Cubic critical

Percentage Accurate: 52.5% → 84.3%

Time: 14.9s

Alternatives: 17

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: 52.5% 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: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3 + \frac{{b}^{2}}{c}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+28)
   (* (/ b a) -0.6666666666666666)
   (if (<= b 4.4e-97)
     (/ (- (sqrt (* c (+ (* a -3.0) (/ (pow b 2.0) c)))) b) (* a 3.0))
     (/ 1.0 (* b (- (/ (* a 1.5) (pow b 2.0)) (/ 2.0 c)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+28) {
		tmp = (b / a) * -0.6666666666666666;
	} else if (b <= 4.4e-97) {
		tmp = (sqrt((c * ((a * -3.0) + (pow(b, 2.0) / c)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / pow(b, 2.0)) - (2.0 / 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 <= (-4.5d+28)) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else if (b <= 4.4d-97) then
        tmp = (sqrt((c * ((a * (-3.0d0)) + ((b ** 2.0d0) / c)))) - b) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (b * (((a * 1.5d0) / (b ** 2.0d0)) - (2.0d0 / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+28) {
		tmp = (b / a) * -0.6666666666666666;
	} else if (b <= 4.4e-97) {
		tmp = (Math.sqrt((c * ((a * -3.0) + (Math.pow(b, 2.0) / c)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / Math.pow(b, 2.0)) - (2.0 / c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e+28:
		tmp = (b / a) * -0.6666666666666666
	elif b <= 4.4e-97:
		tmp = (math.sqrt((c * ((a * -3.0) + (math.pow(b, 2.0) / c)))) - b) / (a * 3.0)
	else:
		tmp = 1.0 / (b * (((a * 1.5) / math.pow(b, 2.0)) - (2.0 / c)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e+28)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	elseif (b <= 4.4e-97)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(Float64(a * -3.0) + Float64((b ^ 2.0) / c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(Float64(a * 1.5) / (b ^ 2.0)) - Float64(2.0 / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.5e+28)
		tmp = (b / a) * -0.6666666666666666;
	elseif (b <= 4.4e-97)
		tmp = (sqrt((c * ((a * -3.0) + ((b ^ 2.0) / c)))) - b) / (a * 3.0);
	else
		tmp = 1.0 / (b * (((a * 1.5) / (b ^ 2.0)) - (2.0 / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+28], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], If[LessEqual[b, 4.4e-97], N[(N[(N[Sqrt[N[(c * N[(N[(a * -3.0), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.4999999999999997e28

   1. Initial program 56.9%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in b around -inf 92.9%

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

      1. *-commutative 92.9%

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

   7. Simplified 92.9%

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

   if -4.4999999999999997e28 < b < 4.3999999999999998e-97

   1. Initial program 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in c around inf 85.7%

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

   if 4.3999999999999998e-97 < b

   1. Initial program 16.9%

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

      1. add-cube-cbrt 16.9%

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

      2. pow3 16.9%

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

   4. Applied egg-rr 16.9%

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

      1. clear-num 16.8%

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

      2. inv-pow 16.8%

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

      3. neg-mul-1 16.8%

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

      4. fma-define 16.8%

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

      5. pow2 16.8%

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

      6. rem-cube-cbrt 16.9%

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

      7. associate-*l* 16.9%

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

   6. Applied egg-rr 16.9%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 16.9%

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

      2. associate-/l* 16.9%

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

      3. unpow2 16.9%

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

      4. *-commutative 16.9%

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

      5. fmm-undef 16.9%

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

      6. distribute-rgt-neg-in 16.9%

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

      7. metadata-eval 16.9%

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

      8. rem-cube-cbrt 16.9%

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

      9. associate-*r* 16.9%

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

      10. rem-cube-cbrt 16.9%

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

   8. Simplified 16.9%

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

   9. Taylor expanded in b around inf 82.6%

      \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)}} \]
   10. Step-by-step derivation

       1. associate-*r/ 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\color{blue}{\frac{1.5 \cdot a}{{b}^{2}}} - 2 \cdot \frac{1}{c}\right)} \]

       2. *-commutative 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\frac{\color{blue}{a \cdot 1.5}}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)} \]

       3. associate-*r/ 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \color{blue}{\frac{2 \cdot 1}{c}}\right)} \]

       4. metadata-eval 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{\color{blue}{2}}{c}\right)} \]

   11. Simplified 82.6%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3 + \frac{{b}^{2}}{c}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+28)
   (* (/ b a) -0.6666666666666666)
   (if (<= b 1.95e-95)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (/ 1.0 (* b (- (/ (* a 1.5) (pow b 2.0)) (/ 2.0 c)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+28) {
		tmp = (b / a) * -0.6666666666666666;
	} else if (b <= 1.95e-95) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / pow(b, 2.0)) - (2.0 / 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 <= (-4.5d+28)) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else if (b <= 1.95d-95) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (b * (((a * 1.5d0) / (b ** 2.0d0)) - (2.0d0 / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+28) {
		tmp = (b / a) * -0.6666666666666666;
	} else if (b <= 1.95e-95) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / Math.pow(b, 2.0)) - (2.0 / c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e+28:
		tmp = (b / a) * -0.6666666666666666
	elif b <= 1.95e-95:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = 1.0 / (b * (((a * 1.5) / math.pow(b, 2.0)) - (2.0 / c)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e+28)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	elseif (b <= 1.95e-95)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(Float64(a * 1.5) / (b ^ 2.0)) - Float64(2.0 / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.5e+28)
		tmp = (b / a) * -0.6666666666666666;
	elseif (b <= 1.95e-95)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = 1.0 / (b * (((a * 1.5) / (b ^ 2.0)) - (2.0 / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+28], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], If[LessEqual[b, 1.95e-95], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.4999999999999997e28

   1. Initial program 56.9%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in b around -inf 92.9%

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

      1. *-commutative 92.9%

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

   7. Simplified 92.9%

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

   if -4.4999999999999997e28 < b < 1.95e-95

   1. Initial program 85.7%

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

   if 1.95e-95 < b

   1. Initial program 16.9%

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

      1. add-cube-cbrt 16.9%

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

      2. pow3 16.9%

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

   4. Applied egg-rr 16.9%

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

      1. clear-num 16.8%

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

      2. inv-pow 16.8%

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

      3. neg-mul-1 16.8%

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

      4. fma-define 16.8%

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

      5. pow2 16.8%

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

      6. rem-cube-cbrt 16.9%

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

      7. associate-*l* 16.9%

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

   6. Applied egg-rr 16.9%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 16.9%

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

      2. associate-/l* 16.9%

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

      3. unpow2 16.9%

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

      4. *-commutative 16.9%

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

      5. fmm-undef 16.9%

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

      6. distribute-rgt-neg-in 16.9%

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

      7. metadata-eval 16.9%

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

      8. rem-cube-cbrt 16.9%

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

      9. associate-*r* 16.9%

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

      10. rem-cube-cbrt 16.9%

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

   8. Simplified 16.9%

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

   9. Taylor expanded in b around inf 82.6%

      \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)}} \]
   10. Step-by-step derivation

       1. associate-*r/ 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\color{blue}{\frac{1.5 \cdot a}{{b}^{2}}} - 2 \cdot \frac{1}{c}\right)} \]

       2. *-commutative 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\frac{\color{blue}{a \cdot 1.5}}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)} \]

       3. associate-*r/ 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \color{blue}{\frac{2 \cdot 1}{c}}\right)} \]

       4. metadata-eval 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{\color{blue}{2}}{c}\right)} \]

   11. Simplified 82.6%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+28)
   (* (/ b a) -0.6666666666666666)
   (if (<= b 4.8e-99)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (/ 1.0 (* b (- (/ (* a 1.5) (pow b 2.0)) (/ 2.0 c)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+28) {
		tmp = (b / a) * -0.6666666666666666;
	} else if (b <= 4.8e-99) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / pow(b, 2.0)) - (2.0 / 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 <= (-4.5d+28)) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else if (b <= 4.8d-99) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (b * (((a * 1.5d0) / (b ** 2.0d0)) - (2.0d0 / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+28) {
		tmp = (b / a) * -0.6666666666666666;
	} else if (b <= 4.8e-99) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / Math.pow(b, 2.0)) - (2.0 / c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e+28:
		tmp = (b / a) * -0.6666666666666666
	elif b <= 4.8e-99:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = 1.0 / (b * (((a * 1.5) / math.pow(b, 2.0)) - (2.0 / c)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e+28)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	elseif (b <= 4.8e-99)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(Float64(a * 1.5) / (b ^ 2.0)) - Float64(2.0 / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.5e+28)
		tmp = (b / a) * -0.6666666666666666;
	elseif (b <= 4.8e-99)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = 1.0 / (b * (((a * 1.5) / (b ^ 2.0)) - (2.0 / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+28], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], If[LessEqual[b, 4.8e-99], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.4999999999999997e28

   1. Initial program 56.9%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in b around -inf 92.9%

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

      1. *-commutative 92.9%

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

   7. Simplified 92.9%

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

   if -4.4999999999999997e28 < b < 4.8000000000000001e-99

   1. Initial program 85.7%

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

      1. sqr-neg 85.7%

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

      2. sqr-neg 85.7%

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

      3. associate-*l* 85.6%

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

   3. Simplified 85.6%

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

   if 4.8000000000000001e-99 < b

   1. Initial program 16.9%

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

      1. add-cube-cbrt 16.9%

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

      2. pow3 16.9%

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

   4. Applied egg-rr 16.9%

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

      1. clear-num 16.8%

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

      2. inv-pow 16.8%

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

      3. neg-mul-1 16.8%

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

      4. fma-define 16.8%

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

      5. pow2 16.8%

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

      6. rem-cube-cbrt 16.9%

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

      7. associate-*l* 16.9%

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

   6. Applied egg-rr 16.9%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 16.9%

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

      2. associate-/l* 16.9%

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

      3. unpow2 16.9%

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

      4. *-commutative 16.9%

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

      5. fmm-undef 16.9%

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

      6. distribute-rgt-neg-in 16.9%

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

      7. metadata-eval 16.9%

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

      8. rem-cube-cbrt 16.9%

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

      9. associate-*r* 16.9%

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

      10. rem-cube-cbrt 16.9%

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

   8. Simplified 16.9%

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

   9. Taylor expanded in b around inf 82.6%

      \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)}} \]
   10. Step-by-step derivation

       1. associate-*r/ 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\color{blue}{\frac{1.5 \cdot a}{{b}^{2}}} - 2 \cdot \frac{1}{c}\right)} \]

       2. *-commutative 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\frac{\color{blue}{a \cdot 1.5}}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)} \]

       3. associate-*r/ 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \color{blue}{\frac{2 \cdot 1}{c}}\right)} \]

       4. metadata-eval 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{\color{blue}{2}}{c}\right)} \]

   11. Simplified 82.6%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-97}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.8e-97)
   (/ (* b -2.0) (* a 3.0))
   (if (<= b 6.4e-96)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ 1.0 (* b (- (/ (* a 1.5) (pow b 2.0)) (/ 2.0 c)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-97) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 6.4e-96) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / pow(b, 2.0)) - (2.0 / 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 <= (-1.8d-97)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else if (b <= 6.4d-96) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (b * (((a * 1.5d0) / (b ** 2.0d0)) - (2.0d0 / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.8e-97) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 6.4e-96) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (b * (((a * 1.5) / Math.pow(b, 2.0)) - (2.0 / c)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.8e-97:
		tmp = (b * -2.0) / (a * 3.0)
	elif b <= 6.4e-96:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = 1.0 / (b * (((a * 1.5) / math.pow(b, 2.0)) - (2.0 / c)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.8e-97)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	elseif (b <= 6.4e-96)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(b * Float64(Float64(Float64(a * 1.5) / (b ^ 2.0)) - Float64(2.0 / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.8e-97)
		tmp = (b * -2.0) / (a * 3.0);
	elseif (b <= 6.4e-96)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = 1.0 / (b * (((a * 1.5) / (b ^ 2.0)) - (2.0 / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.8e-97], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.4e-96], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(b * N[(N[(N[(a * 1.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(2.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.79999999999999999e-97

   1. Initial program 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in b around -inf 86.9%

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

      1. *-commutative 86.9%

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

   7. Simplified 86.9%

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

   if -1.79999999999999999e-97 < b < 6.40000000000000023e-96

   1. Initial program 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in b around 0 74.1%

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

      1. associate-*r* 74.2%

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

      2. *-commutative 74.2%

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

   7. Simplified 74.2%

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

   if 6.40000000000000023e-96 < b

   1. Initial program 16.9%

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

      1. add-cube-cbrt 16.9%

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

      2. pow3 16.9%

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

   4. Applied egg-rr 16.9%

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

      1. clear-num 16.8%

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

      2. inv-pow 16.8%

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

      3. neg-mul-1 16.8%

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

      4. fma-define 16.8%

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

      5. pow2 16.8%

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

      6. rem-cube-cbrt 16.9%

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

      7. associate-*l* 16.9%

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

   6. Applied egg-rr 16.9%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right)}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 16.9%

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

      2. associate-/l* 16.9%

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

      3. unpow2 16.9%

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

      4. *-commutative 16.9%

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

      5. fmm-undef 16.9%

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

      6. distribute-rgt-neg-in 16.9%

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

      7. metadata-eval 16.9%

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

      8. rem-cube-cbrt 16.9%

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

      9. associate-*r* 16.9%

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

      10. rem-cube-cbrt 16.9%

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

   8. Simplified 16.9%

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

   9. Taylor expanded in b around inf 82.6%

      \[\leadsto \frac{1}{\color{blue}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)}} \]
   10. Step-by-step derivation

       1. associate-*r/ 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\color{blue}{\frac{1.5 \cdot a}{{b}^{2}}} - 2 \cdot \frac{1}{c}\right)} \]

       2. *-commutative 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\frac{\color{blue}{a \cdot 1.5}}{{b}^{2}} - 2 \cdot \frac{1}{c}\right)} \]

       3. associate-*r/ 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \color{blue}{\frac{2 \cdot 1}{c}}\right)} \]

       4. metadata-eval 82.6%

          \[\leadsto \frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{\color{blue}{2}}{c}\right)} \]

   11. Simplified 82.6%

       \[\leadsto \frac{1}{\color{blue}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-97}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b \cdot \left(\frac{a \cdot 1.5}{{b}^{2}} - \frac{2}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-156}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\frac{1}{-0.3333333333333333 \cdot \frac{a}{c}}} \cdot \left(--0.3333333333333333\right)\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-137}:\\ \;\;\;\;-0.3333333333333333 \cdot \sqrt{c \cdot \frac{-3}{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e-156)
   (/ (* b -2.0) (* a 3.0))
   (if (<= b 1.8e-245)
     (* (sqrt (/ 1.0 (* -0.3333333333333333 (/ a c)))) (- -0.3333333333333333))
     (if (<= b 3.5e-137)
       (* -0.3333333333333333 (sqrt (* c (/ -3.0 a))))
       (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e-156) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 1.8e-245) {
		tmp = sqrt((1.0 / (-0.3333333333333333 * (a / c)))) * -(-0.3333333333333333);
	} else if (b <= 3.5e-137) {
		tmp = -0.3333333333333333 * sqrt((c * (-3.0 / 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 <= (-7d-156)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else if (b <= 1.8d-245) then
        tmp = sqrt((1.0d0 / ((-0.3333333333333333d0) * (a / c)))) * -(-0.3333333333333333d0)
    else if (b <= 3.5d-137) then
        tmp = (-0.3333333333333333d0) * sqrt((c * ((-3.0d0) / 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 <= -7e-156) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 1.8e-245) {
		tmp = Math.sqrt((1.0 / (-0.3333333333333333 * (a / c)))) * -(-0.3333333333333333);
	} else if (b <= 3.5e-137) {
		tmp = -0.3333333333333333 * Math.sqrt((c * (-3.0 / a)));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7e-156:
		tmp = (b * -2.0) / (a * 3.0)
	elif b <= 1.8e-245:
		tmp = math.sqrt((1.0 / (-0.3333333333333333 * (a / c)))) * -(-0.3333333333333333)
	elif b <= 3.5e-137:
		tmp = -0.3333333333333333 * math.sqrt((c * (-3.0 / a)))
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7e-156)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	elseif (b <= 1.8e-245)
		tmp = Float64(sqrt(Float64(1.0 / Float64(-0.3333333333333333 * Float64(a / c)))) * Float64(-(-0.3333333333333333)));
	elseif (b <= 3.5e-137)
		tmp = Float64(-0.3333333333333333 * sqrt(Float64(c * Float64(-3.0 / 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 <= -7e-156)
		tmp = (b * -2.0) / (a * 3.0);
	elseif (b <= 1.8e-245)
		tmp = sqrt((1.0 / (-0.3333333333333333 * (a / c)))) * -(-0.3333333333333333);
	elseif (b <= 3.5e-137)
		tmp = -0.3333333333333333 * sqrt((c * (-3.0 / a)));
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7e-156], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-245], N[(N[Sqrt[N[(1.0 / N[(-0.3333333333333333 * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (--0.3333333333333333)), $MachinePrecision], If[LessEqual[b, 3.5e-137], N[(-0.3333333333333333 * N[Sqrt[N[(c * N[(-3.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.9999999999999999e-156

   1. Initial program 69.2%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 82.2%

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

      1. *-commutative 82.2%

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

   7. Simplified 82.2%

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

   if -6.9999999999999999e-156 < b < 1.8e-245

   1. Initial program 76.9%

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

      1. add-cube-cbrt 76.3%

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

      2. pow3 76.4%

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

   4. Applied egg-rr 76.4%

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

   5. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

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

      3. rem-square-sqrt 49.0%

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

      4. rem-cube-cbrt 49.3%

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

   7. Simplified 49.3%

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

      1. clear-num 49.3%

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

      2. inv-pow 49.3%

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

      3. *-un-lft-identity 49.3%

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

      4. *-commutative 49.3%

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

      5. times-frac 49.5%

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

      6. metadata-eval 49.5%

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

   9. Applied egg-rr 49.5%

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

       1. unpow-1 49.5%

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

   11. Simplified 49.5%

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

   if 1.8e-245 < b < 3.5000000000000001e-137

   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. Step-by-step derivation

      1. add-cube-cbrt 79.6%

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

      2. pow3 79.6%

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

   4. Applied egg-rr 79.6%

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

   5. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

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

      3. rem-square-sqrt 24.2%

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

      4. rem-cube-cbrt 24.4%

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

   7. Simplified 24.4%

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

      1. mul-1-neg 24.4%

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

      2. distribute-rgt-neg-out 24.4%

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

      3. add-sqr-sqrt 24.4%

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

      4. sqr-neg 24.4%

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

      5. mul-1-neg 24.4%

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

      6. mul-1-neg 24.4%

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

      7. sqrt-unprod 0.2%

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

      8. add-sqr-sqrt 57.5%

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

      9. associate-*r* 57.5%

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

      10. metadata-eval 57.5%

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

      11. associate-/l* 57.6%

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

   9. Applied egg-rr 57.6%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \sqrt{c \cdot \frac{-3}{a}}} \]
   10. Step-by-step derivation

       1. distribute-lft-neg-in 57.6%

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

       2. metadata-eval 57.6%

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

   11. Simplified 57.6%

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

   if 3.5000000000000001e-137 < b

   1. Initial program 20.5%

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

   3. Simplified 20.5%

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

   5. Taylor expanded in b around inf 79.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-156}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\frac{1}{-0.3333333333333333 \cdot \frac{a}{c}}} \cdot \left(--0.3333333333333333\right)\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-137}:\\ \;\;\;\;-0.3333333333333333 \cdot \sqrt{c \cdot \frac{-3}{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-158}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 7.1 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{c \cdot -3}{a}} \cdot \left(--0.3333333333333333\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-139}:\\ \;\;\;\;-0.3333333333333333 \cdot \sqrt{c \cdot \frac{-3}{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.3e-158)
   (/ (* b -2.0) (* a 3.0))
   (if (<= b 7.1e-246)
     (* (sqrt (/ (* c -3.0) a)) (- -0.3333333333333333))
     (if (<= b 5e-139)
       (* -0.3333333333333333 (sqrt (* c (/ -3.0 a))))
       (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.3e-158) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 7.1e-246) {
		tmp = sqrt(((c * -3.0) / a)) * -(-0.3333333333333333);
	} else if (b <= 5e-139) {
		tmp = -0.3333333333333333 * sqrt((c * (-3.0 / 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 <= (-1.3d-158)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else if (b <= 7.1d-246) then
        tmp = sqrt(((c * (-3.0d0)) / a)) * -(-0.3333333333333333d0)
    else if (b <= 5d-139) then
        tmp = (-0.3333333333333333d0) * sqrt((c * ((-3.0d0) / 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 <= -1.3e-158) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 7.1e-246) {
		tmp = Math.sqrt(((c * -3.0) / a)) * -(-0.3333333333333333);
	} else if (b <= 5e-139) {
		tmp = -0.3333333333333333 * Math.sqrt((c * (-3.0 / a)));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.3e-158:
		tmp = (b * -2.0) / (a * 3.0)
	elif b <= 7.1e-246:
		tmp = math.sqrt(((c * -3.0) / a)) * -(-0.3333333333333333)
	elif b <= 5e-139:
		tmp = -0.3333333333333333 * math.sqrt((c * (-3.0 / a)))
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.3e-158)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	elseif (b <= 7.1e-246)
		tmp = Float64(sqrt(Float64(Float64(c * -3.0) / a)) * Float64(-(-0.3333333333333333)));
	elseif (b <= 5e-139)
		tmp = Float64(-0.3333333333333333 * sqrt(Float64(c * Float64(-3.0 / 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 <= -1.3e-158)
		tmp = (b * -2.0) / (a * 3.0);
	elseif (b <= 7.1e-246)
		tmp = sqrt(((c * -3.0) / a)) * -(-0.3333333333333333);
	elseif (b <= 5e-139)
		tmp = -0.3333333333333333 * sqrt((c * (-3.0 / a)));
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.3e-158], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.1e-246], N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] / a), $MachinePrecision]], $MachinePrecision] * (--0.3333333333333333)), $MachinePrecision], If[LessEqual[b, 5e-139], N[(-0.3333333333333333 * N[Sqrt[N[(c * N[(-3.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.3e-158

   1. Initial program 69.2%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 82.2%

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

      1. *-commutative 82.2%

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

   7. Simplified 82.2%

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

   if -1.3e-158 < b < 7.10000000000000036e-246

   1. Initial program 76.9%

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

      1. add-cube-cbrt 76.3%

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

      2. pow3 76.4%

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

   4. Applied egg-rr 76.4%

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

   5. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

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

      3. rem-square-sqrt 49.0%

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

      4. rem-cube-cbrt 49.3%

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

   7. Simplified 49.3%

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

   if 7.10000000000000036e-246 < b < 5.00000000000000034e-139

   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. Step-by-step derivation

      1. add-cube-cbrt 79.6%

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

      2. pow3 79.6%

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

   4. Applied egg-rr 79.6%

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

   5. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

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

      3. rem-square-sqrt 24.2%

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

      4. rem-cube-cbrt 24.4%

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

   7. Simplified 24.4%

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

      1. mul-1-neg 24.4%

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

      2. distribute-rgt-neg-out 24.4%

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

      3. add-sqr-sqrt 24.4%

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

      4. sqr-neg 24.4%

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

      5. mul-1-neg 24.4%

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

      6. mul-1-neg 24.4%

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

      7. sqrt-unprod 0.2%

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

      8. add-sqr-sqrt 57.5%

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

      9. associate-*r* 57.5%

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

      10. metadata-eval 57.5%

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

      11. associate-/l* 57.6%

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

   9. Applied egg-rr 57.6%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \sqrt{c \cdot \frac{-3}{a}}} \]
   10. Step-by-step derivation

       1. distribute-lft-neg-in 57.6%

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

       2. metadata-eval 57.6%

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

   11. Simplified 57.6%

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

   if 5.00000000000000034e-139 < b

   1. Initial program 20.5%

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

   3. Simplified 20.5%

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

   5. Taylor expanded in b around inf 79.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-158}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 7.1 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{c \cdot -3}{a}} \cdot \left(--0.3333333333333333\right)\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-139}:\\ \;\;\;\;-0.3333333333333333 \cdot \sqrt{c \cdot \frac{-3}{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{c \cdot \left(\frac{-3}{a} \cdot 0.1111111111111111\right)}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-147}:\\ \;\;\;\;-0.3333333333333333 \cdot \sqrt{c \cdot \frac{-3}{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.8e-156)
   (/ (* b -2.0) (* a 3.0))
   (if (<= b 5.4e-269)
     (sqrt (* c (* (/ -3.0 a) 0.1111111111111111)))
     (if (<= b 8.2e-147)
       (* -0.3333333333333333 (sqrt (* c (/ -3.0 a))))
       (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.8e-156) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 5.4e-269) {
		tmp = sqrt((c * ((-3.0 / a) * 0.1111111111111111)));
	} else if (b <= 8.2e-147) {
		tmp = -0.3333333333333333 * sqrt((c * (-3.0 / 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 <= (-6.8d-156)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else if (b <= 5.4d-269) then
        tmp = sqrt((c * (((-3.0d0) / a) * 0.1111111111111111d0)))
    else if (b <= 8.2d-147) then
        tmp = (-0.3333333333333333d0) * sqrt((c * ((-3.0d0) / 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 <= -6.8e-156) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 5.4e-269) {
		tmp = Math.sqrt((c * ((-3.0 / a) * 0.1111111111111111)));
	} else if (b <= 8.2e-147) {
		tmp = -0.3333333333333333 * Math.sqrt((c * (-3.0 / a)));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.8e-156:
		tmp = (b * -2.0) / (a * 3.0)
	elif b <= 5.4e-269:
		tmp = math.sqrt((c * ((-3.0 / a) * 0.1111111111111111)))
	elif b <= 8.2e-147:
		tmp = -0.3333333333333333 * math.sqrt((c * (-3.0 / a)))
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.8e-156)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	elseif (b <= 5.4e-269)
		tmp = sqrt(Float64(c * Float64(Float64(-3.0 / a) * 0.1111111111111111)));
	elseif (b <= 8.2e-147)
		tmp = Float64(-0.3333333333333333 * sqrt(Float64(c * Float64(-3.0 / 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 <= -6.8e-156)
		tmp = (b * -2.0) / (a * 3.0);
	elseif (b <= 5.4e-269)
		tmp = sqrt((c * ((-3.0 / a) * 0.1111111111111111)));
	elseif (b <= 8.2e-147)
		tmp = -0.3333333333333333 * sqrt((c * (-3.0 / a)));
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.8e-156], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.4e-269], N[Sqrt[N[(c * N[(N[(-3.0 / a), $MachinePrecision] * 0.1111111111111111), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[b, 8.2e-147], N[(-0.3333333333333333 * N[Sqrt[N[(c * N[(-3.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -6.79999999999999981e-156

   1. Initial program 69.2%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 82.2%

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

      1. *-commutative 82.2%

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

   7. Simplified 82.2%

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

   if -6.79999999999999981e-156 < b < 5.40000000000000031e-269

   1. Initial program 74.9%

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

      1. add-cube-cbrt 74.4%

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

      2. pow3 74.4%

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

   4. Applied egg-rr 74.4%

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

   5. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

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

      3. rem-square-sqrt 48.8%

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

      4. rem-cube-cbrt 49.2%

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

   7. Simplified 49.2%

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

      1. add-sqr-sqrt 49.0%

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

      2. sqrt-unprod 49.2%

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

      3. *-commutative 49.2%

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

      4. *-commutative 49.2%

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

      5. swap-sqr 49.1%

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

      6. mul-1-neg 49.1%

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

      7. mul-1-neg 49.1%

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

      8. sqr-neg 49.1%

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

      9. add-sqr-sqrt 49.3%

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

      10. associate-/l* 49.2%

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

      11. metadata-eval 49.2%

          \[\leadsto \sqrt{\left(c \cdot \frac{-3}{a}\right) \cdot \color{blue}{0.1111111111111111}} \]

   9. Applied egg-rr 49.2%

      \[\leadsto \color{blue}{\sqrt{\left(c \cdot \frac{-3}{a}\right) \cdot 0.1111111111111111}} \]
   10. Step-by-step derivation

       1. associate-*l* 49.2%

          \[\leadsto \sqrt{\color{blue}{c \cdot \left(\frac{-3}{a} \cdot 0.1111111111111111\right)}} \]

   11. Simplified 49.2%

       \[\leadsto \color{blue}{\sqrt{c \cdot \left(\frac{-3}{a} \cdot 0.1111111111111111\right)}} \]

   if 5.40000000000000031e-269 < b < 8.1999999999999999e-147

   1. Initial program 82.2%

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

      1. add-cube-cbrt 81.2%

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

      2. pow3 81.4%

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

   4. Applied egg-rr 81.4%

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

   5. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

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

      3. rem-square-sqrt 26.5%

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

      4. rem-cube-cbrt 26.8%

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

   7. Simplified 26.8%

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

      1. mul-1-neg 26.8%

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

      2. distribute-rgt-neg-out 26.8%

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

      3. add-sqr-sqrt 26.8%

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

      4. sqr-neg 26.8%

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

      5. mul-1-neg 26.8%

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

      6. mul-1-neg 26.8%

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

      7. sqrt-unprod 0.2%

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

      8. add-sqr-sqrt 56.8%

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

      9. associate-*r* 56.8%

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

      10. metadata-eval 56.8%

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

      11. associate-/l* 56.9%

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

   9. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \sqrt{c \cdot \frac{-3}{a}}} \]
   10. Step-by-step derivation

       1. distribute-lft-neg-in 56.9%

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

       2. metadata-eval 56.9%

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

   11. Simplified 56.9%

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

   if 8.1999999999999999e-147 < b

   1. Initial program 20.5%

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

   3. Simplified 20.5%

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

   5. Taylor expanded in b around inf 79.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-269}:\\ \;\;\;\;\sqrt{c \cdot \left(\frac{-3}{a} \cdot 0.1111111111111111\right)}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-147}:\\ \;\;\;\;-0.3333333333333333 \cdot \sqrt{c \cdot \frac{-3}{a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-97)
   (/ (* b -2.0) (* a 3.0))
   (if (<= b 2.55e-95)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-97) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 2.55e-95) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} 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 <= (-1.9d-97)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else if (b <= 2.55d-95) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    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 <= -1.9e-97) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 2.55e-95) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.9e-97:
		tmp = (b * -2.0) / (a * 3.0)
	elif b <= 2.55e-95:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e-97)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	elseif (b <= 2.55e-95)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	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 <= -1.9e-97)
		tmp = (b * -2.0) / (a * 3.0);
	elseif (b <= 2.55e-95)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e-97], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.55e-95], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9e-97

   1. Initial program 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in b around -inf 86.9%

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

      1. *-commutative 86.9%

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

   7. Simplified 86.9%

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

   if -1.9e-97 < b < 2.55e-95

   1. Initial program 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in b around 0 74.1%

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

      1. associate-*r* 74.2%

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

      2. *-commutative 74.2%

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

   7. Simplified 74.2%

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

   if 2.55e-95 < b

   1. Initial program 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in b around inf 82.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.2e-98)
   (/ (* b -2.0) (* a 3.0))
   (if (<= b 4.3e-95)
     (/ (- (sqrt (* -3.0 (* a c))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e-98) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 4.3e-95) {
		tmp = (sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	} 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 <= (-7.2d-98)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else if (b <= 4.3d-95) then
        tmp = (sqrt(((-3.0d0) * (a * c))) - b) / (a * 3.0d0)
    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 <= -7.2e-98) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 4.3e-95) {
		tmp = (Math.sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.2e-98:
		tmp = (b * -2.0) / (a * 3.0)
	elif b <= 4.3e-95:
		tmp = (math.sqrt((-3.0 * (a * c))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.2e-98)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	elseif (b <= 4.3e-95)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / Float64(a * 3.0));
	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 <= -7.2e-98)
		tmp = (b * -2.0) / (a * 3.0);
	elseif (b <= 4.3e-95)
		tmp = (sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.2e-98], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.3e-95], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.2000000000000005e-98

   1. Initial program 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in b around -inf 86.9%

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

      1. *-commutative 86.9%

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

   7. Simplified 86.9%

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

   if -7.2000000000000005e-98 < b < 4.29999999999999997e-95

   1. Initial program 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in b around 0 74.1%

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

   if 4.29999999999999997e-95 < b

   1. Initial program 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in b around inf 82.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-95}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.6e-110)
   (/ (* b -2.0) (* a 3.0))
   (if (<= b 1.2e-95)
     (* 0.3333333333333333 (/ (+ b (sqrt (* c (* a -3.0)))) a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.6e-110) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 1.2e-95) {
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -3.0)))) / 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 <= (-7.6d-110)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else if (b <= 1.2d-95) then
        tmp = 0.3333333333333333d0 * ((b + sqrt((c * (a * (-3.0d0))))) / 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 <= -7.6e-110) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 1.2e-95) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.6e-110:
		tmp = (b * -2.0) / (a * 3.0)
	elif b <= 1.2e-95:
		tmp = 0.3333333333333333 * ((b + math.sqrt((c * (a * -3.0)))) / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.6e-110)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	elseif (b <= 1.2e-95)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(c * Float64(a * -3.0)))) / 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 <= -7.6e-110)
		tmp = (b * -2.0) / (a * 3.0);
	elseif (b <= 1.2e-95)
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -3.0)))) / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.6e-110], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.2e-95], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.5999999999999996e-110

   1. Initial program 66.9%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in b around -inf 86.3%

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

      1. *-commutative 86.3%

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

   7. Simplified 86.3%

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

   if -7.5999999999999996e-110 < b < 1.2e-95

   1. Initial program 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in b around 0 74.5%

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

      1. *-un-lft-identity 74.5%

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

      2. times-frac 74.6%

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

      3. metadata-eval 74.6%

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

      4. sub-neg 74.6%

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

      5. metadata-eval 74.6%

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

      6. distribute-lft-neg-in 74.6%

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

      7. associate-*r* 74.6%

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

      8. *-commutative 74.6%

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

      9. distribute-rgt-neg-in 74.6%

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

      10. *-commutative 74.6%

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

      11. distribute-rgt-neg-in 74.6%

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

      12. metadata-eval 74.6%

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

      13. add-sqr-sqrt 34.1%

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

      14. sqrt-unprod 74.2%

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

      15. sqr-neg 74.2%

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

      16. sqrt-prod 40.4%

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

      17. add-sqr-sqrt 72.7%

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

   7. Applied egg-rr 72.7%

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

   if 1.2e-95 < b

   1. Initial program 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in b around inf 82.4%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-110}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-95}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.68 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.4e-107)
   (/ (* b -2.0) (* a 3.0))
   (if (<= b 1.68e-96)
     (/ (sqrt (* a (* c -3.0))) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.4e-107) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 1.68e-96) {
		tmp = sqrt((a * (c * -3.0))) / (a * 3.0);
	} 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 <= (-4.4d-107)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else if (b <= 1.68d-96) then
        tmp = sqrt((a * (c * (-3.0d0)))) / (a * 3.0d0)
    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 <= -4.4e-107) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 1.68e-96) {
		tmp = Math.sqrt((a * (c * -3.0))) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.4e-107:
		tmp = (b * -2.0) / (a * 3.0)
	elif b <= 1.68e-96:
		tmp = math.sqrt((a * (c * -3.0))) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.4e-107)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	elseif (b <= 1.68e-96)
		tmp = Float64(sqrt(Float64(a * Float64(c * -3.0))) / Float64(a * 3.0));
	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 <= -4.4e-107)
		tmp = (b * -2.0) / (a * 3.0);
	elseif (b <= 1.68e-96)
		tmp = sqrt((a * (c * -3.0))) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.4e-107], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.68e-96], N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.40000000000000025e-107

   1. Initial program 66.9%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in b around -inf 86.3%

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

      1. *-commutative 86.3%

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

   7. Simplified 86.3%

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

   if -4.40000000000000025e-107 < b < 1.6800000000000001e-96

   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. Step-by-step derivation

      1. add-cube-cbrt 80.4%

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

      2. pow3 80.5%

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

   4. Applied egg-rr 80.5%

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

   5. Taylor expanded in a around -inf 0.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(\sqrt{a \cdot \left(c \cdot {\left(\sqrt[3]{-3}\right)}^{3}\right)} \cdot {\left(\sqrt{-1}\right)}^{2}\right)}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. mul-1-neg 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 72.6%

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

      5. distribute-lft-neg-in 72.6%

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

      6. metadata-eval 72.6%

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

      7. rem-cube-cbrt 73.0%

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

   7. Simplified 73.0%

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

   if 1.6800000000000001e-96 < b

   1. Initial program 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in b around inf 82.4%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.4 \cdot 10^{-107}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.68 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{c \cdot \left(\frac{-3}{a} \cdot 0.1111111111111111\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e-159)
   (/ (* b -2.0) (* a 3.0))
   (if (<= b 2.7e-179)
     (sqrt (* c (* (/ -3.0 a) 0.1111111111111111)))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-159) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 2.7e-179) {
		tmp = sqrt((c * ((-3.0 / a) * 0.1111111111111111)));
	} 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 <= (-5.5d-159)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    else if (b <= 2.7d-179) then
        tmp = sqrt((c * (((-3.0d0) / a) * 0.1111111111111111d0)))
    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 <= -5.5e-159) {
		tmp = (b * -2.0) / (a * 3.0);
	} else if (b <= 2.7e-179) {
		tmp = Math.sqrt((c * ((-3.0 / a) * 0.1111111111111111)));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.5e-159:
		tmp = (b * -2.0) / (a * 3.0)
	elif b <= 2.7e-179:
		tmp = math.sqrt((c * ((-3.0 / a) * 0.1111111111111111)))
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.5e-159)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	elseif (b <= 2.7e-179)
		tmp = sqrt(Float64(c * Float64(Float64(-3.0 / a) * 0.1111111111111111)));
	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 <= -5.5e-159)
		tmp = (b * -2.0) / (a * 3.0);
	elseif (b <= 2.7e-179)
		tmp = sqrt((c * ((-3.0 / a) * 0.1111111111111111)));
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.5e-159], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.7e-179], N[Sqrt[N[(c * N[(N[(-3.0 / a), $MachinePrecision] * 0.1111111111111111), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.5000000000000003e-159

   1. Initial program 69.2%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 82.2%

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

      1. *-commutative 82.2%

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

   7. Simplified 82.2%

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

   if -5.5000000000000003e-159 < b < 2.69999999999999988e-179

   1. Initial program 76.4%

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

      1. add-cube-cbrt 75.7%

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

      2. pow3 75.8%

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

   4. Applied egg-rr 75.8%

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

   5. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.3333333333333333 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-3}\right)}^{3}}{a}}\right)} \]

      2. unpow2 0.0%

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

      3. rem-square-sqrt 45.9%

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

      4. rem-cube-cbrt 46.2%

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

   7. Simplified 46.2%

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

      1. add-sqr-sqrt 46.1%

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

      2. sqrt-unprod 46.1%

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

      3. *-commutative 46.1%

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

      4. *-commutative 46.1%

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

      5. swap-sqr 46.0%

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

      6. mul-1-neg 46.0%

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

      7. mul-1-neg 46.0%

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

      8. sqr-neg 46.0%

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

      9. add-sqr-sqrt 46.2%

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

      10. associate-/l* 46.0%

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

      11. metadata-eval 46.0%

          \[\leadsto \sqrt{\left(c \cdot \frac{-3}{a}\right) \cdot \color{blue}{0.1111111111111111}} \]

   9. Applied egg-rr 46.0%

      \[\leadsto \color{blue}{\sqrt{\left(c \cdot \frac{-3}{a}\right) \cdot 0.1111111111111111}} \]
   10. Step-by-step derivation

       1. associate-*l* 46.0%

          \[\leadsto \sqrt{\color{blue}{c \cdot \left(\frac{-3}{a} \cdot 0.1111111111111111\right)}} \]

   11. Simplified 46.0%

       \[\leadsto \color{blue}{\sqrt{c \cdot \left(\frac{-3}{a} \cdot 0.1111111111111111\right)}} \]

   if 2.69999999999999988e-179 < b

   1. Initial program 26.0%

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

   3. Simplified 26.0%

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

   5. Taylor expanded in b around inf 74.1%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{c \cdot \left(\frac{-3}{a} \cdot 0.1111111111111111\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ (* b -2.0) (* a 3.0)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (b * -2.0) / (a * 3.0);
	} 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 <= (-1d-310)) then
        tmp = (b * (-2.0d0)) / (a * 3.0d0)
    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 <= -1e-310) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (b * -2.0) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(b * -2.0) / Float64(a * 3.0));
	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 <= -1e-310)
		tmp = (b * -2.0) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in b around -inf 72.9%

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

      1. *-commutative 72.9%

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

   7. Simplified 72.9%

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

   if -9.999999999999969e-311 < b

   1. Initial program 33.9%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in b around inf 62.6%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(b \cdot -2\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (* (* b -2.0) (/ 0.3333333333333333 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (b * -2.0) * (0.3333333333333333 / 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 <= (-1d-310)) then
        tmp = (b * (-2.0d0)) * (0.3333333333333333d0 / 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 <= -1e-310) {
		tmp = (b * -2.0) * (0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (b * -2.0) * (0.3333333333333333 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(Float64(b * -2.0) * Float64(0.3333333333333333 / 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 <= -1e-310)
		tmp = (b * -2.0) * (0.3333333333333333 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(N[(b * -2.0), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.5%

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

      1. add-cube-cbrt 70.4%

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

      2. pow3 70.4%

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

   4. Applied egg-rr 70.4%

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

      1. div-inv 70.3%

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

      2. neg-mul-1 70.3%

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

      3. fma-define 70.3%

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

      4. pow2 70.3%

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

      5. rem-cube-cbrt 70.4%

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

      6. associate-*l* 70.4%

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

   6. Applied egg-rr 70.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{3 \cdot a}} \]
   7. Step-by-step derivation

      1. associate-/r* 70.4%

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

      2. metadata-eval 70.4%

         \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{\color{blue}{0.3333333333333333}}{a} \]

      3. metadata-eval 70.4%

         \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{\color{blue}{0.3333333333333333 \cdot 1}}{a} \]

      4. associate-*r/ 70.4%

         \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{a}\right)} \]

      5. *-commutative 70.4%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right)} \]

      6. associate-*r/ 70.4%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot 1}{a}} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right) \]

      7. metadata-eval 70.4%

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right) \]

      8. unpow2 70.4%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{b \cdot b} - 3 \cdot \left(a \cdot c\right)}\right) \]

      9. *-commutative 70.4%

         \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}\right) \]

      10. fmm-undef 70.4%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 3\right)}}\right) \]

      11. distribute-rgt-neg-in 70.4%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-3\right)}\right)}\right) \]

      12. metadata-eval 70.4%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-3}\right)}\right) \]

      13. rem-cube-cbrt 70.3%

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

      14. associate-*r* 70.3%

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

      15. rem-cube-cbrt 70.5%

          \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot \color{blue}{-3}\right)\right)}\right) \]

   8. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right)} \]

   9. Taylor expanded in b around -inf 72.9%

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

       1. *-commutative 72.9%

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

   11. Simplified 72.9%

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

   if -9.999999999999969e-311 < b

   1. Initial program 33.9%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in b around inf 62.6%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left(b \cdot -2\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (* (/ b a) -0.6666666666666666) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		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 <= (-1d-310)) 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 <= -1e-310) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		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 <= -1e-310)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], 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 < -9.999999999999969e-311

   1. Initial program 70.5%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in b around -inf 72.8%

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

      1. *-commutative 72.8%

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

   7. Simplified 72.8%

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

   if -9.999999999999969e-311 < b

   1. Initial program 33.9%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in b around inf 62.6%

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

Alternative 16: 35.1% 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 53.9%

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

3. Simplified 53.9%

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

5. Taylor expanded in b around inf 29.6%

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

Alternative 17: 11.0% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 53.9%

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

   1. add-cube-cbrt 53.7%

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

   2. pow3 53.7%

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

4. Applied egg-rr 53.7%

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

   1. div-inv 53.7%

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

   2. neg-mul-1 53.7%

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

   3. fma-define 53.7%

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

   4. pow2 53.7%

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

   5. rem-cube-cbrt 53.9%

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

   6. associate-*l* 53.9%

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

6. Applied egg-rr 53.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{3 \cdot a}} \]
7. Step-by-step derivation

   1. associate-/r* 53.9%

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

   2. metadata-eval 53.9%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{\color{blue}{0.3333333333333333}}{a} \]

   3. metadata-eval 53.9%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{\color{blue}{0.3333333333333333 \cdot 1}}{a} \]

   4. associate-*r/ 53.8%

      \[\leadsto \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{a}\right)} \]

   5. *-commutative 53.8%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right)} \]

   6. associate-*r/ 53.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot 1}{a}} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right) \]

   7. metadata-eval 53.9%

      \[\leadsto \frac{\color{blue}{0.3333333333333333}}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right) \]

   8. unpow2 53.9%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{b \cdot b} - 3 \cdot \left(a \cdot c\right)}\right) \]

   9. *-commutative 53.9%

      \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot 3}}\right) \]

   10. fmm-undef 53.8%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 3\right)}}\right) \]

   11. distribute-rgt-neg-in 53.8%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-3\right)}\right)}\right) \]

   12. metadata-eval 53.8%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot \color{blue}{-3}\right)}\right) \]

   13. rem-cube-cbrt 53.7%

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

   14. associate-*r* 53.7%

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

   15. rem-cube-cbrt 53.9%

       \[\leadsto \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot \color{blue}{-3}\right)\right)}\right) \]

8. Simplified 53.9%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right)} \]

9. Taylor expanded in a around 0 8.6%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 8.6%

       \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(b + -1 \cdot b\right)}{a}} \]

    2. distribute-rgt1-in 8.6%

       \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)}}{a} \]

    3. metadata-eval 8.6%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

    4. mul0-lft 8.6%

       \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

    5. metadata-eval 8.6%

       \[\leadsto \frac{\color{blue}{0}}{a} \]

11. Simplified 8.6%

    \[\leadsto \color{blue}{\frac{0}{a}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))