Cubic critical

Average Error: 34.3 → 10.2

Time: 25.2s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

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 -9.5e+95)
   (/ (fma 1.5 (/ c (/ b a)) (* b -2.0)) (* a 3.0))
   (if (<= b 1.55e-75)
     (/ (- (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 <= -9.5e+95) {
		tmp = fma(1.5, (c / (b / a)), (b * -2.0)) / (a * 3.0);
	} else if (b <= 1.55e-75) {
		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 <= -9.5e+95)
		tmp = Float64(fma(1.5, Float64(c / Float64(b / a)), Float64(b * -2.0)) / Float64(a * 3.0));
	elseif (b <= 1.55e-75)
		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, -9.5e+95], N[(N[(1.5 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.55e-75], 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 < -9.5000000000000004e95

   1. Initial program 47.1

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

   2. Taylor expanded in b around -inf 11.0

      \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{c \cdot a}{b} + -2 \cdot b}}{3 \cdot a} \]

   3. Simplified 4.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}}{3 \cdot a} \]
      Proof

      [Start] 11.0	\[ \frac{1.5 \cdot \frac{c \cdot a}{b} + -2 \cdot b}{3 \cdot a} \]
      fma-def [=>] 11.0	\[ \frac{\color{blue}{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, -2 \cdot b\right)}}{3 \cdot a} \]
      associate-/l* [=>] 4.3	\[ \frac{\mathsf{fma}\left(1.5, \color{blue}{\frac{c}{\frac{b}{a}}}, -2 \cdot b\right)}{3 \cdot a} \]
      *-commutative [=>] 4.3	\[ \frac{\mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, \color{blue}{b \cdot -2}\right)}{3 \cdot a} \]

   if -9.5000000000000004e95 < b < 1.55000000000000003e-75

   1. Initial program 13.2

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

   if 1.55000000000000003e-75 < b

   1. Initial program 52.9

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

   2. Simplified 52.9

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

      [Start] 52.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      *-lft-identity [<=] 52.9	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      metadata-eval [<=] 52.9	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      times-frac [<=] 52.9	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
      *-commutative [<=] 52.9	\[ \frac{\color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}}{-1 \cdot \left(3 \cdot a\right)} \]
      times-frac [=>] 52.9	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \frac{-1}{3 \cdot a}} \]
      associate-*r/ [=>] 52.9	\[ \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot -1}{3 \cdot a}} \]

   3. Taylor expanded in b around inf 9.2

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

   4. Simplified 9.2

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

      [Start] 9.2	\[ -0.5 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 9.2	\[ \color{blue}{\frac{-0.5 \cdot c}{b}} \]
      *-commutative [=>] 9.2	\[ \frac{\color{blue}{c \cdot -0.5}}{b} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2
Error	13.5
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-63}:\\ \;\;\;\;\left(-1.5 \cdot \frac{c}{b} + \frac{b}{a} \cdot 2\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-75}:\\ \;\;\;\;\left(\sqrt{-3 \cdot \left(c \cdot a\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3
Error	13.6
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-61}:\\ \;\;\;\;\left(-1.5 \cdot \frac{c}{b} + \frac{b}{a} \cdot 2\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-75}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4
Error	13.5
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-66}:\\ \;\;\;\;\left(-1.5 \cdot \frac{c}{b} + \frac{b}{a} \cdot 2\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5
Error	13.5
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-65}:\\ \;\;\;\;\left(-1.5 \cdot \frac{c}{b} + \frac{b}{a} \cdot 2\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6
Error	22.7
Cost	964

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-1.5 \cdot \frac{c}{b} + \frac{b}{a} \cdot 2\right) \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7
Error	22.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 8
Error	36.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-286}:\\ \;\;\;\;\frac{b}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]

Alternative 9
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{-284}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]

Alternative 10
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-286}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]

Alternative 11
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-289}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]

Alternative 12
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-283}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 13
Error	59.3
Cost	320

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

Reproduce

herbie shell --seed 2023032 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))