)) / 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 <= -1e+154) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2.65e-68) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / 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 <= -1e+154:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 2.65e-68:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / 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 <= -1e+154)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 2.65e-68)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / 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 <= -1e+154)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 2.65e-68)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / 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, -1e+154], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.65e-68], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.00000000000000004e154

   1. Initial program 0.1%

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

   2. Taylor expanded in b around -inf 95.6%

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

   3. Simplified 95.6%

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

      [Start] 95.6	\[ \frac{-2 \cdot b}{3 \cdot a} \]
      *-commutative [=>] 95.6	\[ \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   if -1.00000000000000004e154 < b < 2.65e-68

   1. Initial program 80.1%

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

   if 2.65e-68 < b

   1. Initial program 16.6%

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

   2. Simplified 16.6%

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

      [Start] 16.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      /-rgt-identity [<=] 16.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\frac{3 \cdot a}{1}}} \]
      metadata-eval [<=] 16.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\frac{3 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [<=] 16.6	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{3 \cdot a}} \]
      associate-*r/ [<=] 16.6	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{3 \cdot a}} \]
      *-commutative [=>] 16.6	\[ \color{blue}{\frac{--1}{3 \cdot a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)} \]
      associate-*l/ [=>] 16.6	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{3 \cdot a}} \]
      associate-*r/ [<=] 16.6	\[ \color{blue}{\left(--1\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      metadata-eval [=>] 16.6	\[ \color{blue}{1} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      metadata-eval [<=] 16.6	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      times-frac [<=] 16.6	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
      neg-mul-1 [<=] 16.6	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
      distribute-rgt-neg-in [=>] 16.6	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
      times-frac [=>] 16.6	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
      metadata-eval [=>] 16.6	\[ \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]
      neg-mul-1 [=>] 16.6	\[ -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Taylor expanded in b around inf 86.6%

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

   4. Simplified 86.6%

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

      [Start] 86.6	\[ -0.5 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 86.6	\[ \color{blue}{\frac{-0.5 \cdot c}{b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	83.9%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{+85}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-68}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2
Accuracy	84.3%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+153}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3
Accuracy	78.9%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 3.15 \cdot 10^{-68}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4
Accuracy	78.9%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -8.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-68}:\\ \;\;\;\;\left(\sqrt{\left(a \cdot c\right) \cdot -3} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5
Accuracy	78.9%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6
Accuracy	64.7%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-169}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7
Accuracy	16.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8
Accuracy	37.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+14}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9
Accuracy	64.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-169}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]

Alternative 10
Accuracy	64.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-169}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 11
Accuracy	7.3%
Cost	320

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

Reproduce

herbie shell --seed 2023146 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))