Cubic critical

Average Error: 33.9 → 11.4

Time: 19.4s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{1}{a} \cdot \left(b \cdot -0.6666666666666666\right)\right)\\ \mathbf{elif}\;b \leq 1850000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

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 -3.7e+32)
   (fma 0.5 (/ c b) (* (/ 1.0 a) (* b -0.6666666666666666)))
   (if (<= b 1850000.0)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (/ (* 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 <= -3.7e+32) {
		tmp = fma(0.5, (c / b), ((1.0 / a) * (b * -0.6666666666666666)));
	} else if (b <= 1850000.0) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} 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 <= -3.7e+32)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(1.0 / a) * Float64(b * -0.6666666666666666)));
	elseif (b <= 1850000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return 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, -3.7e+32], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(1.0 / a), $MachinePrecision] * N[(b * -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1850000.0], 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[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7e32

   1. Initial program 35.2

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

   2. Simplified 35.2

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

      [Start] 35.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      *-lft-identity [<=] 35.2	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      metadata-eval [<=] 35.2	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      times-frac [<=] 35.2	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
      *-commutative [<=] 35.2	\[ \frac{\color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}}{-1 \cdot \left(3 \cdot a\right)} \]
      times-frac [=>] 35.3	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \frac{-1}{3 \cdot a}} \]
      associate-*r/ [=>] 35.2	\[ \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot -1}{3 \cdot a}} \]

   3. Taylor expanded in b around -inf 7.5

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

   4. Simplified 7.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{-0.6666666666666666}{\frac{a}{b}}\right)} \]
      Proof

      [Start] 7.5	\[ -0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b} \]
      +-commutative [=>] 7.5	\[ \color{blue}{0.5 \cdot \frac{c}{b} + -0.6666666666666666 \cdot \frac{b}{a}} \]
      fma-def [=>] 7.5	\[ \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, -0.6666666666666666 \cdot \frac{b}{a}\right)} \]
      associate-*r/ [=>] 7.4	\[ \mathsf{fma}\left(0.5, \frac{c}{b}, \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}}\right) \]
      associate-/l* [=>] 7.5	\[ \mathsf{fma}\left(0.5, \frac{c}{b}, \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}}\right) \]

   5. Applied egg-rr 7.5

      \[\leadsto \mathsf{fma}\left(0.5, \frac{c}{b}, \color{blue}{\frac{1}{a} \cdot \left(-0.6666666666666666 \cdot b\right)}\right) \]

   if -3.7e32 < b < 1.85e6

   1. Initial program 17.5

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

   if 1.85e6 < b

   1. Initial program 55.6

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

   2. Simplified 55.6

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

      [Start] 55.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      *-lft-identity [<=] 55.6	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      metadata-eval [<=] 55.6	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      times-frac [<=] 55.6	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
      *-commutative [<=] 55.6	\[ \frac{\color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}}{-1 \cdot \left(3 \cdot a\right)} \]
      times-frac [=>] 55.6	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \frac{-1}{3 \cdot a}} \]
      associate-*r/ [=>] 55.6	\[ \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot -1}{3 \cdot a}} \]

   3. Taylor expanded in b around inf 5.5

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

   4. Simplified 5.5

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

      [Start] 5.5	\[ -0.5 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 5.5	\[ \color{blue}{\frac{-0.5 \cdot c}{b}} \]
      *-commutative [=>] 5.5	\[ \frac{\color{blue}{c \cdot -0.5}}{b} \]

3. Recombined 3 regimes into one program.

4. Final simplification 11.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{1}{a} \cdot \left(b \cdot -0.6666666666666666\right)\right)\\ \mathbf{elif}\;b \leq 1850000:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.4
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{1}{a} \cdot \left(b \cdot -0.6666666666666666\right)\right)\\ \mathbf{elif}\;b \leq 1850000:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2
Error	14.4
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4.05 \cdot 10^{-75}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 0.09:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3
Error	14.4
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-75}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 0.09:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4
Error	14.4
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.38 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{1}{a} \cdot \left(b \cdot -0.6666666666666666\right)\right)\\ \mathbf{elif}\;b \leq 0.145:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5
Error	23.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-178}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]

Alternative 6
Error	22.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-178}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]

Alternative 7
Error	22.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-178}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]

Alternative 8
Error	22.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2.3 \cdot 10^{-178}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 9
Error	45.5
Cost	320

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

Reproduce

herbie shell --seed 2023025 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))