Cubic critical

Percentage Accurate: 51.9% → 84.7%

Time: 11.6s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

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 -9.2e+95)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 1.12e-11)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) 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 <= -9.2e+95) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.12e-11) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} 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 <= (-9.2d+95)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 1.12d-11) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    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 <= -9.2e+95) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.12e-11) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} 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 <= -9.2e+95:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 1.12e-11:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - 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 <= -9.2e+95)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 1.12e-11)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	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 <= -9.2e+95)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 1.12e-11)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	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, -9.2e+95], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.12e-11], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $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 < -9.19999999999999989e95

   1. Initial program 45.0%

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

   2. Simplified 45.0%

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

      [Start] 45.0%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      /-rgt-identity [<=] 45.0%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\frac{3 \cdot a}{1}}} \]
      metadata-eval [<=] 45.0%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\frac{3 \cdot a}{\color{blue}{--1}}} \]
      associate-/r/ [=>] 45.0%	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \left(--1\right)} \]
      metadata-eval [=>] 45.0%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{1} \]
      metadata-eval [<=] 45.0%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{\frac{-1}{-1}} \]
      times-frac [<=] 45.0%	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\left(3 \cdot a\right) \cdot -1}} \]
      *-commutative [<=] 45.0%	\[ \frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\color{blue}{-1 \cdot \left(3 \cdot a\right)}} \]
      times-frac [=>] 45.0%	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \frac{-1}{3 \cdot a}} \]
      associate-/r* [=>] 45.0%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \color{blue}{\frac{\frac{-1}{3}}{a}} \]

   3. Applied egg-rr 47.7%

      \[\leadsto \color{blue}{\frac{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot -0.3333333333333333}{a}} \]
      Step-by-step derivation

      [Start] 45.0%	\[ \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a} \]
      associate-*r/ [=>] 45.0%	\[ \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right) \cdot -0.3333333333333333}{a}} \]
      fma-udef [=>] 45.0%	\[ \frac{\left(b - \sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -3}}\right) \cdot -0.3333333333333333}{a} \]
      associate-*r* [<=] 45.0%	\[ \frac{\left(b - \sqrt{b \cdot b + \color{blue}{a \cdot \left(c \cdot -3\right)}}\right) \cdot -0.3333333333333333}{a} \]
      add-sqr-sqrt [=>] 25.4%	\[ \frac{\left(b - \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -3\right)} \cdot \sqrt{a \cdot \left(c \cdot -3\right)}}}\right) \cdot -0.3333333333333333}{a} \]
      hypot-def [=>] 47.7%	\[ \frac{\left(b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}\right) \cdot -0.3333333333333333}{a} \]

   4. Taylor expanded in b around -inf 93.9%

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

   5. Simplified 93.9%

      \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]
      Step-by-step derivation

      [Start] 93.9%	\[ \frac{-0.6666666666666666 \cdot b}{a} \]
      *-commutative [=>] 93.9%	\[ \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   if -9.19999999999999989e95 < b < 1.1200000000000001e-11

   1. Initial program 78.3%

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

   if 1.1200000000000001e-11 < b

   1. Initial program 16.9%

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

   2. Simplified 16.9%

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

      [Start] 16.9%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      /-rgt-identity [<=] 16.9%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\frac{3 \cdot a}{1}}} \]
      metadata-eval [<=] 16.9%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\frac{3 \cdot a}{\color{blue}{--1}}} \]
      associate-/r/ [=>] 16.9%	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \left(--1\right)} \]
      metadata-eval [=>] 16.9%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{1} \]
      metadata-eval [<=] 16.9%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{\frac{-1}{-1}} \]
      times-frac [<=] 16.9%	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\left(3 \cdot a\right) \cdot -1}} \]
      *-commutative [<=] 16.9%	\[ \frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\color{blue}{-1 \cdot \left(3 \cdot a\right)}} \]
      times-frac [=>] 16.9%	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \frac{-1}{3 \cdot a}} \]
      associate-/r* [=>] 16.9%	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \color{blue}{\frac{\frac{-1}{3}}{a}} \]

   3. Applied egg-rr 33.1%

      \[\leadsto \color{blue}{\frac{\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot -0.3333333333333333}{a}} \]
      Step-by-step derivation

      [Start] 16.9%	\[ \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a} \]
      associate-*r/ [=>] 16.9%	\[ \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right) \cdot -0.3333333333333333}{a}} \]
      fma-udef [=>] 16.9%	\[ \frac{\left(b - \sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -3}}\right) \cdot -0.3333333333333333}{a} \]
      associate-*r* [<=] 16.9%	\[ \frac{\left(b - \sqrt{b \cdot b + \color{blue}{a \cdot \left(c \cdot -3\right)}}\right) \cdot -0.3333333333333333}{a} \]
      add-sqr-sqrt [=>] 11.4%	\[ \frac{\left(b - \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -3\right)} \cdot \sqrt{a \cdot \left(c \cdot -3\right)}}}\right) \cdot -0.3333333333333333}{a} \]
      hypot-def [=>] 33.1%	\[ \frac{\left(b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}\right) \cdot -0.3333333333333333}{a} \]

   4. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]

   5. Simplified 91.9%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
      Step-by-step derivation

      [Start] 0.0%	\[ 0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b} \]
      associate-*r/ [=>] 0.0%	\[ \color{blue}{\frac{0.16666666666666666 \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}{b}} \]
      *-commutative [=>] 0.0%	\[ \frac{0.16666666666666666 \cdot \color{blue}{\left({\left(\sqrt{-3}\right)}^{2} \cdot c\right)}}{b} \]
      associate-*r* [=>] 0.0%	\[ \frac{\color{blue}{\left(0.16666666666666666 \cdot {\left(\sqrt{-3}\right)}^{2}\right) \cdot c}}{b} \]
      unpow2 [=>] 0.0%	\[ \frac{\left(0.16666666666666666 \cdot \color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)}\right) \cdot c}{b} \]
      rem-square-sqrt [=>] 91.9%	\[ \frac{\left(0.16666666666666666 \cdot \color{blue}{-3}\right) \cdot c}{b} \]
      metadata-eval [=>] 91.9%	\[ \frac{\color{blue}{-0.5} \cdot c}{b} \]
      *-commutative [=>] 91.9%	\[ \frac{\color{blue}{c \cdot -0.5}}{b} \]

3. Recombined 3 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.7%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2
Accuracy	84.7%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+95}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3
Accuracy	79.8%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4
Accuracy	79.8%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5
Accuracy	68.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2.9 \cdot 10^{-308}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6
Accuracy	68.1%
Cost	452

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

Alternative 7
Accuracy	68.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 8
Accuracy	34.7%
Cost	320

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

Reproduce

herbie shell --seed 2023167 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))