Cubic critical

Percentage Accurate: 53.0% → 84.4%

Time: 19.1s

Alternatives: 16

Speedup: 11.6×

Specification

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

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

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
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]

Initial Program: 53.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

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

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
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]

Alternative 1: 84.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{+161}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1100 \lor \neg \left(b \leq 15000000\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}{t\_0} - \frac{{b}^{2}}{t\_0}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ b (hypot b (sqrt (* c (* a -3.0)))))))
   (if (<= b -1.6e+161)
     (/ (* b -2.0) (* 3.0 a))
     (if (<= b 8.6e-130)
       (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
       (if (or (<= b 1100.0) (not (<= b 15000000.0)))
         (/ (* c -0.5) b)
         (/
          (- (/ (fma c (* a -3.0) (pow b 2.0)) t_0) (/ (pow b 2.0) t_0))
          (* 3.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = b + hypot(b, sqrt((c * (a * -3.0))));
	double tmp;
	if (b <= -1.6e+161) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 8.6e-130) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else if ((b <= 1100.0) || !(b <= 15000000.0)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = ((fma(c, (a * -3.0), pow(b, 2.0)) / t_0) - (pow(b, 2.0) / t_0)) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b + hypot(b, sqrt(Float64(c * Float64(a * -3.0)))))
	tmp = 0.0
	if (b <= -1.6e+161)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 8.6e-130)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	elseif ((b <= 1100.0) || !(b <= 15000000.0))
		tmp = Float64(Float64(c * -0.5) / b);
	else
		tmp = Float64(Float64(Float64(fma(c, Float64(a * -3.0), (b ^ 2.0)) / t_0) - Float64((b ^ 2.0) / t_0)) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.6e+161], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-130], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 1100.0], N[Not[LessEqual[b, 15000000.0]], $MachinePrecision]], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision], N[(N[(N[(N[(c * N[(a * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(N[Power[b, 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.60000000000000001e161

   1. Initial program 23.9%

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

   3. Taylor expanded in b around -inf 95.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative 95.3%

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

   5. Simplified 95.3%

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

   if -1.60000000000000001e161 < b < 8.60000000000000058e-130

   1. Initial program 86.3%

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

   if 8.60000000000000058e-130 < b < 1100 or 1.5e7 < b

   1. Initial program 18.6%

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

   3. Taylor expanded in b around inf 85.7%

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

      1. associate-*r/ 85.7%

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

   5. Simplified 85.7%

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

   if 1100 < b < 1.5e7

   1. Initial program 99.4%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)} - \frac{{b}^{2}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \frac{\frac{\color{blue}{b \cdot b + c \cdot \left(a \cdot -3\right)}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)} - \frac{{b}^{2}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)}}{3 \cdot a} \]

      2. unpow2 100.0%

         \[\leadsto \frac{\frac{\color{blue}{{b}^{2}} + c \cdot \left(a \cdot -3\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)} - \frac{{b}^{2}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)}}{3 \cdot a} \]

      3. +-commutative 100.0%

         \[\leadsto \frac{\frac{\color{blue}{c \cdot \left(a \cdot -3\right) + {b}^{2}}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)} - \frac{{b}^{2}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)}}{3 \cdot a} \]

      4. fma-def 100.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)} - \frac{{b}^{2}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)}}{3 \cdot a} \]

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)} - \frac{{b}^{2}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)}}}{3 \cdot a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+161}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1100 \lor \neg \left(b \leq 15000000\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)} - \frac{{b}^{2}}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)}}{3 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-130} \lor \neg \left(b \leq 1100\right) \land b \leq 11000000:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.2e-117)
   (+ (/ -0.6666666666666666 (/ a b)) (* 0.5 (/ c b)))
   (if (or (<= b 8.6e-130) (and (not (<= b 1100.0)) (<= b 11000000.0)))
     (* 0.3333333333333333 (/ (+ b (sqrt (* c (* a -3.0)))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.2e-117) {
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	} else if ((b <= 8.6e-130) || (!(b <= 1100.0) && (b <= 11000000.0))) {
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -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
    real(8) :: tmp
    if (b <= (-6.2d-117)) then
        tmp = ((-0.6666666666666666d0) / (a / b)) + (0.5d0 * (c / b))
    else if ((b <= 8.6d-130) .or. (.not. (b <= 1100.0d0)) .and. (b <= 11000000.0d0)) then
        tmp = 0.3333333333333333d0 * ((b + sqrt((c * (a * (-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) {
	double tmp;
	if (b <= -6.2e-117) {
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	} else if ((b <= 8.6e-130) || (!(b <= 1100.0) && (b <= 11000000.0))) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.2e-117:
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b))
	elif (b <= 8.6e-130) or (not (b <= 1100.0) and (b <= 11000000.0)):
		tmp = 0.3333333333333333 * ((b + math.sqrt((c * (a * -3.0)))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.2e-117)
		tmp = Float64(Float64(-0.6666666666666666 / Float64(a / b)) + Float64(0.5 * Float64(c / b)));
	elseif ((b <= 8.6e-130) || (!(b <= 1100.0) && (b <= 11000000.0)))
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(c * Float64(a * -3.0)))) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.2e-117)
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	elseif ((b <= 8.6e-130) || (~((b <= 1100.0)) && (b <= 11000000.0)))
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -3.0)))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.2e-117], N[(N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 8.6e-130], And[N[Not[LessEqual[b, 1100.0]], $MachinePrecision], LessEqual[b, 11000000.0]]], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.20000000000000022e-117

   1. Initial program 64.1%

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

   3. Taylor expanded in b around -inf 87.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. clear-num 87.0%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} + 0.5 \cdot \frac{c}{b} \]

      2. un-div-inv 87.2%

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

   5. Applied egg-rr 87.2%

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

   if -6.20000000000000022e-117 < b < 8.60000000000000058e-130 or 1100 < b < 1.1e7

   1. Initial program 82.1%

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

   3. Taylor expanded in b around 0 81.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 81.5%

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

      2. times-frac 81.3%

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

      3. metadata-eval 81.3%

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

      4. add-sqr-sqrt 40.3%

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

      5. sqrt-unprod 80.6%

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

      6. sqr-neg 80.6%

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

      7. sqrt-unprod 41.0%

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

      8. add-sqr-sqrt 80.3%

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

      9. associate-*r* 80.3%

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

      10. *-commutative 80.3%

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

      11. *-commutative 80.3%

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

   5. Applied egg-rr 80.3%

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

   if 8.60000000000000058e-130 < b < 1100 or 1.1e7 < b

   1. Initial program 18.6%

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

   3. Taylor expanded in b around inf 85.7%

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

      1. associate-*r/ 85.7%

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

   5. Simplified 85.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-130} \lor \neg \left(b \leq 1100\right) \land b \leq 11000000:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b + \sqrt{c \cdot \left(a \cdot -3\right)}\\ \mathbf{if}\;b \leq -1.15 \cdot 10^{-110}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-130}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t\_0}{a}\\ \mathbf{elif}\;b \leq 1100 \lor \neg \left(b \leq 12000000\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{0.3333333333333333}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ b (sqrt (* c (* a -3.0))))))
   (if (<= b -1.15e-110)
     (+ (/ -0.6666666666666666 (/ a b)) (* 0.5 (/ c b)))
     (if (<= b 8.6e-130)
       (* 0.3333333333333333 (/ t_0 a))
       (if (or (<= b 1100.0) (not (<= b 12000000.0)))
         (/ (* c -0.5) b)
         (* t_0 (/ 0.3333333333333333 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = b + sqrt((c * (a * -3.0)));
	double tmp;
	if (b <= -1.15e-110) {
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	} else if (b <= 8.6e-130) {
		tmp = 0.3333333333333333 * (t_0 / a);
	} else if ((b <= 1100.0) || !(b <= 12000000.0)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = t_0 * (0.3333333333333333 / a);
	}
	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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b + sqrt((c * (a * (-3.0d0))))
    if (b <= (-1.15d-110)) then
        tmp = ((-0.6666666666666666d0) / (a / b)) + (0.5d0 * (c / b))
    else if (b <= 8.6d-130) then
        tmp = 0.3333333333333333d0 * (t_0 / a)
    else if ((b <= 1100.0d0) .or. (.not. (b <= 12000000.0d0))) then
        tmp = (c * (-0.5d0)) / b
    else
        tmp = t_0 * (0.3333333333333333d0 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b + Math.sqrt((c * (a * -3.0)));
	double tmp;
	if (b <= -1.15e-110) {
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	} else if (b <= 8.6e-130) {
		tmp = 0.3333333333333333 * (t_0 / a);
	} else if ((b <= 1100.0) || !(b <= 12000000.0)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = t_0 * (0.3333333333333333 / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b + math.sqrt((c * (a * -3.0)))
	tmp = 0
	if b <= -1.15e-110:
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b))
	elif b <= 8.6e-130:
		tmp = 0.3333333333333333 * (t_0 / a)
	elif (b <= 1100.0) or not (b <= 12000000.0):
		tmp = (c * -0.5) / b
	else:
		tmp = t_0 * (0.3333333333333333 / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b + sqrt(Float64(c * Float64(a * -3.0))))
	tmp = 0.0
	if (b <= -1.15e-110)
		tmp = Float64(Float64(-0.6666666666666666 / Float64(a / b)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 8.6e-130)
		tmp = Float64(0.3333333333333333 * Float64(t_0 / a));
	elseif ((b <= 1100.0) || !(b <= 12000000.0))
		tmp = Float64(Float64(c * -0.5) / b);
	else
		tmp = Float64(t_0 * Float64(0.3333333333333333 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b + sqrt((c * (a * -3.0)));
	tmp = 0.0;
	if (b <= -1.15e-110)
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	elseif (b <= 8.6e-130)
		tmp = 0.3333333333333333 * (t_0 / a);
	elseif ((b <= 1100.0) || ~((b <= 12000000.0)))
		tmp = (c * -0.5) / b;
	else
		tmp = t_0 * (0.3333333333333333 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.15e-110], N[(N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e-130], N[(0.3333333333333333 * N[(t$95$0 / a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 1100.0], N[Not[LessEqual[b, 12000000.0]], $MachinePrecision]], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision], N[(t$95$0 * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.1500000000000001e-110

   1. Initial program 64.1%

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

   3. Taylor expanded in b around -inf 87.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. clear-num 87.0%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} + 0.5 \cdot \frac{c}{b} \]

      2. un-div-inv 87.2%

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

   5. Applied egg-rr 87.2%

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

   if -1.1500000000000001e-110 < b < 8.60000000000000058e-130

   1. Initial program 80.4%

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

   3. Taylor expanded in b around 0 79.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 79.7%

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

      2. times-frac 79.6%

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

      3. metadata-eval 79.6%

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

      4. add-sqr-sqrt 44.3%

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

      5. sqrt-unprod 78.9%

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

      6. sqr-neg 78.9%

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

      7. sqrt-unprod 35.4%

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

      8. add-sqr-sqrt 78.5%

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

      9. associate-*r* 78.5%

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

      10. *-commutative 78.5%

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

      11. *-commutative 78.5%

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

   5. Applied egg-rr 78.5%

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

   if 8.60000000000000058e-130 < b < 1100 or 1.2e7 < b

   1. Initial program 18.6%

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

   3. Taylor expanded in b around inf 85.7%

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

      1. associate-*r/ 85.7%

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

   5. Simplified 85.7%

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

   if 1100 < b < 1.2e7

   1. Initial program 99.4%

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

   3. Taylor expanded in b around 0 99.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 59.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{-3 \cdot \left(a \cdot c\right)}}{3 \cdot a}\right)\right)} \]

      2. expm1-udef 2.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{-3 \cdot \left(a \cdot c\right)}}{3 \cdot a}\right)} - 1} \]

      3. div-inv 2.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\left(-b\right) + \sqrt{-3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{3 \cdot a}}\right)} - 1 \]

      4. add-sqr-sqrt 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{-3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      5. sqrt-unprod 2.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{-3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      6. sqr-neg 2.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\sqrt{\color{blue}{b \cdot b}} + \sqrt{-3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      7. sqrt-unprod 2.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{-3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      8. add-sqr-sqrt 2.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{b} + \sqrt{-3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      9. associate-*r* 2.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(b + \sqrt{\color{blue}{\left(-3 \cdot a\right) \cdot c}}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      10. *-commutative 2.5%

          \[\leadsto e^{\mathsf{log1p}\left(\left(b + \sqrt{\color{blue}{\left(a \cdot -3\right)} \cdot c}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      11. *-commutative 2.5%

          \[\leadsto e^{\mathsf{log1p}\left(\left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}\right) \cdot \frac{1}{3 \cdot a}\right)} - 1 \]

      12. associate-/r* 2.5%

          \[\leadsto e^{\mathsf{log1p}\left(\left(b + \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \color{blue}{\frac{\frac{1}{3}}{a}}\right)} - 1 \]

      13. metadata-eval 2.5%

          \[\leadsto e^{\mathsf{log1p}\left(\left(b + \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{\color{blue}{0.3333333333333333}}{a}\right)} - 1 \]

   5. Applied egg-rr 2.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(b + \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{0.3333333333333333}{a}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 59.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(b + \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{0.3333333333333333}{a}\right)\right)} \]

      2. expm1-log1p 98.8%

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

      3. *-commutative 98.8%

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

   7. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -3\right)}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-110}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-130}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{elif}\;b \leq 1100 \lor \neg \left(b \leq 12000000\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(b + \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{0.3333333333333333}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-96}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-132} \lor \neg \left(b \leq 600\right) \land b \leq 11000000:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.02e-96)
   (+ (/ -0.6666666666666666 (/ a b)) (* 0.5 (/ c b)))
   (if (or (<= b 4.1e-132) (and (not (<= b 600.0)) (<= b 11000000.0)))
     (/ (- (sqrt (* a (* c -3.0))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.02e-96) {
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	} else if ((b <= 4.1e-132) || (!(b <= 600.0) && (b <= 11000000.0))) {
		tmp = (sqrt((a * (c * -3.0))) - b) / (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
    real(8) :: tmp
    if (b <= (-1.02d-96)) then
        tmp = ((-0.6666666666666666d0) / (a / b)) + (0.5d0 * (c / b))
    else if ((b <= 4.1d-132) .or. (.not. (b <= 600.0d0)) .and. (b <= 11000000.0d0)) then
        tmp = (sqrt((a * (c * (-3.0d0)))) - b) / (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) {
	double tmp;
	if (b <= -1.02e-96) {
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	} else if ((b <= 4.1e-132) || (!(b <= 600.0) && (b <= 11000000.0))) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.02e-96:
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b))
	elif (b <= 4.1e-132) or (not (b <= 600.0) and (b <= 11000000.0)):
		tmp = (math.sqrt((a * (c * -3.0))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.02e-96)
		tmp = Float64(Float64(-0.6666666666666666 / Float64(a / b)) + Float64(0.5 * Float64(c / b)));
	elseif ((b <= 4.1e-132) || (!(b <= 600.0) && (b <= 11000000.0)))
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.02e-96)
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	elseif ((b <= 4.1e-132) || (~((b <= 600.0)) && (b <= 11000000.0)))
		tmp = (sqrt((a * (c * -3.0))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.02e-96], N[(N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 4.1e-132], And[N[Not[LessEqual[b, 600.0]], $MachinePrecision], LessEqual[b, 11000000.0]]], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.02000000000000007e-96

   1. Initial program 63.4%

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

   3. Taylor expanded in b around -inf 87.8%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. clear-num 87.8%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} + 0.5 \cdot \frac{c}{b} \]

      2. un-div-inv 87.9%

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

   5. Applied egg-rr 87.9%

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

   if -1.02000000000000007e-96 < b < 4.10000000000000007e-132 or 600 < b < 1.1e7

   1. Initial program 82.7%

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

   3. Taylor expanded in b around 0 80.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. +-commutative 80.7%

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

      2. unsub-neg 80.7%

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

      3. associate-*r* 80.7%

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

      4. *-commutative 80.7%

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

      5. *-commutative 80.7%

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

   5. Applied egg-rr 80.7%

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

      1. *-commutative 80.7%

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

      2. associate-*l* 80.7%

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

      3. *-commutative 80.7%

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

   7. Simplified 80.7%

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

   if 4.10000000000000007e-132 < b < 600 or 1.1e7 < b

   1. Initial program 18.6%

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

   3. Taylor expanded in b around inf 85.7%

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

      1. associate-*r/ 85.7%

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

   5. Simplified 85.7%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-96}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-132} \lor \neg \left(b \leq 600\right) \land b \leq 11000000:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-96}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-137} \lor \neg \left(b \leq 1100\right) \land b \leq 40000000:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.02e-96)
   (+ (/ -0.6666666666666666 (/ a b)) (* 0.5 (/ c b)))
   (if (or (<= b 3e-137) (and (not (<= b 1100.0)) (<= b 40000000.0)))
     (/ (- (sqrt (* -3.0 (* a c))) b) (* 3.0 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.02e-96) {
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	} else if ((b <= 3e-137) || (!(b <= 1100.0) && (b <= 40000000.0))) {
		tmp = (sqrt((-3.0 * (a * c))) - b) / (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
    real(8) :: tmp
    if (b <= (-1.02d-96)) then
        tmp = ((-0.6666666666666666d0) / (a / b)) + (0.5d0 * (c / b))
    else if ((b <= 3d-137) .or. (.not. (b <= 1100.0d0)) .and. (b <= 40000000.0d0)) then
        tmp = (sqrt(((-3.0d0) * (a * c))) - b) / (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) {
	double tmp;
	if (b <= -1.02e-96) {
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	} else if ((b <= 3e-137) || (!(b <= 1100.0) && (b <= 40000000.0))) {
		tmp = (Math.sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.02e-96:
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b))
	elif (b <= 3e-137) or (not (b <= 1100.0) and (b <= 40000000.0)):
		tmp = (math.sqrt((-3.0 * (a * c))) - b) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.02e-96)
		tmp = Float64(Float64(-0.6666666666666666 / Float64(a / b)) + Float64(0.5 * Float64(c / b)));
	elseif ((b <= 3e-137) || (!(b <= 1100.0) && (b <= 40000000.0)))
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.02e-96)
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	elseif ((b <= 3e-137) || (~((b <= 1100.0)) && (b <= 40000000.0)))
		tmp = (sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.02e-96], N[(N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 3e-137], And[N[Not[LessEqual[b, 1100.0]], $MachinePrecision], LessEqual[b, 40000000.0]]], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.02000000000000007e-96

   1. Initial program 63.4%

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

   3. Taylor expanded in b around -inf 87.8%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. clear-num 87.8%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} + 0.5 \cdot \frac{c}{b} \]

      2. un-div-inv 87.9%

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

   5. Applied egg-rr 87.9%

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

   if -1.02000000000000007e-96 < b < 2.9999999999999998e-137 or 1100 < b < 4e7

   1. Initial program 82.7%

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

   3. Taylor expanded in b around 0 80.7%

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

   if 2.9999999999999998e-137 < b < 1100 or 4e7 < b

   1. Initial program 18.6%

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

   3. Taylor expanded in b around inf 85.7%

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

      1. associate-*r/ 85.7%

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

   5. Simplified 85.7%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-96}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-137} \lor \neg \left(b \leq 1100\right) \land b \leq 40000000:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-96}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1100 \lor \neg \left(b \leq 11000000\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.02e-96)
   (+ (/ -0.6666666666666666 (/ a b)) (* 0.5 (/ c b)))
   (if (<= b 1.6e-133)
     (/ (- (sqrt (* c (* a -3.0))) b) (* 3.0 a))
     (if (or (<= b 1100.0) (not (<= b 11000000.0)))
       (/ (* c -0.5) b)
       (/ (- (sqrt (* -3.0 (* a c))) b) (* 3.0 a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.02e-96) {
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	} else if (b <= 1.6e-133) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else if ((b <= 1100.0) || !(b <= 11000000.0)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = (sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	}
	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
    real(8) :: tmp
    if (b <= (-1.02d-96)) then
        tmp = ((-0.6666666666666666d0) / (a / b)) + (0.5d0 * (c / b))
    else if (b <= 1.6d-133) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (3.0d0 * a)
    else if ((b <= 1100.0d0) .or. (.not. (b <= 11000000.0d0))) then
        tmp = (c * (-0.5d0)) / b
    else
        tmp = (sqrt(((-3.0d0) * (a * c))) - b) / (3.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.02e-96) {
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	} else if (b <= 1.6e-133) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else if ((b <= 1100.0) || !(b <= 11000000.0)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = (Math.sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.02e-96:
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b))
	elif b <= 1.6e-133:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (3.0 * a)
	elif (b <= 1100.0) or not (b <= 11000000.0):
		tmp = (c * -0.5) / b
	else:
		tmp = (math.sqrt((-3.0 * (a * c))) - b) / (3.0 * a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.02e-96)
		tmp = Float64(Float64(-0.6666666666666666 / Float64(a / b)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.6e-133)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(3.0 * a));
	elseif ((b <= 1100.0) || !(b <= 11000000.0))
		tmp = Float64(Float64(c * -0.5) / b);
	else
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / Float64(3.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.02e-96)
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	elseif (b <= 1.6e-133)
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	elseif ((b <= 1100.0) || ~((b <= 11000000.0)))
		tmp = (c * -0.5) / b;
	else
		tmp = (sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.02e-96], N[(N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.6e-133], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 1100.0], N[Not[LessEqual[b, 11000000.0]], $MachinePrecision]], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.02000000000000007e-96

   1. Initial program 63.4%

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

   3. Taylor expanded in b around -inf 87.8%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. clear-num 87.8%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} + 0.5 \cdot \frac{c}{b} \]

      2. un-div-inv 87.9%

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

   5. Applied egg-rr 87.9%

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

   if -1.02000000000000007e-96 < b < 1.60000000000000006e-133

   1. Initial program 81.1%

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

   3. Taylor expanded in b around 0 78.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-*r* 79.0%

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

      2. *-commutative 79.0%

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

      3. *-commutative 79.0%

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

   5. Simplified 79.0%

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

   if 1.60000000000000006e-133 < b < 1100 or 1.1e7 < b

   1. Initial program 18.6%

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

   3. Taylor expanded in b around inf 85.7%

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

      1. associate-*r/ 85.7%

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

   5. Simplified 85.7%

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

   if 1100 < b < 1.1e7

   1. Initial program 99.4%

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

   3. Taylor expanded in b around 0 99.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-96}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1100 \lor \neg \left(b \leq 11000000\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+161}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1100 \lor \neg \left(b \leq 21000000\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e+161)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 5.2e-130)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (if (or (<= b 1100.0) (not (<= b 21000000.0)))
       (/ (* c -0.5) b)
       (/ (- (sqrt (* -3.0 (* a c))) b) (* 3.0 a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e+161) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 5.2e-130) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else if ((b <= 1100.0) || !(b <= 21000000.0)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = (sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	}
	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
    real(8) :: tmp
    if (b <= (-1.6d+161)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 5.2d-130) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else if ((b <= 1100.0d0) .or. (.not. (b <= 21000000.0d0))) then
        tmp = (c * (-0.5d0)) / b
    else
        tmp = (sqrt(((-3.0d0) * (a * c))) - b) / (3.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e+161) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 5.2e-130) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else if ((b <= 1100.0) || !(b <= 21000000.0)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = (Math.sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.6e+161:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 5.2e-130:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	elif (b <= 1100.0) or not (b <= 21000000.0):
		tmp = (c * -0.5) / b
	else:
		tmp = (math.sqrt((-3.0 * (a * c))) - b) / (3.0 * a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e+161)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 5.2e-130)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	elseif ((b <= 1100.0) || !(b <= 21000000.0))
		tmp = Float64(Float64(c * -0.5) / b);
	else
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / Float64(3.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.6e+161)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 5.2e-130)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	elseif ((b <= 1100.0) || ~((b <= 21000000.0)))
		tmp = (c * -0.5) / b;
	else
		tmp = (sqrt((-3.0 * (a * c))) - b) / (3.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e+161], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-130], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 1100.0], N[Not[LessEqual[b, 21000000.0]], $MachinePrecision]], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.60000000000000001e161

   1. Initial program 23.9%

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

   3. Taylor expanded in b around -inf 95.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative 95.3%

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

   5. Simplified 95.3%

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

   if -1.60000000000000001e161 < b < 5.2000000000000001e-130

   1. Initial program 86.3%

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

   if 5.2000000000000001e-130 < b < 1100 or 2.1e7 < b

   1. Initial program 18.6%

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

   3. Taylor expanded in b around inf 85.7%

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

      1. associate-*r/ 85.7%

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

   5. Simplified 85.7%

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

   if 1100 < b < 2.1e7

   1. Initial program 99.4%

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

   3. Taylor expanded in b around 0 99.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+161}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1100 \lor \neg \left(b \leq 21000000\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.4% accurate, 7.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (+ (* 0.5 (/ c b)) (* -0.6666666666666666 (/ b a)))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / 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
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (0.5d0 * (c / b)) + ((-0.6666666666666666d0) * (b / a))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(0.5 * Float64(c / b)) + Float64(-0.6666666666666666 * Float64(b / a)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (0.5 * (c / b)) + (-0.6666666666666666 * (b / a));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 67.6%

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

   3. Taylor expanded in b around -inf 69.6%

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

   if -4.999999999999985e-310 < b

   1. Initial program 32.6%

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

   3. Taylor expanded in b around inf 69.4%

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

      1. associate-*r/ 69.4%

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

   5. Simplified 69.4%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

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

Alternative 9: 66.4% accurate, 7.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (+ (/ -0.6666666666666666 (/ a b)) (* 0.5 (/ c b)))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	} 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
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = ((-0.6666666666666666d0) / (a / b)) + (0.5d0 * (c / b))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(-0.6666666666666666 / Float64(a / b)) + Float64(0.5 * Float64(c / b)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (-0.6666666666666666 / (a / b)) + (0.5 * (c / b));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 67.6%

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

   3. Taylor expanded in b around -inf 69.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. clear-num 69.6%

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{a}{b}}} + 0.5 \cdot \frac{c}{b} \]

      2. un-div-inv 69.7%

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

   5. Applied egg-rr 69.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 32.6%

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

   3. Taylor expanded in b around inf 69.4%

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

      1. associate-*r/ 69.4%

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

   5. Simplified 69.4%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.6%

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

Alternative 10: 66.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-298}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.5e-298) (/ (* b -2.0) (* 3.0 a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.5e-298) {
		tmp = (b * -2.0) / (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
    real(8) :: tmp
    if (b <= 1.5d-298) then
        tmp = (b * (-2.0d0)) / (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) {
	double tmp;
	if (b <= 1.5e-298) {
		tmp = (b * -2.0) / (3.0 * a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.5e-298:
		tmp = (b * -2.0) / (3.0 * a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.5e-298)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.5e-298)
		tmp = (b * -2.0) / (3.0 * a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.5e-298], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5e-298

   1. Initial program 67.0%

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

   3. Taylor expanded in b around -inf 68.7%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative 68.7%

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

   5. Simplified 68.7%

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

   if 1.5e-298 < b

   1. Initial program 32.9%

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

   3. Taylor expanded in b around inf 70.0%

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

      1. associate-*r/ 70.0%

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

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-298}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-293}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.8e-293) (/ (/ (* b -2.0) 3.0) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.8e-293) {
		tmp = ((b * -2.0) / 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
    real(8) :: tmp
    if (b <= 5.8d-293) then
        tmp = ((b * (-2.0d0)) / 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) {
	double tmp;
	if (b <= 5.8e-293) {
		tmp = ((b * -2.0) / 3.0) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.8e-293:
		tmp = ((b * -2.0) / 3.0) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.8e-293)
		tmp = Float64(Float64(Float64(b * -2.0) / 3.0) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.8e-293)
		tmp = ((b * -2.0) / 3.0) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.8e-293], N[(N[(N[(b * -2.0), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.7999999999999999e-293

   1. Initial program 67.0%

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

   3. Taylor expanded in b around -inf 68.7%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative 68.7%

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

   5. Simplified 68.7%

      \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. add-cube-cbrt 67.8%

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

      2. *-commutative 67.8%

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

      3. times-frac 67.7%

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

      4. pow2 67.7%

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

   7. Applied egg-rr 67.7%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{b \cdot -2}\right)}^{2}}{a} \cdot \frac{\sqrt[3]{b \cdot -2}}{3}} \]
   8. Step-by-step derivation

      1. associate-*l/ 67.7%

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

      2. associate-*r/ 67.7%

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

      3. unpow2 67.7%

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

      4. rem-3cbrt-lft 68.7%

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

   9. Simplified 68.7%

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

   if 5.7999999999999999e-293 < b

   1. Initial program 32.9%

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

   3. Taylor expanded in b around inf 70.0%

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

      1. associate-*r/ 70.0%

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

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{-293}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.006:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.006) (* b (/ -0.6666666666666666 a)) (* 0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.006) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.5 * (c / 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
    real(8) :: tmp
    if (b <= 0.006d0) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = 0.5d0 * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.006) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.006:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = 0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.006)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.006)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = 0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.006], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0060000000000000001

   1. Initial program 65.1%

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

   3. Taylor expanded in b around -inf 51.7%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative 51.7%

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

   5. Simplified 51.7%

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

   6. Taylor expanded in b around 0 51.5%

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

      1. *-commutative 51.5%

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

      2. associate-*l/ 51.6%

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

      3. associate-*r/ 51.6%

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

   8. Simplified 51.6%

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

   if 0.0060000000000000001 < b

   1. Initial program 18.3%

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

   3. Taylor expanded in b around -inf 2.6%

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

   4. Taylor expanded in b around 0 25.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.006:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-302}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.8e-302) (* b (/ -0.6666666666666666 a)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.8e-302) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 / (b / c);
	}
	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
    real(8) :: tmp
    if (b <= 2.8d-302) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.8e-302) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.8e-302:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.8e-302)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.8e-302)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.8e-302], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8e-302

   1. Initial program 67.0%

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

   3. Taylor expanded in b around -inf 68.7%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative 68.7%

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

   5. Simplified 68.7%

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

   6. Taylor expanded in b around 0 68.5%

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

      1. *-commutative 68.5%

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

      2. associate-*l/ 68.6%

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

      3. associate-*r/ 68.6%

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

   8. Simplified 68.6%

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

   if 2.8e-302 < b

   1. Initial program 32.9%

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

   3. Taylor expanded in b around inf 70.0%

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

      1. associate-*r/ 70.0%

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

      2. associate-/l* 69.7%

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

   5. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{b}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-302}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-302}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.8e-302) (* b (/ -0.6666666666666666 a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.8e-302) {
		tmp = b * (-0.6666666666666666 / 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
    real(8) :: tmp
    if (b <= 2.8d-302) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.8e-302) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.8e-302:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.8e-302)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.8e-302)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.8e-302], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8e-302

   1. Initial program 67.0%

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

   3. Taylor expanded in b around -inf 68.7%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative 68.7%

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

   5. Simplified 68.7%

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

   6. Taylor expanded in b around 0 68.5%

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

      1. *-commutative 68.5%

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

      2. associate-*l/ 68.6%

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

      3. associate-*r/ 68.6%

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

   8. Simplified 68.6%

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

   if 2.8e-302 < b

   1. Initial program 32.9%

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

   3. Taylor expanded in b around inf 70.0%

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

      1. associate-*r/ 70.0%

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

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-302}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{-301}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.5e-301) (/ (* b -0.6666666666666666) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5e-301) {
		tmp = (b * -0.6666666666666666) / 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
    real(8) :: tmp
    if (b <= 2.5d-301) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.5e-301) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.5e-301:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.5e-301)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.5e-301)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.5e-301], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.50000000000000006e-301

   1. Initial program 67.0%

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

   3. Taylor expanded in b around -inf 68.5%

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

      1. *-commutative 68.5%

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

   5. Simplified 68.5%

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

      1. associate-*l/ 68.6%

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

   7. Applied egg-rr 68.6%

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

   if 2.50000000000000006e-301 < b

   1. Initial program 32.9%

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

   3. Taylor expanded in b around inf 70.0%

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

      1. associate-*r/ 70.0%

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

   5. Simplified 70.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{-301}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 10.9% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* 0.5 (/ c b)))

C

double code(double a, double b, double c) {
	return 0.5 * (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.5d0 * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return 0.5 * (c / b);
}

Python

def code(a, b, c):
	return 0.5 * (c / b)

Julia

function code(a, b, c)
	return Float64(0.5 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.5 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

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

3. Taylor expanded in b around -inf 36.0%

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

4. Taylor expanded in b around 0 10.3%

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

5. Final simplification 10.3%

   \[\leadsto 0.5 \cdot \frac{c}{b} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))