Cubic critical

Average Accuracy: 47.1% → 83.6%

Time: 24.2s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

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 -8.8e+43)
   (/ -0.6666666666666666 (/ a b))
   (if (<= b 1.9e-71)
     (/ (- (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 <= -8.8e+43) {
		tmp = -0.6666666666666666 / (a / b);
	} else if (b <= 1.9e-71) {
		tmp = (sqrt(((b * b) + (c * (a * -3.0)))) - 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 <= (-8.8d+43)) then
        tmp = (-0.6666666666666666d0) / (a / b)
    else if (b <= 1.9d-71) then
        tmp = (sqrt(((b * b) + (c * (a * (-3.0d0))))) - 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 <= -8.8e+43) {
		tmp = -0.6666666666666666 / (a / b);
	} else if (b <= 1.9e-71) {
		tmp = (Math.sqrt(((b * b) + (c * (a * -3.0)))) - 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 <= -8.8e+43:
		tmp = -0.6666666666666666 / (a / b)
	elif b <= 1.9e-71:
		tmp = (math.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 <= -8.8e+43)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	elseif (b <= 1.9e-71)
		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

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 <= -8.8e+43)
		tmp = -0.6666666666666666 / (a / b);
	elseif (b <= 1.9e-71)
		tmp = (sqrt(((b * b) + (c * (a * -3.0)))) - 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, -8.8e+43], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.9e-71], 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 < -8.80000000000000002e43

   1. Initial program 43.4%

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

   2. Applied egg-rr 43.4%

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

      [Start] 43.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-/r* [=>] 43.5	\[ \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}} \]
      div-inv [=>] 43.4	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3} \cdot \frac{1}{a}} \]

   3. Simplified 43.5%

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

      [Start] 43.4	\[ \frac{b - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}{-3} \cdot \frac{1}{a} \]
      *-commutative [<=] 43.4	\[ \color{blue}{\frac{1}{a} \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}{-3}} \]
      associate-*l/ [=>] 43.5	\[ \color{blue}{\frac{1 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}{-3}}{a}} \]
      *-lft-identity [=>] 43.5	\[ \frac{\color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(a \cdot c\right)\right)}}{-3}}}{a} \]
      fma-def [<=] 43.5	\[ \frac{\frac{b - \sqrt{\color{blue}{b \cdot b + -3 \cdot \left(a \cdot c\right)}}}{-3}}{a} \]
      +-commutative [=>] 43.5	\[ \frac{\frac{b - \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right) + b \cdot b}}}{-3}}{a} \]
      fma-def [=>] 43.5	\[ \frac{\frac{b - \sqrt{\color{blue}{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}}}{-3}}{a} \]
      *-commutative [=>] 43.5	\[ \frac{\frac{b - \sqrt{\mathsf{fma}\left(-3, \color{blue}{c \cdot a}, b \cdot b\right)}}{-3}}{a} \]

   4. Taylor expanded in b around -inf 90.1%

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

   5. Simplified 90.1%

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

      [Start] 90.1	\[ \frac{\frac{2 \cdot b}{-3}}{a} \]
      *-commutative [=>] 90.1	\[ \frac{\frac{\color{blue}{b \cdot 2}}{-3}}{a} \]

   6. Applied egg-rr 40.7%

      \[\leadsto \color{blue}{e^{\log \left(-0.6666666666666666 \cdot \frac{b}{a}\right)}} \]
      Proof

      [Start] 90.1	\[ \frac{\frac{b \cdot 2}{-3}}{a} \]
      add-exp-log [=>] 40.7	\[ \color{blue}{e^{\log \left(\frac{\frac{b \cdot 2}{-3}}{a}\right)}} \]
      associate-/l/ [=>] 40.7	\[ e^{\log \color{blue}{\left(\frac{b \cdot 2}{a \cdot -3}\right)}} \]
      times-frac [=>] 40.7	\[ e^{\log \color{blue}{\left(\frac{b}{a} \cdot \frac{2}{-3}\right)}} \]
      metadata-eval [=>] 40.7	\[ e^{\log \left(\frac{b}{a} \cdot \color{blue}{-0.6666666666666666}\right)} \]
      metadata-eval [<=] 40.7	\[ e^{\log \left(\frac{b}{a} \cdot \color{blue}{\left(2 \cdot -0.3333333333333333\right)}\right)} \]
      metadata-eval [<=] 40.7	\[ e^{\log \left(\frac{b}{a} \cdot \left(2 \cdot \color{blue}{\frac{1}{-3}}\right)\right)} \]
      *-commutative [=>] 40.7	\[ e^{\log \color{blue}{\left(\left(2 \cdot \frac{1}{-3}\right) \cdot \frac{b}{a}\right)}} \]
      metadata-eval [=>] 40.7	\[ e^{\log \left(\left(2 \cdot \color{blue}{-0.3333333333333333}\right) \cdot \frac{b}{a}\right)} \]
      metadata-eval [=>] 40.7	\[ e^{\log \left(\color{blue}{-0.6666666666666666} \cdot \frac{b}{a}\right)} \]

   7. Applied egg-rr 90.0%

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

      [Start] 40.7	\[ e^{\log \left(-0.6666666666666666 \cdot \frac{b}{a}\right)} \]
      add-exp-log [<=] 90.0	\[ \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
      associate-*r/ [=>] 90.1	\[ \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]
      associate-/l* [=>] 90.0	\[ \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} \]

   if -8.80000000000000002e43 < b < 1.89999999999999996e-71

   1. Initial program 77.5%

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

   if 1.89999999999999996e-71 < b

   1. Initial program 17.2%

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

   2. Simplified 17.2%

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

      [Start] 17.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      remove-double-neg [<=] 17.2	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
      sub-neg [<=] 17.2	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      div-sub [=>] 16.4	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      neg-mul-1 [=>] 16.4	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-*l/ [<=] 15.2	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      distribute-frac-neg [=>] 15.2	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      fma-neg [=>] 11.1	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
      /-rgt-identity [<=] 11.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      metadata-eval [<=] 11.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      associate-/l* [<=] 11.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      *-commutative [<=] 11.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      neg-mul-1 [<=] 11.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      fma-neg [<=] 15.2	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      neg-mul-1 [=>] 15.2	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   3. Taylor expanded in b around inf 86.5%

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

   4. Applied egg-rr 86.5%

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

      [Start] 86.5	\[ -0.5 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 86.5	\[ \color{blue}{\frac{-0.5 \cdot c}{b}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	83.7%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+54}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{elif}\;b \leq 10^{-73}:\\ \;\;\;\;\frac{\frac{b - \sqrt{b \cdot b + -3 \cdot \left(a \cdot c\right)}}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2
Accuracy	78.8%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-65}:\\ \;\;\;\;\left(-1.5 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-64}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3
Accuracy	78.7%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-65}:\\ \;\;\;\;\left(-1.5 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-59}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4
Accuracy	79.0%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-66}:\\ \;\;\;\;\left(-1.5 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5
Accuracy	78.8%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-65}:\\ \;\;\;\;\left(-1.5 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6
Accuracy	64.4%
Cost	964

\[\begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-294}:\\ \;\;\;\;\left(-1.5 \cdot \frac{c}{b} + 2 \cdot \frac{b}{a}\right) \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7
Accuracy	64.4%
Cost	836

\[\begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-294}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 8
Accuracy	64.5%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 10^{-281}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 9
Accuracy	42.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-284}:\\ \;\;\;\;\frac{b}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10
Accuracy	64.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{-281}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11
Accuracy	64.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-288}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 12
Accuracy	37.7%
Cost	320

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

Alternative 13
Accuracy	12.3%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023137 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))