Cubic critical

Percentage Accurate: 51.5% → 84.2%

Time: 20.3s

Alternatives: 11

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: 51.5% 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: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\\ \mathbf{if}\;b \leq -9.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-131}:\\ \;\;\;\;\frac{t_0 - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-76} \lor \neg \left(b \leq 2.05 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3 \cdot \frac{a}{b + t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma b b (* a (* c -3.0))))))
   (if (<= b -9.8e+126)
     (/ (* b -2.0) (* 3.0 a))
     (if (<= b 2e-131)
       (/ (- t_0 b) (* 3.0 a))
       (if (or (<= b 1.12e-76) (not (<= b 2.05e-11)))
         (/ (* c -0.5) b)
         (/ 1.0 (* 3.0 (/ a (+ b t_0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(b, b, (a * (c * -3.0))));
	double tmp;
	if (b <= -9.8e+126) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2e-131) {
		tmp = (t_0 - b) / (3.0 * a);
	} else if ((b <= 1.12e-76) || !(b <= 2.05e-11)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = 1.0 / (3.0 * (a / (b + t_0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(b, b, Float64(a * Float64(c * -3.0))))
	tmp = 0.0
	if (b <= -9.8e+126)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 2e-131)
		tmp = Float64(Float64(t_0 - b) / Float64(3.0 * a));
	elseif ((b <= 1.12e-76) || !(b <= 2.05e-11))
		tmp = Float64(Float64(c * -0.5) / b);
	else
		tmp = Float64(1.0 / Float64(3.0 * Float64(a / Float64(b + t_0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -9.8e+126], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-131], N[(N[(t$95$0 - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 1.12e-76], N[Not[LessEqual[b, 2.05e-11]], $MachinePrecision]], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision], N[(1.0 / N[(3.0 * N[(a / N[(b + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -9.80000000000000002e126

   1. Initial program 52.5%

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

   3. Taylor expanded in b around -inf 98.1%

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

      1. *-commutative 98.1%

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

   5. Simplified 98.1%

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

   if -9.80000000000000002e126 < b < 2e-131

   1. Initial program 89.7%

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

      1. +-commutative 89.7%

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

      2. sqr-neg 89.7%

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

      3. unsub-neg 89.7%

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

      4. div-sub 89.7%

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

      5. --rgt-identity 89.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0}}{3 \cdot a} - \frac{b}{3 \cdot a} \]

      6. div-sub 89.7%

         \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0\right) - b}{3 \cdot a}} \]

   3. Simplified 89.7%

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

   if 2e-131 < b < 1.12e-76 or 2.05e-11 < b

   1. Initial program 12.8%

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

   3. Taylor expanded in b around inf 90.2%

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

      1. associate-*r/ 90.2%

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

   5. Simplified 90.2%

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

   if 1.12e-76 < b < 2.05e-11

   1. Initial program 67.3%

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

      1. prod-diff 67.0%

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

      2. *-commutative 67.0%

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

      3. fma-neg 67.0%

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

      4. prod-diff 67.0%

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

      5. *-commutative 67.0%

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

      6. fma-neg 67.0%

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

      7. associate-+l+ 67.3%

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

      8. fma-neg 67.3%

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

      9. *-commutative 67.3%

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

      10. distribute-rgt-neg-in 67.3%

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

      11. *-commutative 67.3%

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

      12. distribute-rgt-neg-in 67.3%

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

      13. metadata-eval 67.3%

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

   4. Applied egg-rr 66.8%

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

      1. clear-num 67.3%

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

      2. inv-pow 67.3%

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

   6. Applied egg-rr 67.0%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{b + \sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(c \cdot a, -3, a \cdot \left(c \cdot 3\right)\right), \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}}\right)}^{-1}} \]

   8. Simplified 67.6%

      \[\leadsto \color{blue}{\frac{1}{3 \cdot \frac{a}{b + \sqrt{0 + \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-131}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-76} \lor \neg \left(b \leq 2.05 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3 \cdot \frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot -3\right)\\ \mathbf{if}\;b \leq -5 \cdot 10^{+126}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, t_0\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-77} \lor \neg \left(b \leq 6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{t_0}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c -3.0))))
   (if (<= b -5e+126)
     (/ (* b -2.0) (* 3.0 a))
     (if (<= b 2.4e-132)
       (/ (- (sqrt (fma b b t_0)) b) (* 3.0 a))
       (if (or (<= b 1.9e-77) (not (<= b 6e-12)))
         (/ (* c -0.5) b)
         (* 0.3333333333333333 (/ (sqrt t_0) a)))))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * -3.0);
	double tmp;
	if (b <= -5e+126) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2.4e-132) {
		tmp = (sqrt(fma(b, b, t_0)) - b) / (3.0 * a);
	} else if ((b <= 1.9e-77) || !(b <= 6e-12)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = 0.3333333333333333 * (sqrt(t_0) / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * -3.0))
	tmp = 0.0
	if (b <= -5e+126)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 2.4e-132)
		tmp = Float64(Float64(sqrt(fma(b, b, t_0)) - b) / Float64(3.0 * a));
	elseif ((b <= 1.9e-77) || !(b <= 6e-12))
		tmp = Float64(Float64(c * -0.5) / b);
	else
		tmp = Float64(0.3333333333333333 * Float64(sqrt(t_0) / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5e+126], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-132], N[(N[(N[Sqrt[N[(b * b + t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 1.9e-77], N[Not[LessEqual[b, 6e-12]], $MachinePrecision]], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision], N[(0.3333333333333333 * N[(N[Sqrt[t$95$0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.99999999999999977e126

   1. Initial program 52.5%

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

   3. Taylor expanded in b around -inf 98.1%

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

      1. *-commutative 98.1%

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

   5. Simplified 98.1%

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

   if -4.99999999999999977e126 < b < 2.40000000000000015e-132

   1. Initial program 89.7%

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

      1. +-commutative 89.7%

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

      2. sqr-neg 89.7%

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

      3. unsub-neg 89.7%

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

      4. div-sub 89.7%

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

      5. --rgt-identity 89.7%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0}}{3 \cdot a} - \frac{b}{3 \cdot a} \]

      6. div-sub 89.7%

         \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0\right) - b}{3 \cdot a}} \]

   3. Simplified 89.7%

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

   if 2.40000000000000015e-132 < b < 1.8999999999999999e-77 or 6.0000000000000003e-12 < b

   1. Initial program 12.8%

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

   3. Taylor expanded in b around inf 90.2%

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

      1. associate-*r/ 90.2%

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

   5. Simplified 90.2%

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

   if 1.8999999999999999e-77 < b < 6.0000000000000003e-12

   1. Initial program 67.3%

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

      1. prod-diff 67.0%

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

      2. *-commutative 67.0%

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

      3. fma-neg 67.0%

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

      4. prod-diff 67.0%

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

      5. *-commutative 67.0%

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

      6. fma-neg 67.0%

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

      7. associate-+l+ 67.3%

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

      8. fma-neg 67.3%

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

      9. *-commutative 67.3%

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

      10. distribute-rgt-neg-in 67.3%

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

      11. *-commutative 67.3%

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

      12. distribute-rgt-neg-in 67.3%

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

      13. metadata-eval 67.3%

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

   4. Applied egg-rr 66.8%

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

   5. Taylor expanded in b around 0 67.3%

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

      1. associate-*l/ 67.1%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)}}{a}} \]

      2. *-lft-identity 67.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{-6 \cdot \left(a \cdot c\right) + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      3. *-commutative 67.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -6} + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-rgt-out 67.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot -6 + \color{blue}{\left(a \cdot c\right) \cdot \left(-3 + 6\right)}}}{a} \]

      5. metadata-eval 67.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot -6 + \left(a \cdot c\right) \cdot \color{blue}{3}}}{a} \]

      6. distribute-lft-out 67.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}}}{a} \]

      7. metadata-eval 67.3%

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

      8. associate-*r* 67.6%

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

   7. Simplified 67.6%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+126}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-77} \lor \neg \left(b \leq 6 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-131} \lor \neg \left(b \leq 1.6 \cdot 10^{-77}\right) \land b \leq 4 \cdot 10^{-12}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-24)
   (/ (* b -2.0) (* 3.0 a))
   (if (or (<= b 2e-131) (and (not (<= b 1.6e-77)) (<= b 4e-12)))
     (* 0.3333333333333333 (/ (sqrt (* a (* c -3.0))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-24) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if ((b <= 2e-131) || (!(b <= 1.6e-77) && (b <= 4e-12))) {
		tmp = 0.3333333333333333 * (sqrt((a * (c * -3.0))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.8d-24)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if ((b <= 2d-131) .or. (.not. (b <= 1.6d-77)) .and. (b <= 4d-12)) then
        tmp = 0.3333333333333333d0 * (sqrt((a * (c * (-3.0d0)))) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-24) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if ((b <= 2e-131) || (!(b <= 1.6e-77) && (b <= 4e-12))) {
		tmp = 0.3333333333333333 * (Math.sqrt((a * (c * -3.0))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.8e-24:
		tmp = (b * -2.0) / (3.0 * a)
	elif (b <= 2e-131) or (not (b <= 1.6e-77) and (b <= 4e-12)):
		tmp = 0.3333333333333333 * (math.sqrt((a * (c * -3.0))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-24)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif ((b <= 2e-131) || (!(b <= 1.6e-77) && (b <= 4e-12)))
		tmp = Float64(0.3333333333333333 * Float64(sqrt(Float64(a * Float64(c * -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 <= -3.8e-24)
		tmp = (b * -2.0) / (3.0 * a);
	elseif ((b <= 2e-131) || (~((b <= 1.6e-77)) && (b <= 4e-12)))
		tmp = 0.3333333333333333 * (sqrt((a * (c * -3.0))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-24], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 2e-131], And[N[Not[LessEqual[b, 1.6e-77]], $MachinePrecision], LessEqual[b, 4e-12]]], N[(0.3333333333333333 * N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.80000000000000026e-24

   1. Initial program 69.0%

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

   3. Taylor expanded in b around -inf 94.4%

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

      1. *-commutative 94.4%

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

   5. Simplified 94.4%

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

   if -3.80000000000000026e-24 < b < 2e-131 or 1.6e-77 < b < 3.99999999999999992e-12

   1. Initial program 82.5%

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

      1. prod-diff 82.3%

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

      2. *-commutative 82.3%

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

      3. fma-neg 82.3%

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

      4. prod-diff 82.3%

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

      5. *-commutative 82.3%

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

      6. fma-neg 82.3%

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

      7. associate-+l+ 82.3%

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

      8. fma-neg 82.3%

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

      9. *-commutative 82.3%

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

      10. distribute-rgt-neg-in 82.3%

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

      11. *-commutative 82.3%

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

      12. distribute-rgt-neg-in 82.3%

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

      13. metadata-eval 82.3%

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

   4. Applied egg-rr 82.1%

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

   5. Taylor expanded in b around 0 67.7%

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

      1. associate-*l/ 67.7%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)}}{a}} \]

      2. *-lft-identity 67.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{-6 \cdot \left(a \cdot c\right) + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      3. *-commutative 67.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -6} + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-rgt-out 67.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot -6 + \color{blue}{\left(a \cdot c\right) \cdot \left(-3 + 6\right)}}}{a} \]

      5. metadata-eval 67.7%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot -6 + \left(a \cdot c\right) \cdot \color{blue}{3}}}{a} \]

      6. distribute-lft-out 67.9%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}}}{a} \]

      7. metadata-eval 67.9%

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

      8. associate-*r* 68.0%

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

   7. Simplified 68.0%

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

   if 2e-131 < b < 1.6e-77 or 3.99999999999999992e-12 < b

   1. Initial program 12.8%

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

   3. Taylor expanded in b around inf 90.2%

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

      1. associate-*r/ 90.2%

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

   5. Simplified 90.2%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-24}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-131} \lor \neg \left(b \leq 1.6 \cdot 10^{-77}\right) \land b \leq 4 \cdot 10^{-12}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(c \cdot -3\right)}\\ \mathbf{if}\;b \leq -2.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-131}:\\ \;\;\;\;\frac{t_0 - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-77} \lor \neg \left(b \leq 4 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* a (* c -3.0)))))
   (if (<= b -2.5e-23)
     (/ (* b -2.0) (* 3.0 a))
     (if (<= b 2e-131)
       (/ (- t_0 b) (* 3.0 a))
       (if (or (<= b 6.5e-77) (not (<= b 4e-12)))
         (/ (* c -0.5) b)
         (* 0.3333333333333333 (/ t_0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (c * -3.0)));
	double tmp;
	if (b <= -2.5e-23) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2e-131) {
		tmp = (t_0 - b) / (3.0 * a);
	} else if ((b <= 6.5e-77) || !(b <= 4e-12)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = 0.3333333333333333 * (t_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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a * (c * (-3.0d0))))
    if (b <= (-2.5d-23)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 2d-131) then
        tmp = (t_0 - b) / (3.0d0 * a)
    else if ((b <= 6.5d-77) .or. (.not. (b <= 4d-12))) then
        tmp = (c * (-0.5d0)) / b
    else
        tmp = 0.3333333333333333d0 * (t_0 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((a * (c * -3.0)));
	double tmp;
	if (b <= -2.5e-23) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2e-131) {
		tmp = (t_0 - b) / (3.0 * a);
	} else if ((b <= 6.5e-77) || !(b <= 4e-12)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = 0.3333333333333333 * (t_0 / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((a * (c * -3.0)))
	tmp = 0
	if b <= -2.5e-23:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 2e-131:
		tmp = (t_0 - b) / (3.0 * a)
	elif (b <= 6.5e-77) or not (b <= 4e-12):
		tmp = (c * -0.5) / b
	else:
		tmp = 0.3333333333333333 * (t_0 / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(a * Float64(c * -3.0)))
	tmp = 0.0
	if (b <= -2.5e-23)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 2e-131)
		tmp = Float64(Float64(t_0 - b) / Float64(3.0 * a));
	elseif ((b <= 6.5e-77) || !(b <= 4e-12))
		tmp = Float64(Float64(c * -0.5) / b);
	else
		tmp = Float64(0.3333333333333333 * Float64(t_0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt((a * (c * -3.0)));
	tmp = 0.0;
	if (b <= -2.5e-23)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 2e-131)
		tmp = (t_0 - b) / (3.0 * a);
	elseif ((b <= 6.5e-77) || ~((b <= 4e-12)))
		tmp = (c * -0.5) / b;
	else
		tmp = 0.3333333333333333 * (t_0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.5e-23], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-131], N[(N[(t$95$0 - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[b, 6.5e-77], N[Not[LessEqual[b, 4e-12]], $MachinePrecision]], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision], N[(0.3333333333333333 * N[(t$95$0 / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.5000000000000001e-23

   1. Initial program 69.0%

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

   3. Taylor expanded in b around -inf 94.4%

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

      1. *-commutative 94.4%

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

   5. Simplified 94.4%

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

   if -2.5000000000000001e-23 < b < 2e-131

   1. Initial program 85.2%

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

      1. prod-diff 85.0%

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

      2. *-commutative 85.0%

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

      3. fma-neg 85.0%

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

      4. prod-diff 85.0%

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

      5. *-commutative 85.0%

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

      6. fma-neg 85.0%

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

      7. associate-+l+ 85.0%

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

      8. fma-neg 85.0%

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

      9. *-commutative 85.0%

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

      10. distribute-rgt-neg-in 85.0%

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

      11. *-commutative 85.0%

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

      12. distribute-rgt-neg-in 85.0%

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

      13. metadata-eval 85.0%

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

   4. Applied egg-rr 84.9%

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

   5. Taylor expanded in b around 0 70.4%

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

      1. mul-1-neg 70.4%

         \[\leadsto \frac{\sqrt{-6 \cdot \left(a \cdot c\right) + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)} + \color{blue}{\left(-b\right)}}{3 \cdot a} \]

      2. unsub-neg 70.4%

         \[\leadsto \frac{\color{blue}{\sqrt{-6 \cdot \left(a \cdot c\right) + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)} - b}}{3 \cdot a} \]

      3. *-commutative 70.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -6} + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)} - b}{3 \cdot a} \]

      4. distribute-rgt-out 70.4%

         \[\leadsto \frac{\sqrt{\left(a \cdot c\right) \cdot -6 + \color{blue}{\left(a \cdot c\right) \cdot \left(-3 + 6\right)}} - b}{3 \cdot a} \]

      5. metadata-eval 70.4%

         \[\leadsto \frac{\sqrt{\left(a \cdot c\right) \cdot -6 + \left(a \cdot c\right) \cdot \color{blue}{3}} - b}{3 \cdot a} \]

      6. distribute-lft-out 70.7%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}} - b}{3 \cdot a} \]

      7. metadata-eval 70.7%

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

      8. associate-*r* 70.7%

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

   7. Simplified 70.7%

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

   if 2e-131 < b < 6.4999999999999999e-77 or 3.99999999999999992e-12 < b

   1. Initial program 12.8%

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

   3. Taylor expanded in b around inf 90.2%

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

      1. associate-*r/ 90.2%

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

   5. Simplified 90.2%

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

   if 6.4999999999999999e-77 < b < 3.99999999999999992e-12

   1. Initial program 67.3%

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

      1. prod-diff 67.0%

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

      2. *-commutative 67.0%

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

      3. fma-neg 67.0%

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

      4. prod-diff 67.0%

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

      5. *-commutative 67.0%

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

      6. fma-neg 67.0%

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

      7. associate-+l+ 67.3%

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

      8. fma-neg 67.3%

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

      9. *-commutative 67.3%

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

      10. distribute-rgt-neg-in 67.3%

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

      11. *-commutative 67.3%

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

      12. distribute-rgt-neg-in 67.3%

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

      13. metadata-eval 67.3%

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

   4. Applied egg-rr 66.8%

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

   5. Taylor expanded in b around 0 67.3%

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

      1. associate-*l/ 67.1%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)}}{a}} \]

      2. *-lft-identity 67.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{-6 \cdot \left(a \cdot c\right) + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      3. *-commutative 67.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -6} + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-rgt-out 67.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot -6 + \color{blue}{\left(a \cdot c\right) \cdot \left(-3 + 6\right)}}}{a} \]

      5. metadata-eval 67.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot -6 + \left(a \cdot c\right) \cdot \color{blue}{3}}}{a} \]

      6. distribute-lft-out 67.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}}}{a} \]

      7. metadata-eval 67.3%

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

      8. associate-*r* 67.6%

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

   7. Simplified 67.6%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-131}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-77} \lor \neg \left(b \leq 4 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+126}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-131}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-77} \lor \neg \left(b \leq 5.1 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e+126)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 2e-131)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (if (or (<= b 2.65e-77) (not (<= b 5.1e-12)))
       (/ (* c -0.5) b)
       (* 0.3333333333333333 (/ (sqrt (* a (* c -3.0))) a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e+126) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2e-131) {
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else if ((b <= 2.65e-77) || !(b <= 5.1e-12)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = 0.3333333333333333 * (sqrt((a * (c * -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 <= (-6d+126)) then
        tmp = (b * (-2.0d0)) / (3.0d0 * a)
    else if (b <= 2d-131) then
        tmp = (sqrt(((b * b) - ((3.0d0 * a) * c))) - b) / (3.0d0 * a)
    else if ((b <= 2.65d-77) .or. (.not. (b <= 5.1d-12))) then
        tmp = (c * (-0.5d0)) / b
    else
        tmp = 0.3333333333333333d0 * (sqrt((a * (c * (-3.0d0)))) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e+126) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 2e-131) {
		tmp = (Math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	} else if ((b <= 2.65e-77) || !(b <= 5.1e-12)) {
		tmp = (c * -0.5) / b;
	} else {
		tmp = 0.3333333333333333 * (Math.sqrt((a * (c * -3.0))) / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e+126:
		tmp = (b * -2.0) / (3.0 * a)
	elif b <= 2e-131:
		tmp = (math.sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)
	elif (b <= 2.65e-77) or not (b <= 5.1e-12):
		tmp = (c * -0.5) / b
	else:
		tmp = 0.3333333333333333 * (math.sqrt((a * (c * -3.0))) / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e+126)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 2e-131)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a));
	elseif ((b <= 2.65e-77) || !(b <= 5.1e-12))
		tmp = Float64(Float64(c * -0.5) / b);
	else
		tmp = Float64(0.3333333333333333 * Float64(sqrt(Float64(a * Float64(c * -3.0))) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6e+126)
		tmp = (b * -2.0) / (3.0 * a);
	elseif (b <= 2e-131)
		tmp = (sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a);
	elseif ((b <= 2.65e-77) || ~((b <= 5.1e-12)))
		tmp = (c * -0.5) / b;
	else
		tmp = 0.3333333333333333 * (sqrt((a * (c * -3.0))) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e+126], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-131], 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, 2.65e-77], N[Not[LessEqual[b, 5.1e-12]], $MachinePrecision]], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision], N[(0.3333333333333333 * N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.0000000000000005e126

   1. Initial program 52.5%

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

   3. Taylor expanded in b around -inf 98.1%

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

      1. *-commutative 98.1%

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

   5. Simplified 98.1%

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

   if -6.0000000000000005e126 < b < 2e-131

   1. Initial program 89.7%

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

   if 2e-131 < b < 2.65000000000000007e-77 or 5.09999999999999968e-12 < b

   1. Initial program 12.8%

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

   3. Taylor expanded in b around inf 90.2%

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

      1. associate-*r/ 90.2%

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

   5. Simplified 90.2%

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

   if 2.65000000000000007e-77 < b < 5.09999999999999968e-12

   1. Initial program 67.3%

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

      1. prod-diff 67.0%

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

      2. *-commutative 67.0%

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

      3. fma-neg 67.0%

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

      4. prod-diff 67.0%

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

      5. *-commutative 67.0%

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

      6. fma-neg 67.0%

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

      7. associate-+l+ 67.3%

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

      8. fma-neg 67.3%

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

      9. *-commutative 67.3%

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

      10. distribute-rgt-neg-in 67.3%

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

      11. *-commutative 67.3%

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

      12. distribute-rgt-neg-in 67.3%

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

      13. metadata-eval 67.3%

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

   4. Applied egg-rr 66.8%

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

   5. Taylor expanded in b around 0 67.3%

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

      1. associate-*l/ 67.1%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)}}{a}} \]

      2. *-lft-identity 67.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{-6 \cdot \left(a \cdot c\right) + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      3. *-commutative 67.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -6} + \left(-3 \cdot \left(a \cdot c\right) + 6 \cdot \left(a \cdot c\right)\right)}}{a} \]

      4. distribute-rgt-out 67.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot -6 + \color{blue}{\left(a \cdot c\right) \cdot \left(-3 + 6\right)}}}{a} \]

      5. metadata-eval 67.1%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot -6 + \left(a \cdot c\right) \cdot \color{blue}{3}}}{a} \]

      6. distribute-lft-out 67.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}}}{a} \]

      7. metadata-eval 67.3%

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

      8. associate-*r* 67.6%

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

   7. Simplified 67.6%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+126}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-131}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-77} \lor \neg \left(b \leq 5.1 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.9% accurate, 9.7× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (* (* b -2.0) (/ 0.3333333333333333 a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (b * -2.0) * (0.3333333333333333 / 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 = (b * (-2.0d0)) * (0.3333333333333333d0 / 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 = (b * -2.0) * (0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (b * -2.0) * (0.3333333333333333 / 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(b * -2.0) * Float64(0.3333333333333333 / 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 = (b * -2.0) * (0.3333333333333333 / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(b * -2.0), $MachinePrecision] * N[(0.3333333333333333 / a), $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 75.9%

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

   4. Applied egg-rr 72.3%

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

      1. sub-neg 72.3%

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

      2. distribute-rgt-out-- 72.4%

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

   6. Simplified 72.4%

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

   7. Taylor expanded in b around -inf 69.9%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(-2 \cdot b\right)} \]
   8. Step-by-step derivation

      1. *-commutative 69.9%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(b \cdot -2\right)} \]

   9. Simplified 69.9%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \color{blue}{\left(b \cdot -2\right)} \]

   if -4.999999999999985e-310 < b

   1. Initial program 27.9%

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

   3. Taylor expanded in b around inf 72.2%

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

      1. associate-*r/ 72.2%

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

   5. Simplified 72.2%

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

4. Final simplification 71.0%

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

Alternative 7: 68.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2e-309) (/ (* b -2.0) (* 3.0 a)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e-309) {
		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 <= 2d-309) 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 <= 2e-309) {
		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 <= 2e-309:
		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 <= 2e-309)
		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 <= 2e-309)
		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, 2e-309], 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 < 1.9999999999999988e-309

   1. Initial program 75.9%

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

   3. Taylor expanded in b around -inf 70.0%

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

      1. *-commutative 70.0%

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

   5. Simplified 70.0%

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

   if 1.9999999999999988e-309 < b

   1. Initial program 27.9%

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

   3. Taylor expanded in b around inf 72.2%

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

      1. associate-*r/ 72.2%

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

   5. Simplified 72.2%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.0% accurate, 11.6× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9e-9) (* (/ b a) -0.6666666666666666) (* 0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9e-9) {
		tmp = (b / a) * -0.6666666666666666;
	} 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 <= 9d-9) then
        tmp = (b / a) * (-0.6666666666666666d0)
    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 <= 9e-9) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9e-9:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = 0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9e-9)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	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 <= 9e-9)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = 0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9e-9], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.99999999999999953e-9

   1. Initial program 71.8%

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

   3. Taylor expanded in b around -inf 54.0%

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

      1. *-commutative 54.0%

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

   5. Simplified 54.0%

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

   if 8.99999999999999953e-9 < b

   1. Initial program 11.4%

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

   3. Taylor expanded in b around -inf 2.1%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. +-commutative 2.1%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} + -1 \cdot b\right)}}{3 \cdot a} \]

      2. mul-1-neg 2.1%

         \[\leadsto \frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} + \color{blue}{\left(-b\right)}\right)}{3 \cdot a} \]

      3. unsub-neg 2.1%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}}{3 \cdot a} \]

      4. associate-/l* 2.4%

         \[\leadsto \frac{\left(-b\right) + \left(1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}} - b\right)}{3 \cdot a} \]

      5. associate-*r/ 2.4%

         \[\leadsto \frac{\left(-b\right) + \left(\color{blue}{\frac{1.5 \cdot a}{\frac{b}{c}}} - b\right)}{3 \cdot a} \]

   5. Simplified 2.4%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(\frac{1.5 \cdot a}{\frac{b}{c}} - b\right)}}{3 \cdot a} \]

   6. Taylor expanded in b around 0 28.2%

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

4. Final simplification 46.1%

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

Alternative 9: 67.6% accurate, 11.6× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (b / a) * -0.6666666666666666;
	} 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 <= (-5d-310)) then
        tmp = (b / a) * (-0.6666666666666666d0)
    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 <= -5e-310) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	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 <= -5e-310)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 75.9%

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

   3. Taylor expanded in b around -inf 69.8%

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

      1. *-commutative 69.8%

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

   5. Simplified 69.8%

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

   if -4.999999999999985e-310 < b

   1. Initial program 27.9%

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

   3. Taylor expanded in b around inf 72.2%

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

      1. associate-*r/ 72.2%

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

      2. associate-/l* 71.5%

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

   5. Simplified 71.5%

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

4. Final simplification 70.6%

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

Alternative 10: 67.9% accurate, 11.6× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (b / a) * -0.6666666666666666;
	} 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 = (b / a) * (-0.6666666666666666d0)
    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 = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	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 = (b / a) * -0.6666666666666666;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $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 75.9%

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

   3. Taylor expanded in b around -inf 69.8%

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

      1. *-commutative 69.8%

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

   5. Simplified 69.8%

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

   if -4.999999999999985e-310 < b

   1. Initial program 27.9%

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

   3. Taylor expanded in b around inf 72.2%

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

      1. associate-*r/ 72.2%

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

   5. Simplified 72.2%

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

4. Final simplification 70.9%

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

Alternative 11: 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 53.2%

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

3. Taylor expanded in b around -inf 36.7%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
4. Step-by-step derivation

   1. +-commutative 36.7%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} + -1 \cdot b\right)}}{3 \cdot a} \]

   2. mul-1-neg 36.7%

      \[\leadsto \frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} + \color{blue}{\left(-b\right)}\right)}{3 \cdot a} \]

   3. unsub-neg 36.7%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}}{3 \cdot a} \]

   4. associate-/l* 37.9%

      \[\leadsto \frac{\left(-b\right) + \left(1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}} - b\right)}{3 \cdot a} \]

   5. associate-*r/ 37.9%

      \[\leadsto \frac{\left(-b\right) + \left(\color{blue}{\frac{1.5 \cdot a}{\frac{b}{c}}} - b\right)}{3 \cdot a} \]

5. Simplified 37.9%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(\frac{1.5 \cdot a}{\frac{b}{c}} - b\right)}}{3 \cdot a} \]

6. Taylor expanded in b around 0 10.9%

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

7. Final simplification 10.9%

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

Reproduce

herbie shell --seed 2024022 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))