Cubic critical

Average Error: 34.1 → 10.7

Time: 26.4s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{b}{a \cdot -3} \cdot 2\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{b - \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

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 -3.1e+95)
   (* (/ b (* a -3.0)) 2.0)
   (if (<= b 2.9e-131)
     (/ (/ (- b (sqrt (- (* b b) (* (* a 3.0) c)))) -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 <= -3.1e+95) {
		tmp = (b / (a * -3.0)) * 2.0;
	} else if (b <= 2.9e-131) {
		tmp = ((b - sqrt(((b * b) - ((a * 3.0) * c)))) / -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 <= (-3.1d+95)) then
        tmp = (b / (a * (-3.0d0))) * 2.0d0
    else if (b <= 2.9d-131) then
        tmp = ((b - sqrt(((b * b) - ((a * 3.0d0) * c)))) / (-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 <= -3.1e+95) {
		tmp = (b / (a * -3.0)) * 2.0;
	} else if (b <= 2.9e-131) {
		tmp = ((b - Math.sqrt(((b * b) - ((a * 3.0) * c)))) / -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 <= -3.1e+95:
		tmp = (b / (a * -3.0)) * 2.0
	elif b <= 2.9e-131:
		tmp = ((b - math.sqrt(((b * b) - ((a * 3.0) * c)))) / -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 <= -3.1e+95)
		tmp = Float64(Float64(b / Float64(a * -3.0)) * 2.0);
	elseif (b <= 2.9e-131)
		tmp = Float64(Float64(Float64(b - sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c)))) / -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 <= -3.1e+95)
		tmp = (b / (a * -3.0)) * 2.0;
	elseif (b <= 2.9e-131)
		tmp = ((b - sqrt(((b * b) - ((a * 3.0) * c)))) / -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, -3.1e+95], N[(N[(b / N[(a * -3.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[b, 2.9e-131], N[(N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.1000000000000003e95

   1. Initial program 47.5

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

   2. Taylor expanded in b around -inf 4.0

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

   3. Simplified 4.0

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

      [Start] 4.0	\[ \frac{-2 \cdot b}{3 \cdot a} \]
      *-commutative [=>] 4.0	\[ \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   4. Applied egg-rr 4.0

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

   if -3.1000000000000003e95 < b < 2.9000000000000002e-131

   1. Initial program 11.4

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

   2. Applied egg-rr 11.5

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

   3. Applied egg-rr 11.4

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

   if 2.9000000000000002e-131 < b

   1. Initial program 50.6

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

   2. Simplified 50.7

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

      [Start] 50.6	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      remove-double-neg [<=] 50.6	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
      sub-neg [<=] 50.6	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      div-sub [=>] 51.2	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      neg-mul-1 [=>] 51.2	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-*l/ [<=] 51.8	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      distribute-frac-neg [=>] 51.8	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      fma-neg [=>] 54.0	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
      /-rgt-identity [<=] 54.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      metadata-eval [<=] 54.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      associate-/l* [<=] 54.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      *-commutative [<=] 54.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      neg-mul-1 [<=] 54.0	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      fma-neg [<=] 51.8	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      neg-mul-1 [=>] 51.8	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   3. Taylor expanded in b around inf 12.4

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

   4. Applied egg-rr 12.4

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

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{b}{a \cdot -3} \cdot 2\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{b - \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.7
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+95}:\\ \;\;\;\;\frac{b}{a \cdot -3} \cdot 2\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-131}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2
Error	14.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{-43}:\\ \;\;\;\;\frac{b}{a \cdot -3} \cdot 2\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-131}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{\left(a \cdot -3\right) \cdot c} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3
Error	14.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{b}{a \cdot -3} \cdot 2\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-131}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{-3 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4
Error	14.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{b}{a \cdot -3} \cdot 2\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5
Error	14.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-43}:\\ \;\;\;\;\frac{b}{a \cdot -3} \cdot 2\\ \mathbf{elif}\;b \leq 1.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{b - \sqrt{-3 \cdot \left(a \cdot c\right)}}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6
Error	14.3
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-43}:\\ \;\;\;\;\frac{b}{a \cdot -3} \cdot 2\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{b - \sqrt{-3 \cdot \left(a \cdot c\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7
Error	22.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-265}:\\ \;\;\;\;\frac{b}{a \cdot -3} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 8
Error	36.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{-280}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{-271}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10
Error	22.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-241}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-273}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-263}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 13
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{-253}:\\ \;\;\;\;\frac{b}{\frac{a}{-0.6666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 14
Error	39.9
Cost	320

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

Reproduce

herbie shell --seed 2023011 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))