Cubic critical

Percentage Accurate: 51.5% → 85.2%

Time: 14.1s

Alternatives: 15

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e+89)
   (/ (/ b a) -1.5)
   (if (<= b 9.5e-44)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e+89) {
		tmp = (b / a) / -1.5;
	} else if (b <= 9.5e-44) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.8d+89)) then
        tmp = (b / a) / (-1.5d0)
    else if (b <= 9.5d-44) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e+89) {
		tmp = (b / a) / -1.5;
	} else if (b <= 9.5e-44) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.8e+89:
		tmp = (b / a) / -1.5
	elif b <= 9.5e-44:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e+89)
		tmp = Float64(Float64(b / a) / -1.5);
	elseif (b <= 9.5e-44)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.8e+89)
		tmp = (b / a) / -1.5;
	elseif (b <= 9.5e-44)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e+89], N[(N[(b / a), $MachinePrecision] / -1.5), $MachinePrecision], If[LessEqual[b, 9.5e-44], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.80000000000000023e89

   1. Initial program 46.3%

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

      1. sqr-neg 46.3%

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

      2. sqr-neg 46.3%

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

      3. associate-*l* 46.3%

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

   3. Simplified 46.3%

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

   5. Taylor expanded in b around -inf 93.8%

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

      1. *-commutative 93.8%

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

   7. Simplified 93.8%

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

      1. associate-*l/ 93.9%

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

      2. clear-num 93.8%

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

   9. Applied egg-rr 93.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{b \cdot -0.6666666666666666}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 93.7%

          \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(b \cdot -0.6666666666666666\right)} \]

   11. Simplified 93.7%

       \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(b \cdot -0.6666666666666666\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 93.8%

          \[\leadsto \color{blue}{\left(\frac{1}{a} \cdot b\right) \cdot -0.6666666666666666} \]

       2. associate-*l/ 93.8%

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

       3. *-un-lft-identity 93.8%

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

       4. metadata-eval 93.8%

          \[\leadsto \frac{b}{a} \cdot \color{blue}{\frac{1}{-1.5}} \]

       5. div-inv 93.9%

          \[\leadsto \color{blue}{\frac{\frac{b}{a}}{-1.5}} \]

   13. Applied egg-rr 93.9%

       \[\leadsto \color{blue}{\frac{\frac{b}{a}}{-1.5}} \]

   if -3.80000000000000023e89 < b < 9.49999999999999924e-44

   1. Initial program 83.5%

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

   if 9.49999999999999924e-44 < b

   1. Initial program 16.0%

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

      1. sqr-neg 16.0%

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

      2. sqr-neg 16.0%

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

      3. associate-*l* 16.0%

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

   3. Simplified 16.0%

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

   6. Applied egg-rr 27.0%

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

   7. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   8. Step-by-step derivation

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 91.9%

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

   9. Simplified 91.9%

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

   10. Taylor expanded in c around 0 92.1%

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

       1. *-commutative 92.1%

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

   12. Simplified 92.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e+88)
   (/ (/ b a) -1.5)
   (if (<= b 1.45e-43)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e+88) {
		tmp = (b / a) / -1.5;
	} else if (b <= 1.45e-43) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.5d+88)) then
        tmp = (b / a) / (-1.5d0)
    else if (b <= 1.45d-43) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e+88) {
		tmp = (b / a) / -1.5;
	} else if (b <= 1.45e-43) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.5e+88:
		tmp = (b / a) / -1.5
	elif b <= 1.45e-43:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e+88)
		tmp = Float64(Float64(b / a) / -1.5);
	elseif (b <= 1.45e-43)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.5e+88)
		tmp = (b / a) / -1.5;
	elseif (b <= 1.45e-43)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e+88], N[(N[(b / a), $MachinePrecision] / -1.5), $MachinePrecision], If[LessEqual[b, 1.45e-43], 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[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.49999999999999999e88

   1. Initial program 46.3%

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

      1. sqr-neg 46.3%

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

      2. sqr-neg 46.3%

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

      3. associate-*l* 46.3%

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

   3. Simplified 46.3%

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

   5. Taylor expanded in b around -inf 93.8%

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

      1. *-commutative 93.8%

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

   7. Simplified 93.8%

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

      1. associate-*l/ 93.9%

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

      2. clear-num 93.8%

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

   9. Applied egg-rr 93.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{b \cdot -0.6666666666666666}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 93.7%

          \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(b \cdot -0.6666666666666666\right)} \]

   11. Simplified 93.7%

       \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(b \cdot -0.6666666666666666\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 93.8%

          \[\leadsto \color{blue}{\left(\frac{1}{a} \cdot b\right) \cdot -0.6666666666666666} \]

       2. associate-*l/ 93.8%

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

       3. *-un-lft-identity 93.8%

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

       4. metadata-eval 93.8%

          \[\leadsto \frac{b}{a} \cdot \color{blue}{\frac{1}{-1.5}} \]

       5. div-inv 93.9%

          \[\leadsto \color{blue}{\frac{\frac{b}{a}}{-1.5}} \]

   13. Applied egg-rr 93.9%

       \[\leadsto \color{blue}{\frac{\frac{b}{a}}{-1.5}} \]

   if -2.49999999999999999e88 < b < 1.4500000000000001e-43

   1. Initial program 83.5%

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

      1. sqr-neg 83.5%

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

      2. sqr-neg 83.5%

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

      3. associate-*l* 83.5%

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

   3. Simplified 83.5%

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

   if 1.4500000000000001e-43 < b

   1. Initial program 16.0%

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

      1. sqr-neg 16.0%

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

      2. sqr-neg 16.0%

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

      3. associate-*l* 16.0%

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

   3. Simplified 16.0%

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

   6. Applied egg-rr 27.0%

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

   7. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   8. Step-by-step derivation

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 91.9%

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

   9. Simplified 91.9%

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

   10. Taylor expanded in c around 0 92.1%

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

       1. *-commutative 92.1%

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

   12. Simplified 92.1%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{b}{a}}{-1.5}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e-92)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 1.32e-68)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-92) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.32e-68) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.2d-92)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 1.32d-68) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-92) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.32e-68) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.2e-92:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 1.32e-68:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e-92)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 1.32e-68)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.2e-92)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 1.32e-68)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e-92], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.32e-68], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.2000000000000005e-92

   1. Initial program 64.5%

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

      1. sqr-neg 64.5%

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

      2. sqr-neg 64.5%

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

      3. associate-*l* 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in b around -inf 85.7%

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

      1. *-commutative 85.7%

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

   7. Simplified 85.7%

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

      1. associate-*l/ 85.8%

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

   9. Applied egg-rr 85.8%

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

   if -8.2000000000000005e-92 < b < 1.32e-68

   1. Initial program 76.2%

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

      1. sqr-neg 76.2%

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

      2. sqr-neg 76.2%

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

      3. associate-*l* 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in b around 0 76.0%

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

      1. *-commutative 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*r* 76.0%

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

   7. Simplified 76.0%

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

   if 1.32e-68 < b

   1. Initial program 17.4%

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

      1. sqr-neg 17.4%

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

      2. sqr-neg 17.4%

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

      3. associate-*l* 17.4%

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

   3. Simplified 17.4%

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

   6. Applied egg-rr 27.2%

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

   7. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   8. Step-by-step derivation

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 90.5%

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

   9. Simplified 90.5%

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

   10. Taylor expanded in c around 0 90.8%

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

       1. *-commutative 90.8%

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

   12. Simplified 90.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.9 \cdot 10^{-91}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.9e-91)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 5.8e-65)
     (/ (- (sqrt (* (* a c) -3.0)) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.9e-91) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 5.8e-65) {
		tmp = (sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.9d-91)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 5.8d-65) then
        tmp = (sqrt(((a * c) * (-3.0d0))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.9e-91) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 5.8e-65) {
		tmp = (Math.sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.9e-91:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 5.8e-65:
		tmp = (math.sqrt(((a * c) * -3.0)) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.9e-91)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 5.8e-65)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) * -3.0)) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.9e-91)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 5.8e-65)
		tmp = (sqrt(((a * c) * -3.0)) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.9e-91], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 5.8e-65], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.90000000000000025e-91

   1. Initial program 64.5%

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

      1. sqr-neg 64.5%

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

      2. sqr-neg 64.5%

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

      3. associate-*l* 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in b around -inf 85.7%

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

      1. *-commutative 85.7%

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

   7. Simplified 85.7%

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

      1. associate-*l/ 85.8%

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

   9. Applied egg-rr 85.8%

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

   if -5.90000000000000025e-91 < b < 5.7999999999999996e-65

   1. Initial program 76.2%

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

      1. sqr-neg 76.2%

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

      2. sqr-neg 76.2%

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

      3. associate-*l* 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in b around 0 76.0%

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

   if 5.7999999999999996e-65 < b

   1. Initial program 17.4%

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

      1. sqr-neg 17.4%

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

      2. sqr-neg 17.4%

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

      3. associate-*l* 17.4%

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

   3. Simplified 17.4%

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

   6. Applied egg-rr 27.2%

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

   7. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   8. Step-by-step derivation

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 90.5%

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

   9. Simplified 90.5%

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

   10. Taylor expanded in c around 0 90.8%

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

       1. *-commutative 90.8%

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

   12. Simplified 90.8%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.9 \cdot 10^{-91}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-90}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-90)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 1.35e-65)
     (/ (/ (+ b (sqrt (* c (* a -3.0)))) 3.0) a)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-90) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.35e-65) {
		tmp = ((b + sqrt((c * (a * -3.0)))) / 3.0) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.9d-90)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 1.35d-65) then
        tmp = ((b + sqrt((c * (a * (-3.0d0))))) / 3.0d0) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-90) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1.35e-65) {
		tmp = ((b + Math.sqrt((c * (a * -3.0)))) / 3.0) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.9e-90:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 1.35e-65:
		tmp = ((b + math.sqrt((c * (a * -3.0)))) / 3.0) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.9e-90)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 1.35e-65)
		tmp = Float64(Float64(Float64(b + sqrt(Float64(c * Float64(a * -3.0)))) / 3.0) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.9e-90)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 1.35e-65)
		tmp = ((b + sqrt((c * (a * -3.0)))) / 3.0) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.9e-90], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.35e-65], N[(N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9e-90

   1. Initial program 64.5%

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

      1. sqr-neg 64.5%

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

      2. sqr-neg 64.5%

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

      3. associate-*l* 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in b around -inf 85.7%

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

      1. *-commutative 85.7%

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

   7. Simplified 85.7%

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

      1. associate-*l/ 85.8%

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

   9. Applied egg-rr 85.8%

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

   if -1.9e-90 < b < 1.3499999999999999e-65

   1. Initial program 76.2%

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

      1. sqr-neg 76.2%

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

      2. sqr-neg 76.2%

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

      3. associate-*l* 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in b around 0 76.0%

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

      1. *-commutative 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*r* 76.0%

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

   7. Simplified 76.0%

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

      1. clear-num 76.0%

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

      2. inv-pow 76.0%

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

      3. *-un-lft-identity 76.0%

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

      4. times-frac 75.7%

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

      5. metadata-eval 75.7%

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

      6. add-sqr-sqrt 43.9%

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

      7. sqrt-unprod 74.8%

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

      8. sqr-neg 74.8%

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

      9. sqrt-unprod 31.3%

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

      10. add-sqr-sqrt 74.9%

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

   9. Applied egg-rr 74.9%

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

       1. unpow-1 74.9%

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

       2. associate-*r/ 75.2%

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

       3. *-commutative 75.2%

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

   11. Simplified 75.2%

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

       1. clear-num 75.2%

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

       2. *-un-lft-identity 75.2%

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

       3. times-frac 75.0%

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

       4. *-commutative 75.0%

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

       5. associate-*r* 74.9%

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

       6. *-commutative 74.9%

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

   13. Applied egg-rr 74.9%

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

       1. associate-*l/ 75.3%

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

       2. *-lft-identity 75.3%

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

       3. associate-*r* 75.2%

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

       4. *-commutative 75.2%

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

       5. associate-*l* 75.2%

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

   15. Simplified 75.2%

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

   if 1.3499999999999999e-65 < b

   1. Initial program 17.4%

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

      1. sqr-neg 17.4%

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

      2. sqr-neg 17.4%

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

      3. associate-*l* 17.4%

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

   3. Simplified 17.4%

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

   6. Applied egg-rr 27.2%

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

   7. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   8. Step-by-step derivation

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 90.5%

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

   9. Simplified 90.5%

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

   10. Taylor expanded in c around 0 90.8%

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

       1. *-commutative 90.8%

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

   12. Simplified 90.8%

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

Alternative 6: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-92}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-92)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 3.8e-65)
     (/ (+ b (sqrt (* c (* a -3.0)))) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-92) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 3.8e-65) {
		tmp = (b + sqrt((c * (a * -3.0)))) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-9d-92)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 3.8d-65) then
        tmp = (b + sqrt((c * (a * (-3.0d0))))) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-92) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 3.8e-65) {
		tmp = (b + Math.sqrt((c * (a * -3.0)))) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9e-92:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 3.8e-65:
		tmp = (b + math.sqrt((c * (a * -3.0)))) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9e-92)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 3.8e-65)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -3.0)))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9e-92)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 3.8e-65)
		tmp = (b + sqrt((c * (a * -3.0)))) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9e-92], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.8e-65], N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.0000000000000001e-92

   1. Initial program 64.5%

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

      1. sqr-neg 64.5%

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

      2. sqr-neg 64.5%

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

      3. associate-*l* 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in b around -inf 85.7%

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

      1. *-commutative 85.7%

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

   7. Simplified 85.7%

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

      1. associate-*l/ 85.8%

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

   9. Applied egg-rr 85.8%

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

   if -9.0000000000000001e-92 < b < 3.8000000000000002e-65

   1. Initial program 76.2%

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

      1. sqr-neg 76.2%

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

      2. sqr-neg 76.2%

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

      3. associate-*l* 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in b around 0 76.0%

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

      1. *-commutative 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*r* 76.0%

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

   7. Simplified 76.0%

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

      1. *-un-lft-identity 76.0%

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

      2. add-sqr-sqrt 44.1%

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

      3. sqrt-unprod 75.1%

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

      4. sqr-neg 75.1%

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

      5. sqrt-unprod 31.4%

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

      6. add-sqr-sqrt 75.2%

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

      7. *-commutative 75.2%

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

   9. Applied egg-rr 75.2%

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

       1. *-lft-identity 75.2%

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

   11. Simplified 75.2%

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

   if 3.8000000000000002e-65 < b

   1. Initial program 17.4%

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

      1. sqr-neg 17.4%

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

      2. sqr-neg 17.4%

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

      3. associate-*l* 17.4%

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

   3. Simplified 17.4%

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

   6. Applied egg-rr 27.2%

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

   7. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   8. Step-by-step derivation

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 90.5%

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

   9. Simplified 90.5%

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

   10. Taylor expanded in c around 0 90.8%

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

       1. *-commutative 90.8%

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

   12. Simplified 90.8%

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

Alternative 7: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(b + \sqrt{a \cdot \left(c \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e-92)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 8.2e-66)
     (* (/ 0.3333333333333333 a) (+ b (sqrt (* a (* c -3.0)))))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-92) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 8.2e-66) {
		tmp = (0.3333333333333333 / a) * (b + sqrt((a * (c * -3.0))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.2d-92)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 8.2d-66) then
        tmp = (0.3333333333333333d0 / a) * (b + sqrt((a * (c * (-3.0d0)))))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-92) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 8.2e-66) {
		tmp = (0.3333333333333333 / a) * (b + Math.sqrt((a * (c * -3.0))));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.2e-92:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 8.2e-66:
		tmp = (0.3333333333333333 / a) * (b + math.sqrt((a * (c * -3.0))))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e-92)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 8.2e-66)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(b + sqrt(Float64(a * Float64(c * -3.0)))));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.2e-92)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 8.2e-66)
		tmp = (0.3333333333333333 / a) * (b + sqrt((a * (c * -3.0))));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e-92], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 8.2e-66], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.2000000000000005e-92

   1. Initial program 64.5%

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

      1. sqr-neg 64.5%

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

      2. sqr-neg 64.5%

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

      3. associate-*l* 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in b around -inf 85.7%

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

      1. *-commutative 85.7%

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

   7. Simplified 85.7%

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

      1. associate-*l/ 85.8%

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

   9. Applied egg-rr 85.8%

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

   if -8.2000000000000005e-92 < b < 8.19999999999999996e-66

   1. Initial program 76.2%

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

      1. sqr-neg 76.2%

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

      2. sqr-neg 76.2%

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

      3. associate-*l* 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in b around 0 76.0%

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

      1. *-commutative 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*r* 76.0%

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

   7. Simplified 76.0%

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

      1. clear-num 76.0%

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

      2. inv-pow 76.0%

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

      3. *-un-lft-identity 76.0%

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

      4. times-frac 75.7%

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

      5. metadata-eval 75.7%

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

      6. add-sqr-sqrt 43.9%

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

      7. sqrt-unprod 74.8%

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

      8. sqr-neg 74.8%

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

      9. sqrt-unprod 31.3%

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

      10. add-sqr-sqrt 74.9%

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

   9. Applied egg-rr 74.9%

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

       1. unpow-1 74.9%

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

       2. associate-*r/ 75.2%

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

       3. *-commutative 75.2%

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

   11. Simplified 75.2%

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

       1. associate-/r/ 75.1%

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

       2. *-un-lft-identity 75.1%

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

       3. *-commutative 75.1%

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

       4. associate-/r* 75.0%

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

       5. metadata-eval 75.0%

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

       6. *-un-lft-identity 75.0%

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

       7. *-commutative 75.0%

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

       8. associate-*r* 75.1%

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

       9. *-commutative 75.1%

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

   13. Applied egg-rr 75.1%

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

   if 8.19999999999999996e-66 < b

   1. Initial program 17.4%

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

      1. sqr-neg 17.4%

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

      2. sqr-neg 17.4%

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

      3. associate-*l* 17.4%

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

   3. Simplified 17.4%

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

   6. Applied egg-rr 27.2%

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

   7. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   8. Step-by-step derivation

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 90.5%

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

   9. Simplified 90.5%

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

   10. Taylor expanded in c around 0 90.8%

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

       1. *-commutative 90.8%

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

   12. Simplified 90.8%

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

Alternative 8: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-66}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e-92)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 2.7e-66)
     (* 0.3333333333333333 (/ (+ b (sqrt (* a (* c -3.0)))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-92) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 2.7e-66) {
		tmp = 0.3333333333333333 * ((b + sqrt((a * (c * -3.0)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.2d-92)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 2.7d-66) then
        tmp = 0.3333333333333333d0 * ((b + sqrt((a * (c * (-3.0d0))))) / a)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-92) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 2.7e-66) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((a * (c * -3.0)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.2e-92:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 2.7e-66:
		tmp = 0.3333333333333333 * ((b + math.sqrt((a * (c * -3.0)))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e-92)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 2.7e-66)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(a * Float64(c * -3.0)))) / a));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.2e-92)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 2.7e-66)
		tmp = 0.3333333333333333 * ((b + sqrt((a * (c * -3.0)))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e-92], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.7e-66], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.2000000000000005e-92

   1. Initial program 64.5%

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

      1. sqr-neg 64.5%

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

      2. sqr-neg 64.5%

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

      3. associate-*l* 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in b around -inf 85.7%

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

      1. *-commutative 85.7%

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

   7. Simplified 85.7%

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

      1. associate-*l/ 85.8%

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

   9. Applied egg-rr 85.8%

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

   if -8.2000000000000005e-92 < b < 2.69999999999999996e-66

   1. Initial program 76.2%

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

      1. sqr-neg 76.2%

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

      2. sqr-neg 76.2%

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

      3. associate-*l* 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in b around 0 76.0%

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

      1. *-commutative 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*r* 76.0%

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

   7. Simplified 76.0%

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

      1. clear-num 76.0%

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

      2. inv-pow 76.0%

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

      3. *-un-lft-identity 76.0%

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

      4. times-frac 75.7%

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

      5. metadata-eval 75.7%

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

      6. add-sqr-sqrt 43.9%

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

      7. sqrt-unprod 74.8%

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

      8. sqr-neg 74.8%

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

      9. sqrt-unprod 31.3%

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

      10. add-sqr-sqrt 74.9%

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

   9. Applied egg-rr 74.9%

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

       1. unpow-1 74.9%

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

       2. associate-*r/ 75.2%

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

       3. *-commutative 75.2%

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

   11. Simplified 75.2%

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

       1. associate-/r/ 75.1%

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

       2. distribute-lft-in 75.1%

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

       3. *-commutative 75.1%

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

       4. associate-/r* 75.1%

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

       5. metadata-eval 75.1%

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

       6. *-commutative 75.1%

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

       7. associate-/r* 75.0%

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

       8. metadata-eval 75.0%

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

       9. *-commutative 75.0%

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

       10. associate-*r* 75.1%

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

       11. *-commutative 75.1%

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

   13. Applied egg-rr 75.1%

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

       1. distribute-lft-in 75.1%

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

       2. associate-*l/ 75.2%

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

       3. associate-*r/ 75.1%

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

   15. Simplified 75.1%

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

   if 2.69999999999999996e-66 < b

   1. Initial program 17.4%

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

      1. sqr-neg 17.4%

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

      2. sqr-neg 17.4%

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

      3. associate-*l* 17.4%

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

   3. Simplified 17.4%

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

   6. Applied egg-rr 27.2%

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

   7. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   8. Step-by-step derivation

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 90.5%

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

   9. Simplified 90.5%

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

   10. Taylor expanded in c around 0 90.8%

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

       1. *-commutative 90.8%

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

   12. Simplified 90.8%

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

Alternative 9: 67.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-223}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-223) (/ (* b -0.6666666666666666) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-223) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3d-223) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-223) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-223:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-223)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3e-223)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-223], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999991e-223

   1. Initial program 70.3%

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

      1. sqr-neg 70.3%

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

      2. sqr-neg 70.3%

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

      3. associate-*l* 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

   7. Simplified 66.3%

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

      1. associate-*l/ 66.3%

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

   9. Applied egg-rr 66.3%

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

   if 2.99999999999999991e-223 < b

   1. Initial program 25.7%

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

      1. sqr-neg 25.7%

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

      2. sqr-neg 25.7%

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

      3. associate-*l* 25.7%

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

   3. Simplified 25.7%

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

   6. Applied egg-rr 33.5%

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

   7. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   8. Step-by-step derivation

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 79.9%

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

   9. Simplified 79.9%

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

   10. Taylor expanded in c around 0 80.1%

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

       1. *-commutative 80.1%

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

   12. Simplified 80.1%

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

Alternative 10: 67.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-223}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-223) (/ (* b -0.6666666666666666) a) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-223) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3d-223) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-223) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-223:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-223)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3e-223)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-223], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999991e-223

   1. Initial program 70.3%

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

      1. sqr-neg 70.3%

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

      2. sqr-neg 70.3%

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

      3. associate-*l* 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

   7. Simplified 66.3%

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

      1. associate-*l/ 66.3%

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

   9. Applied egg-rr 66.3%

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

   if 2.99999999999999991e-223 < b

   1. Initial program 25.7%

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

      1. sqr-neg 25.7%

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

      2. sqr-neg 25.7%

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

      3. associate-*l* 25.7%

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

   3. Simplified 25.7%

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

   5. Taylor expanded in b around inf 80.1%

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

      1. *-commutative 80.1%

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

   7. Simplified 80.1%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-223}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-223}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-223) (/ b (* a -1.5)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-223) {
		tmp = b / (a * -1.5);
	} 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 <= 3d-223) then
        tmp = b / (a * (-1.5d0))
    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 <= 3e-223) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-223:
		tmp = b / (a * -1.5)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-223)
		tmp = Float64(b / Float64(a * -1.5));
	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 <= 3e-223)
		tmp = b / (a * -1.5);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-223], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999991e-223

   1. Initial program 70.3%

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

      1. sqr-neg 70.3%

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

      2. sqr-neg 70.3%

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

      3. associate-*l* 70.3%

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

   3. Simplified 70.3%

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

   6. Applied egg-rr 66.7%

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

   7. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

      2. associate-*l/ 66.3%

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

      3. associate-/l* 66.2%

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

   9. Simplified 66.2%

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

       1. clear-num 66.2%

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

       2. un-div-inv 66.3%

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

       3. div-inv 66.3%

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

       4. metadata-eval 66.3%

          \[\leadsto \frac{b}{a \cdot \color{blue}{-1.5}} \]

   11. Applied egg-rr 66.3%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if 2.99999999999999991e-223 < b

   1. Initial program 25.7%

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

      1. sqr-neg 25.7%

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

      2. sqr-neg 25.7%

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

      3. associate-*l* 25.7%

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

   3. Simplified 25.7%

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

   5. Taylor expanded in b around inf 80.1%

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

      1. *-commutative 80.1%

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

   7. Simplified 80.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-223}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-223}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-223) (* (/ b a) -0.6666666666666666) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-223) {
		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 <= 3d-223) 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 <= 3e-223) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-223:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-223)
		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 <= 3e-223)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-223], 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 < 2.99999999999999991e-223

   1. Initial program 70.3%

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

      1. sqr-neg 70.3%

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

      2. sqr-neg 70.3%

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

      3. associate-*l* 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

   7. Simplified 66.3%

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

   if 2.99999999999999991e-223 < b

   1. Initial program 25.7%

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

      1. sqr-neg 25.7%

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

      2. sqr-neg 25.7%

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

      3. associate-*l* 25.7%

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

   3. Simplified 25.7%

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

   5. Taylor expanded in b around inf 80.1%

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

      1. *-commutative 80.1%

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

   7. Simplified 80.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-223}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-223}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-223) (* (/ b a) -0.6666666666666666) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-223) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3d-223) then
        tmp = (b / a) * (-0.6666666666666666d0)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-223) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-223:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-223)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3e-223)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-223], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999991e-223

   1. Initial program 70.3%

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

      1. sqr-neg 70.3%

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

      2. sqr-neg 70.3%

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

      3. associate-*l* 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

   7. Simplified 66.3%

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

   if 2.99999999999999991e-223 < b

   1. Initial program 25.7%

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

      1. sqr-neg 25.7%

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

      2. sqr-neg 25.7%

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

      3. associate-*l* 25.7%

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

   3. Simplified 25.7%

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

   6. Applied egg-rr 33.5%

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

      1. un-div-inv 33.5%

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

      2. div-inv 33.6%

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

      3. metadata-eval 33.6%

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

   8. Applied egg-rr 33.6%

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

   9. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   10. Step-by-step derivation

       1. *-commutative 0.0%

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

       2. associate-/l* 0.0%

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

       3. associate-*r* 0.0%

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

       4. *-commutative 0.0%

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

       5. associate-*r/ 0.0%

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

       6. unpow2 0.0%

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

       7. rem-square-sqrt 79.9%

          \[\leadsto c \cdot \frac{0.16666666666666666 \cdot \color{blue}{-3}}{b} \]

       8. metadata-eval 79.9%

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

   11. Simplified 79.9%

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

Alternative 14: 67.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-223}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-223) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-223) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3d-223) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-223) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-223:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-223)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3e-223)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-223], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999991e-223

   1. Initial program 70.3%

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

      1. sqr-neg 70.3%

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

      2. sqr-neg 70.3%

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

      3. associate-*l* 70.3%

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

   3. Simplified 70.3%

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

   6. Applied egg-rr 66.7%

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

   7. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

      2. associate-*l/ 66.3%

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

      3. associate-/l* 66.2%

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

   9. Simplified 66.2%

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

   if 2.99999999999999991e-223 < b

   1. Initial program 25.7%

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

      1. sqr-neg 25.7%

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

      2. sqr-neg 25.7%

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

      3. associate-*l* 25.7%

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

   3. Simplified 25.7%

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

   6. Applied egg-rr 33.5%

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

      1. un-div-inv 33.5%

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

      2. div-inv 33.6%

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

      3. metadata-eval 33.6%

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

   8. Applied egg-rr 33.6%

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

   9. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   10. Step-by-step derivation

       1. *-commutative 0.0%

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

       2. associate-/l* 0.0%

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

       3. associate-*r* 0.0%

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

       4. *-commutative 0.0%

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

       5. associate-*r/ 0.0%

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

       6. unpow2 0.0%

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

       7. rem-square-sqrt 79.9%

          \[\leadsto c \cdot \frac{0.16666666666666666 \cdot \color{blue}{-3}}{b} \]

       8. metadata-eval 79.9%

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

   11. Simplified 79.9%

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

Alternative 15: 35.3% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ b \cdot \frac{-0.6666666666666666}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* b (/ -0.6666666666666666 a)))

C

double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / 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 * ((-0.6666666666666666d0) / a)
end function

Java

public static double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / a);
}

Python

def code(a, b, c):
	return b * (-0.6666666666666666 / a)

Julia

function code(a, b, c)
	return Float64(b * Float64(-0.6666666666666666 / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = b * (-0.6666666666666666 / a);
end

Wolfram

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

Derivation

1. Initial program 48.9%

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

   1. sqr-neg 48.9%

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

   2. sqr-neg 48.9%

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

   3. associate-*l* 48.9%

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

3. Simplified 48.9%

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

6. Applied egg-rr 50.7%

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

7. Taylor expanded in b around -inf 35.7%

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

   1. *-commutative 35.7%

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

   2. associate-*l/ 35.7%

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

   3. associate-/l* 35.7%

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

9. Simplified 35.7%

   \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))