Cubic critical

Percentage Accurate: 51.3% → 83.2%

Time: 19.9s

Alternatives: 14

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: 51.3% 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: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-1.5}{b}}\\ \mathbf{elif}\;b \leq 0.102:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e-15)
   (/ (/ 1.0 a) (/ -1.5 b))
   (if (<= b 0.102)
     (/ (- (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 <= -3.2e-15) {
		tmp = (1.0 / a) / (-1.5 / b);
	} else if (b <= 0.102) {
		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 <= (-3.2d-15)) then
        tmp = (1.0d0 / a) / ((-1.5d0) / b)
    else if (b <= 0.102d0) 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 <= -3.2e-15) {
		tmp = (1.0 / a) / (-1.5 / b);
	} else if (b <= 0.102) {
		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 <= -3.2e-15:
		tmp = (1.0 / a) / (-1.5 / b)
	elif b <= 0.102:
		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 <= -3.2e-15)
		tmp = Float64(Float64(1.0 / a) / Float64(-1.5 / b));
	elseif (b <= 0.102)
		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 <= -3.2e-15)
		tmp = (1.0 / a) / (-1.5 / b);
	elseif (b <= 0.102)
		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, -3.2e-15], N[(N[(1.0 / a), $MachinePrecision] / N[(-1.5 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.102], 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 < -3.1999999999999999e-15

   1. Initial program 68.6%

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

      1. sqr-neg 68.6%

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

      2. sqr-neg 68.6%

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

      3. associate-*l* 68.6%

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

   3. Simplified 68.6%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in b around -inf 93.6%

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

      1. *-commutative 93.6%

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

      2. associate-*l/ 93.5%

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

      3. associate-/l* 93.5%

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

   9. Simplified 93.5%

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

       1. associate-*r/ 93.5%

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

       2. metadata-eval 93.5%

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

       3. associate-/l* 93.7%

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

       4. un-div-inv 93.6%

          \[\leadsto \color{blue}{\frac{b \cdot 2}{-3} \cdot \frac{1}{a}} \]

       5. *-commutative 93.6%

          \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{b \cdot 2}{-3}} \]

       6. clear-num 93.6%

          \[\leadsto \frac{1}{a} \cdot \color{blue}{\frac{1}{\frac{-3}{b \cdot 2}}} \]

       7. un-div-inv 93.7%

          \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{-3}{b \cdot 2}}} \]

       8. *-commutative 93.7%

          \[\leadsto \frac{\frac{1}{a}}{\frac{-3}{\color{blue}{2 \cdot b}}} \]

       9. associate-/r* 93.7%

          \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-3}{2}}{b}}} \]

       10. metadata-eval 93.7%

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

   11. Applied egg-rr 93.7%

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

   if -3.1999999999999999e-15 < b < 0.101999999999999993

   1. Initial program 80.7%

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

   if 0.101999999999999993 < b

   1. Initial program 14.4%

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

      1. sqr-neg 14.4%

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

      2. sqr-neg 14.4%

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

      3. associate-*l* 14.4%

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

   3. Simplified 14.4%

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

   6. Applied egg-rr 19.6%

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

   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 90.8%

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

   9. Simplified 90.8%

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

   10. Taylor expanded in c around 0 91.1%

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

       1. *-commutative 91.1%

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

   12. Simplified 91.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-1.5}{b}}\\ \mathbf{elif}\;b \leq 0.102:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-1.5}{b}}\\ \mathbf{elif}\;b \leq 0.27:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e-15)
   (/ (/ 1.0 a) (/ -1.5 b))
   (if (<= b 0.27)
     (/ (- (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 <= -3.2e-15) {
		tmp = (1.0 / a) / (-1.5 / b);
	} else if (b <= 0.27) {
		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 <= (-3.2d-15)) then
        tmp = (1.0d0 / a) / ((-1.5d0) / b)
    else if (b <= 0.27d0) 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 <= -3.2e-15) {
		tmp = (1.0 / a) / (-1.5 / b);
	} else if (b <= 0.27) {
		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 <= -3.2e-15:
		tmp = (1.0 / a) / (-1.5 / b)
	elif b <= 0.27:
		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 <= -3.2e-15)
		tmp = Float64(Float64(1.0 / a) / Float64(-1.5 / b));
	elseif (b <= 0.27)
		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 <= -3.2e-15)
		tmp = (1.0 / a) / (-1.5 / b);
	elseif (b <= 0.27)
		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, -3.2e-15], N[(N[(1.0 / a), $MachinePrecision] / N[(-1.5 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.27], 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 < -3.1999999999999999e-15

   1. Initial program 68.6%

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

      1. sqr-neg 68.6%

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

      2. sqr-neg 68.6%

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

      3. associate-*l* 68.6%

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

   3. Simplified 68.6%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in b around -inf 93.6%

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

      1. *-commutative 93.6%

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

      2. associate-*l/ 93.5%

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

      3. associate-/l* 93.5%

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

   9. Simplified 93.5%

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

       1. associate-*r/ 93.5%

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

       2. metadata-eval 93.5%

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

       3. associate-/l* 93.7%

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

       4. un-div-inv 93.6%

          \[\leadsto \color{blue}{\frac{b \cdot 2}{-3} \cdot \frac{1}{a}} \]

       5. *-commutative 93.6%

          \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{b \cdot 2}{-3}} \]

       6. clear-num 93.6%

          \[\leadsto \frac{1}{a} \cdot \color{blue}{\frac{1}{\frac{-3}{b \cdot 2}}} \]

       7. un-div-inv 93.7%

          \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{-3}{b \cdot 2}}} \]

       8. *-commutative 93.7%

          \[\leadsto \frac{\frac{1}{a}}{\frac{-3}{\color{blue}{2 \cdot b}}} \]

       9. associate-/r* 93.7%

          \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-3}{2}}{b}}} \]

       10. metadata-eval 93.7%

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

   11. Applied egg-rr 93.7%

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

   if -3.1999999999999999e-15 < b < 0.27000000000000002

   1. Initial program 80.7%

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

      1. sqr-neg 80.7%

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

      2. sqr-neg 80.7%

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

      3. associate-*l* 80.6%

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

   3. Simplified 80.6%

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

   if 0.27000000000000002 < b

   1. Initial program 14.4%

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

      1. sqr-neg 14.4%

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

      2. sqr-neg 14.4%

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

      3. associate-*l* 14.4%

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

   3. Simplified 14.4%

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

   6. Applied egg-rr 19.6%

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

   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 90.8%

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

   9. Simplified 90.8%

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

   10. Taylor expanded in c around 0 91.1%

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

       1. *-commutative 91.1%

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

   12. Simplified 91.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-1.5}{b}}\\ \mathbf{elif}\;b \leq 0.27:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-1.5}{b}}\\ \mathbf{elif}\;b \leq 0.102:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e-104)
   (/ (/ 1.0 a) (/ -1.5 b))
   (if (<= b 0.102)
     (/ (- (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 <= -9.5e-104) {
		tmp = (1.0 / a) / (-1.5 / b);
	} else if (b <= 0.102) {
		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 <= (-9.5d-104)) then
        tmp = (1.0d0 / a) / ((-1.5d0) / b)
    else if (b <= 0.102d0) 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 <= -9.5e-104) {
		tmp = (1.0 / a) / (-1.5 / b);
	} else if (b <= 0.102) {
		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 <= -9.5e-104:
		tmp = (1.0 / a) / (-1.5 / b)
	elif b <= 0.102:
		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 <= -9.5e-104)
		tmp = Float64(Float64(1.0 / a) / Float64(-1.5 / b));
	elseif (b <= 0.102)
		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 <= -9.5e-104)
		tmp = (1.0 / a) / (-1.5 / b);
	elseif (b <= 0.102)
		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, -9.5e-104], N[(N[(1.0 / a), $MachinePrecision] / N[(-1.5 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.102], 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 < -9.5000000000000002e-104

   1. Initial program 73.3%

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

      1. sqr-neg 73.3%

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

      2. sqr-neg 73.3%

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

      3. associate-*l* 73.3%

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

   3. Simplified 73.3%

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

   6. Applied egg-rr 58.2%

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

   7. Taylor expanded in b around -inf 88.3%

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

      1. *-commutative 88.3%

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

      2. associate-*l/ 88.3%

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

      3. associate-/l* 88.2%

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

   9. Simplified 88.2%

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

       1. associate-*r/ 88.3%

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

       2. metadata-eval 88.3%

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

       3. associate-/l* 88.4%

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

       4. un-div-inv 88.3%

          \[\leadsto \color{blue}{\frac{b \cdot 2}{-3} \cdot \frac{1}{a}} \]

       5. *-commutative 88.3%

          \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{b \cdot 2}{-3}} \]

       6. clear-num 88.3%

          \[\leadsto \frac{1}{a} \cdot \color{blue}{\frac{1}{\frac{-3}{b \cdot 2}}} \]

       7. un-div-inv 88.4%

          \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{-3}{b \cdot 2}}} \]

       8. *-commutative 88.4%

          \[\leadsto \frac{\frac{1}{a}}{\frac{-3}{\color{blue}{2 \cdot b}}} \]

       9. associate-/r* 88.4%

          \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-3}{2}}{b}}} \]

       10. metadata-eval 88.4%

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

   11. Applied egg-rr 88.4%

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

   if -9.5000000000000002e-104 < b < 0.101999999999999993

   1. Initial program 78.0%

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

      1. sqr-neg 78.0%

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

      2. sqr-neg 78.0%

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

      3. associate-*l* 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in b around 0 73.3%

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

      1. *-commutative 73.3%

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

      2. *-commutative 73.3%

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

      3. associate-*r* 73.4%

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

   7. Simplified 73.4%

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

   if 0.101999999999999993 < b

   1. Initial program 14.4%

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

      1. sqr-neg 14.4%

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

      2. sqr-neg 14.4%

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

      3. associate-*l* 14.4%

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

   3. Simplified 14.4%

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

   6. Applied egg-rr 19.6%

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

   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 90.8%

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

   9. Simplified 90.8%

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

   10. Taylor expanded in c around 0 91.1%

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

       1. *-commutative 91.1%

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

   12. Simplified 91.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-1.5}{b}}\\ \mathbf{elif}\;b \leq 0.102:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-1.5}{b}}\\ \mathbf{elif}\;b \leq 0.102:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-104)
   (/ (/ 1.0 a) (/ -1.5 b))
   (if (<= b 0.102)
     (/ (- (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 <= -9e-104) {
		tmp = (1.0 / a) / (-1.5 / b);
	} else if (b <= 0.102) {
		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 <= (-9d-104)) then
        tmp = (1.0d0 / a) / ((-1.5d0) / b)
    else if (b <= 0.102d0) 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 <= -9e-104) {
		tmp = (1.0 / a) / (-1.5 / b);
	} else if (b <= 0.102) {
		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 <= -9e-104:
		tmp = (1.0 / a) / (-1.5 / b)
	elif b <= 0.102:
		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 <= -9e-104)
		tmp = Float64(Float64(1.0 / a) / Float64(-1.5 / b));
	elseif (b <= 0.102)
		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 <= -9e-104)
		tmp = (1.0 / a) / (-1.5 / b);
	elseif (b <= 0.102)
		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, -9e-104], N[(N[(1.0 / a), $MachinePrecision] / N[(-1.5 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.102], 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 < -8.9999999999999995e-104

   1. Initial program 73.3%

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

      1. sqr-neg 73.3%

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

      2. sqr-neg 73.3%

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

      3. associate-*l* 73.3%

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

   3. Simplified 73.3%

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

   6. Applied egg-rr 58.2%

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

   7. Taylor expanded in b around -inf 88.3%

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

      1. *-commutative 88.3%

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

      2. associate-*l/ 88.3%

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

      3. associate-/l* 88.2%

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

   9. Simplified 88.2%

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

       1. associate-*r/ 88.3%

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

       2. metadata-eval 88.3%

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

       3. associate-/l* 88.4%

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

       4. un-div-inv 88.3%

          \[\leadsto \color{blue}{\frac{b \cdot 2}{-3} \cdot \frac{1}{a}} \]

       5. *-commutative 88.3%

          \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{b \cdot 2}{-3}} \]

       6. clear-num 88.3%

          \[\leadsto \frac{1}{a} \cdot \color{blue}{\frac{1}{\frac{-3}{b \cdot 2}}} \]

       7. un-div-inv 88.4%

          \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{-3}{b \cdot 2}}} \]

       8. *-commutative 88.4%

          \[\leadsto \frac{\frac{1}{a}}{\frac{-3}{\color{blue}{2 \cdot b}}} \]

       9. associate-/r* 88.4%

          \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-3}{2}}{b}}} \]

       10. metadata-eval 88.4%

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

   11. Applied egg-rr 88.4%

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

   if -8.9999999999999995e-104 < b < 0.101999999999999993

   1. Initial program 78.0%

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

      1. sqr-neg 78.0%

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

      2. sqr-neg 78.0%

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

      3. associate-*l* 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in b around 0 73.3%

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

   if 0.101999999999999993 < b

   1. Initial program 14.4%

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

      1. sqr-neg 14.4%

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

      2. sqr-neg 14.4%

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

      3. associate-*l* 14.4%

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

   3. Simplified 14.4%

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

   6. Applied egg-rr 19.6%

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

   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 90.8%

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

   9. Simplified 90.8%

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

   10. Taylor expanded in c around 0 91.1%

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

       1. *-commutative 91.1%

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

   12. Simplified 91.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-1.5}{b}}\\ \mathbf{elif}\;b \leq 0.102:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-1.5}{b}}\\ \mathbf{elif}\;b \leq 0.102:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-104)
   (/ (/ 1.0 a) (/ -1.5 b))
   (if (<= b 0.102)
     (/ (+ 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 <= -8.5e-104) {
		tmp = (1.0 / a) / (-1.5 / b);
	} else if (b <= 0.102) {
		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 <= (-8.5d-104)) then
        tmp = (1.0d0 / a) / ((-1.5d0) / b)
    else if (b <= 0.102d0) 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 <= -8.5e-104) {
		tmp = (1.0 / a) / (-1.5 / b);
	} else if (b <= 0.102) {
		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 <= -8.5e-104:
		tmp = (1.0 / a) / (-1.5 / b)
	elif b <= 0.102:
		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 <= -8.5e-104)
		tmp = Float64(Float64(1.0 / a) / Float64(-1.5 / b));
	elseif (b <= 0.102)
		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 <= -8.5e-104)
		tmp = (1.0 / a) / (-1.5 / b);
	elseif (b <= 0.102)
		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, -8.5e-104], N[(N[(1.0 / a), $MachinePrecision] / N[(-1.5 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.102], 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 < -8.50000000000000007e-104

   1. Initial program 73.3%

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

      1. sqr-neg 73.3%

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

      2. sqr-neg 73.3%

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

      3. associate-*l* 73.3%

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

   3. Simplified 73.3%

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

   6. Applied egg-rr 58.2%

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

   7. Taylor expanded in b around -inf 88.3%

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

      1. *-commutative 88.3%

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

      2. associate-*l/ 88.3%

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

      3. associate-/l* 88.2%

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

   9. Simplified 88.2%

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

       1. associate-*r/ 88.3%

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

       2. metadata-eval 88.3%

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

       3. associate-/l* 88.4%

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

       4. un-div-inv 88.3%

          \[\leadsto \color{blue}{\frac{b \cdot 2}{-3} \cdot \frac{1}{a}} \]

       5. *-commutative 88.3%

          \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{b \cdot 2}{-3}} \]

       6. clear-num 88.3%

          \[\leadsto \frac{1}{a} \cdot \color{blue}{\frac{1}{\frac{-3}{b \cdot 2}}} \]

       7. un-div-inv 88.4%

          \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{-3}{b \cdot 2}}} \]

       8. *-commutative 88.4%

          \[\leadsto \frac{\frac{1}{a}}{\frac{-3}{\color{blue}{2 \cdot b}}} \]

       9. associate-/r* 88.4%

          \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-3}{2}}{b}}} \]

       10. metadata-eval 88.4%

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

   11. Applied egg-rr 88.4%

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

   if -8.50000000000000007e-104 < b < 0.101999999999999993

   1. Initial program 78.0%

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

      1. sqr-neg 78.0%

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

      2. sqr-neg 78.0%

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

      3. associate-*l* 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in b around 0 73.3%

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

      1. *-un-lft-identity 73.3%

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

      2. add-sqr-sqrt 42.4%

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

      3. sqrt-unprod 73.0%

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

      4. sqr-neg 73.0%

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

      5. sqrt-prod 31.0%

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

      6. add-sqr-sqrt 72.4%

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

      7. associate-*r* 72.5%

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

      8. *-commutative 72.5%

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

      9. *-commutative 72.5%

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

   7. Applied egg-rr 72.5%

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

      1. *-lft-identity 72.5%

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

   9. Simplified 72.5%

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

   if 0.101999999999999993 < b

   1. Initial program 14.4%

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

      1. sqr-neg 14.4%

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

      2. sqr-neg 14.4%

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

      3. associate-*l* 14.4%

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

   3. Simplified 14.4%

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

   6. Applied egg-rr 19.6%

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

   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 90.8%

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

   9. Simplified 90.8%

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

   10. Taylor expanded in c around 0 91.1%

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

       1. *-commutative 91.1%

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

   12. Simplified 91.1%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-1.5}{b}}\\ \mathbf{elif}\;b \leq 0.102:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-1.5}{b}}\\ \mathbf{elif}\;b \leq 0.102:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.8e-104)
   (/ (/ 1.0 a) (/ -1.5 b))
   (if (<= b 0.102)
     (* 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 <= -4.8e-104) {
		tmp = (1.0 / a) / (-1.5 / b);
	} else if (b <= 0.102) {
		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 <= (-4.8d-104)) then
        tmp = (1.0d0 / a) / ((-1.5d0) / b)
    else if (b <= 0.102d0) 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 <= -4.8e-104) {
		tmp = (1.0 / a) / (-1.5 / b);
	} else if (b <= 0.102) {
		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 <= -4.8e-104:
		tmp = (1.0 / a) / (-1.5 / b)
	elif b <= 0.102:
		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 <= -4.8e-104)
		tmp = Float64(Float64(1.0 / a) / Float64(-1.5 / b));
	elseif (b <= 0.102)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(Float64(a * 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 <= -4.8e-104)
		tmp = (1.0 / a) / (-1.5 / b);
	elseif (b <= 0.102)
		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, -4.8e-104], N[(N[(1.0 / a), $MachinePrecision] / N[(-1.5 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.102], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.8000000000000001e-104

   1. Initial program 73.3%

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

      1. sqr-neg 73.3%

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

      2. sqr-neg 73.3%

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

      3. associate-*l* 73.3%

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

   3. Simplified 73.3%

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

   6. Applied egg-rr 58.2%

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

   7. Taylor expanded in b around -inf 88.3%

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

      1. *-commutative 88.3%

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

      2. associate-*l/ 88.3%

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

      3. associate-/l* 88.2%

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

   9. Simplified 88.2%

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

       1. associate-*r/ 88.3%

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

       2. metadata-eval 88.3%

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

       3. associate-/l* 88.4%

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

       4. un-div-inv 88.3%

          \[\leadsto \color{blue}{\frac{b \cdot 2}{-3} \cdot \frac{1}{a}} \]

       5. *-commutative 88.3%

          \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{b \cdot 2}{-3}} \]

       6. clear-num 88.3%

          \[\leadsto \frac{1}{a} \cdot \color{blue}{\frac{1}{\frac{-3}{b \cdot 2}}} \]

       7. un-div-inv 88.4%

          \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{-3}{b \cdot 2}}} \]

       8. *-commutative 88.4%

          \[\leadsto \frac{\frac{1}{a}}{\frac{-3}{\color{blue}{2 \cdot b}}} \]

       9. associate-/r* 88.4%

          \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-3}{2}}{b}}} \]

       10. metadata-eval 88.4%

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

   11. Applied egg-rr 88.4%

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

   if -4.8000000000000001e-104 < b < 0.101999999999999993

   1. Initial program 78.0%

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

      1. sqr-neg 78.0%

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

      2. sqr-neg 78.0%

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

      3. associate-*l* 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in b around 0 73.3%

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

      1. *-un-lft-identity 73.3%

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

      2. *-un-lft-identity 73.3%

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

      3. times-frac 73.3%

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

      4. metadata-eval 73.3%

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

      5. add-sqr-sqrt 42.4%

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

      6. sqrt-unprod 73.0%

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

      7. sqr-neg 73.0%

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

      8. sqrt-prod 31.0%

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

      9. add-sqr-sqrt 72.4%

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

      10. associate-*r* 72.3%

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

      11. *-commutative 72.3%

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

      12. *-commutative 72.3%

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

   7. Applied egg-rr 72.3%

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

      1. *-lft-identity 72.3%

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

   9. Simplified 72.3%

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

   10. Taylor expanded in c around 0 72.4%

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

   if 0.101999999999999993 < b

   1. Initial program 14.4%

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

      1. sqr-neg 14.4%

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

      2. sqr-neg 14.4%

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

      3. associate-*l* 14.4%

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

   3. Simplified 14.4%

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

   6. Applied egg-rr 19.6%

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

   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 90.8%

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

   9. Simplified 90.8%

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

   10. Taylor expanded in c around 0 91.1%

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

       1. *-commutative 91.1%

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

   12. Simplified 91.1%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-1.5}{b}}\\ \mathbf{elif}\;b \leq 0.102:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-1.5}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ (/ 1.0 a) (/ -1.5 b)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (1.0 / a) / (-1.5 / b);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (1.0d0 / a) / ((-1.5d0) / b)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (1.0 / a) / (-1.5 / b);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (1.0 / a) / (-1.5 / b)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(1.0 / a) / Float64(-1.5 / b));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (1.0 / a) / (-1.5 / b);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(1.0 / a), $MachinePrecision] / N[(-1.5 / b), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 76.5%

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

      1. sqr-neg 76.5%

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

      2. sqr-neg 76.5%

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

      3. associate-*l* 76.5%

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

   3. Simplified 76.5%

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

   6. Applied egg-rr 65.2%

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

   7. Taylor expanded in b around -inf 66.2%

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

      1. *-commutative 66.2%

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

      2. associate-*l/ 66.2%

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

      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. associate-*r/ 66.2%

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

       2. metadata-eval 66.2%

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

       3. associate-/l* 66.3%

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

       4. un-div-inv 66.2%

          \[\leadsto \color{blue}{\frac{b \cdot 2}{-3} \cdot \frac{1}{a}} \]

       5. *-commutative 66.2%

          \[\leadsto \color{blue}{\frac{1}{a} \cdot \frac{b \cdot 2}{-3}} \]

       6. clear-num 66.2%

          \[\leadsto \frac{1}{a} \cdot \color{blue}{\frac{1}{\frac{-3}{b \cdot 2}}} \]

       7. un-div-inv 66.3%

          \[\leadsto \color{blue}{\frac{\frac{1}{a}}{\frac{-3}{b \cdot 2}}} \]

       8. *-commutative 66.3%

          \[\leadsto \frac{\frac{1}{a}}{\frac{-3}{\color{blue}{2 \cdot b}}} \]

       9. associate-/r* 66.3%

          \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-3}{2}}{b}}} \]

       10. metadata-eval 66.3%

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

   11. Applied egg-rr 66.3%

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

   if -4.999999999999985e-310 < b

   1. Initial program 30.3%

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

      1. sqr-neg 30.3%

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

      2. sqr-neg 30.3%

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

      3. associate-*l* 30.3%

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

   3. Simplified 30.3%

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

   6. Applied egg-rr 34.0%

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

   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 71.3%

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

   9. Simplified 71.3%

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

   10. Taylor expanded in c around 0 71.6%

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

       1. *-commutative 71.6%

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

   12. Simplified 71.6%

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

Alternative 8: 68.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.5e-308) (/ (* b -2.0) (* a 3.0)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.5e-308) {
		tmp = (b * -2.0) / (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 <= 5.5d-308) then
        tmp = (b * (-2.0d0)) / (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 <= 5.5e-308) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.5e-308:
		tmp = (b * -2.0) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.5e-308)
		tmp = Float64(Float64(b * -2.0) / 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 <= 5.5e-308)
		tmp = (b * -2.0) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.5e-308], N[(N[(b * -2.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5e-308

   1. Initial program 76.5%

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

      1. sqr-neg 76.5%

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

      2. sqr-neg 76.5%

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

      3. associate-*l* 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

   7. Simplified 66.3%

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

   if 5.5e-308 < b

   1. Initial program 30.3%

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

      1. sqr-neg 30.3%

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

      2. sqr-neg 30.3%

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

      3. associate-*l* 30.3%

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

   3. Simplified 30.3%

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

   6. Applied egg-rr 34.0%

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

   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 71.3%

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

   9. Simplified 71.3%

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

   10. Taylor expanded in c around 0 71.6%

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

       1. *-commutative 71.6%

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

   12. Simplified 71.6%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-1.5}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ 1.0 (* a (/ -1.5 b))) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = 1.0 / (a * (-1.5 / b));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = 1.0d0 / (a * ((-1.5d0) / b))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = 1.0 / (a * (-1.5 / b));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = 1.0 / (a * (-1.5 / b))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(1.0 / Float64(a * Float64(-1.5 / b)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = 1.0 / (a * (-1.5 / b));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(1.0 / N[(a * N[(-1.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 76.5%

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

      1. sqr-neg 76.5%

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

      2. sqr-neg 76.5%

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

      3. associate-*l* 76.5%

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

   3. Simplified 76.5%

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

   6. Applied egg-rr 65.2%

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

   7. Taylor expanded in b around -inf 66.2%

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

      1. *-commutative 66.2%

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

      2. associate-*l/ 66.2%

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

      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. associate-*r/ 66.2%

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

       2. metadata-eval 66.2%

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

       3. associate-/l* 66.3%

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

       4. un-div-inv 66.2%

          \[\leadsto \color{blue}{\frac{b \cdot 2}{-3} \cdot \frac{1}{a}} \]

       5. clear-num 66.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{-3}{b \cdot 2}}} \cdot \frac{1}{a} \]

       6. frac-times 66.2%

          \[\leadsto \color{blue}{\frac{1 \cdot 1}{\frac{-3}{b \cdot 2} \cdot a}} \]

       7. metadata-eval 66.2%

          \[\leadsto \frac{\color{blue}{1}}{\frac{-3}{b \cdot 2} \cdot a} \]

       8. *-commutative 66.2%

          \[\leadsto \frac{1}{\frac{-3}{\color{blue}{2 \cdot b}} \cdot a} \]

       9. associate-/r* 66.2%

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-3}{2}}{b}} \cdot a} \]

       10. metadata-eval 66.2%

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

   11. Applied egg-rr 66.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 30.3%

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

      1. sqr-neg 30.3%

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

      2. sqr-neg 30.3%

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

      3. associate-*l* 30.3%

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

   3. Simplified 30.3%

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

   6. Applied egg-rr 34.0%

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

   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 71.3%

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

   9. Simplified 71.3%

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

   10. Taylor expanded in c around 0 71.6%

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

       1. *-commutative 71.6%

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

   12. Simplified 71.6%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{a \cdot \frac{-1.5}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-305}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.8e-305) (* (/ b a) -0.6666666666666666) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.8e-305) {
		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 <= 8.8d-305) 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 <= 8.8e-305) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.8e-305:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.8e-305)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8.8e-305)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.8e-305], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.79999999999999987e-305

   1. Initial program 76.5%

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

      1. sqr-neg 76.5%

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

      2. sqr-neg 76.5%

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

      3. associate-*l* 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in b around -inf 66.2%

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

      1. *-commutative 66.2%

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

   7. Simplified 66.2%

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

   if 8.79999999999999987e-305 < b

   1. Initial program 30.3%

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

      1. sqr-neg 30.3%

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

      2. sqr-neg 30.3%

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

      3. associate-*l* 30.3%

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

   3. Simplified 30.3%

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

   6. Applied egg-rr 34.0%

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

   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 71.3%

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

   9. Simplified 71.3%

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

   10. Taylor expanded in c around 0 71.6%

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

       1. *-commutative 71.6%

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

   12. Simplified 71.6%

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

Alternative 11: 68.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.8e-308) (* (/ b a) -0.6666666666666666) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.8e-308) {
		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 <= 3.8d-308) 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 <= 3.8e-308) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.8e-308:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.8e-308)
		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 <= 3.8e-308)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.8e-308], 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 < 3.79999999999999975e-308

   1. Initial program 76.5%

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

      1. sqr-neg 76.5%

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

      2. sqr-neg 76.5%

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

      3. associate-*l* 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in b around -inf 66.2%

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

      1. *-commutative 66.2%

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

   7. Simplified 66.2%

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

   if 3.79999999999999975e-308 < b

   1. Initial program 30.3%

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

      1. sqr-neg 30.3%

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

      2. sqr-neg 30.3%

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

      3. associate-*l* 30.3%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in b around inf 71.5%

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

      1. *-commutative 71.5%

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

   7. Simplified 71.5%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.3 \cdot 10^{-308}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.3e-308) (* (/ b a) -0.6666666666666666) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.3e-308) {
		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 <= 4.3d-308) 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 <= 4.3e-308) {
		tmp = (b / a) * -0.6666666666666666;
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.3e-308:
		tmp = (b / a) * -0.6666666666666666
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.3e-308)
		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 <= 4.3e-308)
		tmp = (b / a) * -0.6666666666666666;
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.3e-308], 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 < 4.3000000000000002e-308

   1. Initial program 76.5%

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

      1. sqr-neg 76.5%

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

      2. sqr-neg 76.5%

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

      3. associate-*l* 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in b around -inf 66.2%

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

      1. *-commutative 66.2%

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

   7. Simplified 66.2%

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

   if 4.3000000000000002e-308 < b

   1. Initial program 30.3%

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

      1. sqr-neg 30.3%

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

      2. sqr-neg 30.3%

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

      3. associate-*l* 30.3%

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

   3. Simplified 30.3%

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

   6. Applied egg-rr 34.0%

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

   7. Taylor expanded in a around 0 34.0%

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

   8. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   9. 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. unpow2 0.0%

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

      4. rem-square-sqrt 71.3%

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

      5. associate-*l* 71.1%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r/ 0.0%

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

      10. unpow2 0.0%

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

      11. rem-square-sqrt 71.3%

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

      12. metadata-eval 71.3%

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

   10. Simplified 71.3%

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

Alternative 13: 68.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-305}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.8e-305) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.8e-305) {
		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 <= 8.8d-305) 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 <= 8.8e-305) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.8e-305:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.8e-305)
		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 <= 8.8e-305)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.8e-305], 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 < 8.79999999999999987e-305

   1. Initial program 76.5%

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

      1. sqr-neg 76.5%

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

      2. sqr-neg 76.5%

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

      3. associate-*l* 76.5%

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

   3. Simplified 76.5%

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

   6. Applied egg-rr 65.2%

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

   7. Taylor expanded in b around -inf 66.2%

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

      1. *-commutative 66.2%

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

      2. associate-*l/ 66.2%

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

      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 8.79999999999999987e-305 < b

   1. Initial program 30.3%

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

      1. sqr-neg 30.3%

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

      2. sqr-neg 30.3%

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

      3. associate-*l* 30.3%

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

   3. Simplified 30.3%

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

   6. Applied egg-rr 34.0%

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

   7. Taylor expanded in a around 0 34.0%

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

   8. Taylor expanded in b around inf 0.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{c \cdot {\left(\sqrt{-3}\right)}^{2}}{b}} \]
   9. 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. unpow2 0.0%

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

      4. rem-square-sqrt 71.3%

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

      5. associate-*l* 71.1%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow2 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r/ 0.0%

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

      10. unpow2 0.0%

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

      11. rem-square-sqrt 71.3%

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

      12. metadata-eval 71.3%

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

   10. Simplified 71.3%

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

Alternative 14: 36.1% 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 55.6%

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

   1. sqr-neg 55.6%

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

   2. sqr-neg 55.6%

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

   3. associate-*l* 55.6%

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

3. Simplified 55.6%

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

6. Applied egg-rr 51.0%

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

7. Taylor expanded in b around -inf 37.6%

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

   1. *-commutative 37.6%

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

   2. associate-*l/ 37.6%

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

   3. associate-/l* 37.6%

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

9. Simplified 37.6%

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

Reproduce

herbie shell --seed 2024100 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))