Cubic critical

Percentage Accurate: 52.4% → 86.1%

Time: 16.9s

Alternatives: 12

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.4% 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: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.56 \cdot 10^{+152}:\\ \;\;\;\;\frac{b}{-3} \cdot \frac{\mathsf{fma}\left(-1.5 \cdot a, c \cdot {b}^{-2}, 2\right)}{a}\\ \mathbf{elif}\;b \leq 2.02 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.56e+152)
   (* (/ b (- 3.0)) (/ (fma (* -1.5 a) (* c (pow b -2.0)) 2.0) a))
   (if (<= b 2.02e-66)
     (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.56e+152) {
		tmp = (b / -3.0) * (fma((-1.5 * a), (c * pow(b, -2.0)), 2.0) / a);
	} else if (b <= 2.02e-66) {
		tmp = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.56e+152)
		tmp = Float64(Float64(b / Float64(-3.0)) * Float64(fma(Float64(-1.5 * a), Float64(c * (b ^ -2.0)), 2.0) / a));
	elseif (b <= 2.02e-66)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.56e+152], N[(N[(b / (-3.0)), $MachinePrecision] * N[(N[(N[(-1.5 * a), $MachinePrecision] * N[(c * N[Power[b, -2.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.02e-66], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5599999999999999e152

   1. Initial program 39.2%

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

      1. sqr-neg 39.2%

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

      2. sqr-neg 39.2%

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

      3. associate-*l* 39.2%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in b around -inf 86.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-*r* 86.5%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\right) \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{3 \cdot a} \]

      2. mul-1-neg 86.5%

         \[\leadsto \frac{\color{blue}{\left(-b\right)} \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)}{3 \cdot a} \]

      3. associate-/l* 94.8%

         \[\leadsto \frac{\left(-b\right) \cdot \left(2 + -1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{2}}\right)}\right)}{3 \cdot a} \]

   7. Simplified 94.8%

      \[\leadsto \frac{\color{blue}{\left(-b\right) \cdot \left(2 + -1.5 \cdot \left(a \cdot \frac{c}{{b}^{2}}\right)\right)}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. times-frac 94.9%

         \[\leadsto \color{blue}{\frac{-b}{3} \cdot \frac{2 + -1.5 \cdot \left(a \cdot \frac{c}{{b}^{2}}\right)}{a}} \]

      2. +-commutative 94.9%

         \[\leadsto \frac{-b}{3} \cdot \frac{\color{blue}{-1.5 \cdot \left(a \cdot \frac{c}{{b}^{2}}\right) + 2}}{a} \]

      3. associate-*r* 94.9%

         \[\leadsto \frac{-b}{3} \cdot \frac{\color{blue}{\left(-1.5 \cdot a\right) \cdot \frac{c}{{b}^{2}}} + 2}{a} \]

      4. fma-define 94.9%

         \[\leadsto \frac{-b}{3} \cdot \frac{\color{blue}{\mathsf{fma}\left(-1.5 \cdot a, \frac{c}{{b}^{2}}, 2\right)}}{a} \]

      5. div-inv 94.9%

         \[\leadsto \frac{-b}{3} \cdot \frac{\mathsf{fma}\left(-1.5 \cdot a, \color{blue}{c \cdot \frac{1}{{b}^{2}}}, 2\right)}{a} \]

      6. pow-flip 94.9%

         \[\leadsto \frac{-b}{3} \cdot \frac{\mathsf{fma}\left(-1.5 \cdot a, c \cdot \color{blue}{{b}^{\left(-2\right)}}, 2\right)}{a} \]

      7. metadata-eval 94.9%

         \[\leadsto \frac{-b}{3} \cdot \frac{\mathsf{fma}\left(-1.5 \cdot a, c \cdot {b}^{\color{blue}{-2}}, 2\right)}{a} \]

   9. Applied egg-rr 94.9%

      \[\leadsto \color{blue}{\frac{-b}{3} \cdot \frac{\mathsf{fma}\left(-1.5 \cdot a, c \cdot {b}^{-2}, 2\right)}{a}} \]

   if -1.5599999999999999e152 < b < 2.0200000000000001e-66

   1. Initial program 81.9%

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

   if 2.0200000000000001e-66 < b

   1. Initial program 13.4%

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

      1. sqr-neg 13.4%

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

      2. sqr-neg 13.4%

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

      3. associate-*l* 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in b around inf 87.5%

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

      1. *-commutative 87.5%

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

   7. Simplified 87.5%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.56 \cdot 10^{+152}:\\ \;\;\;\;\frac{b}{-3} \cdot \frac{\mathsf{fma}\left(-1.5 \cdot a, c \cdot {b}^{-2}, 2\right)}{a}\\ \mathbf{elif}\;b \leq 2.02 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+150}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e+150)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 7.5e-67)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e+150) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 7.5e-67) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.6d+150)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 7.5d-67) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e+150) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 7.5e-67) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.6e+150:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 7.5e-67:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e+150)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 7.5e-67)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.6e+150)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 7.5e-67)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e+150], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7.5e-67], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.60000000000000008e150

   1. Initial program 39.2%

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

      1. sqr-neg 39.2%

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

      2. sqr-neg 39.2%

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

      3. associate-*l* 39.2%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in b around -inf 94.8%

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

      1. *-commutative 94.8%

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

   7. Simplified 94.8%

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

      1. associate-*l/ 94.9%

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

   9. Applied egg-rr 94.9%

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

   if -1.60000000000000008e150 < b < 7.5000000000000005e-67

   1. Initial program 81.9%

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

      1. sqr-neg 81.9%

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

      2. sqr-neg 81.9%

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

      3. associate-*l* 81.8%

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

   3. Simplified 81.8%

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

   if 7.5000000000000005e-67 < b

   1. Initial program 13.4%

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

      1. sqr-neg 13.4%

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

      2. sqr-neg 13.4%

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

      3. associate-*l* 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in b around inf 87.5%

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

      1. *-commutative 87.5%

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

   7. Simplified 87.5%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+150}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{+150}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.26e+150)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 2.1e-67)
     (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.26e+150) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 2.1e-67) {
		tmp = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.26d+150)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 2.1d-67) then
        tmp = (sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.26e+150) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 2.1e-67) {
		tmp = (Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.26e+150:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 2.1e-67:
		tmp = (math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.26e+150)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 2.1e-67)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.26e+150)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 2.1e-67)
		tmp = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.26e+150], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.1e-67], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.26e150

   1. Initial program 39.2%

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

      1. sqr-neg 39.2%

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

      2. sqr-neg 39.2%

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

      3. associate-*l* 39.2%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in b around -inf 94.8%

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

      1. *-commutative 94.8%

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

   7. Simplified 94.8%

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

      1. associate-*l/ 94.9%

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

   9. Applied egg-rr 94.9%

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

   if -1.26e150 < b < 2.1000000000000002e-67

   1. Initial program 81.9%

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

   if 2.1000000000000002e-67 < b

   1. Initial program 13.4%

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

      1. sqr-neg 13.4%

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

      2. sqr-neg 13.4%

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

      3. associate-*l* 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in b around inf 87.5%

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

      1. *-commutative 87.5%

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

   7. Simplified 87.5%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{+150}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -115000000:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-66}:\\ \;\;\;\;\frac{b + \sqrt{\left(a \cdot c\right) \cdot -3}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -115000000.0)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 2.05e-66)
     (/ (+ b (sqrt (* (* a c) -3.0))) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -115000000.0) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 2.05e-66) {
		tmp = (b + sqrt(((a * c) * -3.0))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-115000000.0d0)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 2.05d-66) then
        tmp = (b + sqrt(((a * c) * (-3.0d0)))) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -115000000.0) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 2.05e-66) {
		tmp = (b + Math.sqrt(((a * c) * -3.0))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -115000000.0:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 2.05e-66:
		tmp = (b + math.sqrt(((a * c) * -3.0))) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -115000000.0)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 2.05e-66)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(a * c) * -3.0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -115000000.0)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 2.05e-66)
		tmp = (b + sqrt(((a * c) * -3.0))) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -115000000.0], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.05e-66], N[(N[(b + N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.15e8

   1. Initial program 58.1%

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

      1. sqr-neg 58.1%

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

      2. sqr-neg 58.1%

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

      3. associate-*l* 58.1%

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

   3. Simplified 58.1%

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

   5. Taylor expanded in b around -inf 78.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-*r* 78.4%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\right) \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{3 \cdot a} \]

      2. mul-1-neg 78.4%

         \[\leadsto \frac{\color{blue}{\left(-b\right)} \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)}{3 \cdot a} \]

      3. associate-/l* 82.8%

         \[\leadsto \frac{\left(-b\right) \cdot \left(2 + -1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{2}}\right)}\right)}{3 \cdot a} \]

   7. Simplified 82.8%

      \[\leadsto \frac{\color{blue}{\left(-b\right) \cdot \left(2 + -1.5 \cdot \left(a \cdot \frac{c}{{b}^{2}}\right)\right)}}{3 \cdot a} \]

   8. Taylor expanded in a around inf 83.0%

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

   if -1.15e8 < b < 2.04999999999999999e-66

   1. Initial program 83.4%

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

      1. sqr-neg 83.4%

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

      2. sqr-neg 83.4%

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

      3. associate-*l* 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in b around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. *-un-lft-identity 75.1%

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

      3. fma-define 75.1%

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

      4. *-commutative 75.1%

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

      5. *-commutative 75.1%

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

      6. associate-*r* 75.2%

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

      7. add-sqr-sqrt 38.1%

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

      8. sqrt-unprod 74.9%

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

      9. sqr-neg 74.9%

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

      10. sqrt-prod 37.0%

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

      11. add-sqr-sqrt 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. fma-undefine 74.0%

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

      2. *-lft-identity 74.0%

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

      3. associate-*r* 73.9%

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

      4. *-commutative 73.9%

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

      5. associate-*l* 73.8%

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

   9. Simplified 73.8%

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

   10. Taylor expanded in a around 0 73.9%

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

   if 2.04999999999999999e-66 < b

   1. Initial program 13.4%

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

      1. sqr-neg 13.4%

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

      2. sqr-neg 13.4%

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

      3. associate-*l* 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in b around inf 87.5%

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

      1. *-commutative 87.5%

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

   7. Simplified 87.5%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -115000000:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 2.05 \cdot 10^{-66}:\\ \;\;\;\;\frac{b + \sqrt{\left(a \cdot c\right) \cdot -3}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -82000000:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -82000000.0)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (if (<= b 2.2e-66)
     (/ (+ b (sqrt (* c (* a -3.0)))) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -82000000.0) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 2.2e-66) {
		tmp = (b + sqrt((c * (a * -3.0)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-82000000.0d0)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else if (b <= 2.2d-66) then
        tmp = (b + sqrt((c * (a * (-3.0d0))))) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -82000000.0) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else if (b <= 2.2e-66) {
		tmp = (b + Math.sqrt((c * (a * -3.0)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -82000000.0:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	elif b <= 2.2e-66:
		tmp = (b + math.sqrt((c * (a * -3.0)))) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -82000000.0)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	elseif (b <= 2.2e-66)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -3.0)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -82000000.0)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	elseif (b <= 2.2e-66)
		tmp = (b + sqrt((c * (a * -3.0)))) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -82000000.0], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.2e-66], N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.2e7

   1. Initial program 58.1%

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

      1. sqr-neg 58.1%

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

      2. sqr-neg 58.1%

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

      3. associate-*l* 58.1%

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

   3. Simplified 58.1%

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

   5. Taylor expanded in b around -inf 78.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-*r* 78.4%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\right) \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{3 \cdot a} \]

      2. mul-1-neg 78.4%

         \[\leadsto \frac{\color{blue}{\left(-b\right)} \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)}{3 \cdot a} \]

      3. associate-/l* 82.8%

         \[\leadsto \frac{\left(-b\right) \cdot \left(2 + -1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{2}}\right)}\right)}{3 \cdot a} \]

   7. Simplified 82.8%

      \[\leadsto \frac{\color{blue}{\left(-b\right) \cdot \left(2 + -1.5 \cdot \left(a \cdot \frac{c}{{b}^{2}}\right)\right)}}{3 \cdot a} \]

   8. Taylor expanded in a around inf 83.0%

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

   if -8.2e7 < b < 2.2000000000000001e-66

   1. Initial program 83.4%

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

      1. sqr-neg 83.4%

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

      2. sqr-neg 83.4%

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

      3. associate-*l* 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in b around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. *-un-lft-identity 75.1%

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

      3. fma-define 75.1%

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

      4. *-commutative 75.1%

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

      5. *-commutative 75.1%

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

      6. associate-*r* 75.2%

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

      7. add-sqr-sqrt 38.1%

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

      8. sqrt-unprod 74.9%

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

      9. sqr-neg 74.9%

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

      10. sqrt-prod 37.0%

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

      11. add-sqr-sqrt 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. fma-undefine 74.0%

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

      2. *-lft-identity 74.0%

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

   9. Simplified 74.0%

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

   if 2.2000000000000001e-66 < b

   1. Initial program 13.4%

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

      1. sqr-neg 13.4%

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

      2. sqr-neg 13.4%

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

      3. associate-*l* 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in b around inf 87.5%

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

      1. *-commutative 87.5%

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

   7. Simplified 87.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -82000000:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -82000000:\\ \;\;\;\;b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{-67}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -82000000.0)
   (* b (- (/ (* c 0.5) (pow b 2.0)) (/ 0.6666666666666666 a)))
   (if (<= b 5.3e-67)
     (/ (+ b (sqrt (* c (* a -3.0)))) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -82000000.0) {
		tmp = b * (((c * 0.5) / pow(b, 2.0)) - (0.6666666666666666 / a));
	} else if (b <= 5.3e-67) {
		tmp = (b + sqrt((c * (a * -3.0)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-82000000.0d0)) then
        tmp = b * (((c * 0.5d0) / (b ** 2.0d0)) - (0.6666666666666666d0 / a))
    else if (b <= 5.3d-67) then
        tmp = (b + sqrt((c * (a * (-3.0d0))))) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -82000000.0) {
		tmp = b * (((c * 0.5) / Math.pow(b, 2.0)) - (0.6666666666666666 / a));
	} else if (b <= 5.3e-67) {
		tmp = (b + Math.sqrt((c * (a * -3.0)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -82000000.0:
		tmp = b * (((c * 0.5) / math.pow(b, 2.0)) - (0.6666666666666666 / a))
	elif b <= 5.3e-67:
		tmp = (b + math.sqrt((c * (a * -3.0)))) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -82000000.0)
		tmp = Float64(b * Float64(Float64(Float64(c * 0.5) / (b ^ 2.0)) - Float64(0.6666666666666666 / a)));
	elseif (b <= 5.3e-67)
		tmp = Float64(Float64(b + sqrt(Float64(c * Float64(a * -3.0)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -82000000.0)
		tmp = b * (((c * 0.5) / (b ^ 2.0)) - (0.6666666666666666 / a));
	elseif (b <= 5.3e-67)
		tmp = (b + sqrt((c * (a * -3.0)))) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -82000000.0], N[(b * N[(N[(N[(c * 0.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.3e-67], N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.2e7

   1. Initial program 58.1%

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

      1. sqr-neg 58.1%

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

      2. sqr-neg 58.1%

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

      3. associate-*l* 58.1%

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

   3. Simplified 58.1%

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

   5. Taylor expanded in b around -inf 78.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-*r* 78.4%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\right) \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{3 \cdot a} \]

      2. mul-1-neg 78.4%

         \[\leadsto \frac{\color{blue}{\left(-b\right)} \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)}{3 \cdot a} \]

      3. associate-/l* 82.8%

         \[\leadsto \frac{\left(-b\right) \cdot \left(2 + -1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{2}}\right)}\right)}{3 \cdot a} \]

   7. Simplified 82.8%

      \[\leadsto \frac{\color{blue}{\left(-b\right) \cdot \left(2 + -1.5 \cdot \left(a \cdot \frac{c}{{b}^{2}}\right)\right)}}{3 \cdot a} \]

   8. Taylor expanded in b around inf 83.0%

      \[\leadsto \color{blue}{b \cdot \left(0.5 \cdot \frac{c}{{b}^{2}} - 0.6666666666666666 \cdot \frac{1}{a}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 83.0%

         \[\leadsto b \cdot \left(\color{blue}{\frac{0.5 \cdot c}{{b}^{2}}} - 0.6666666666666666 \cdot \frac{1}{a}\right) \]

      2. *-commutative 83.0%

         \[\leadsto b \cdot \left(\frac{\color{blue}{c \cdot 0.5}}{{b}^{2}} - 0.6666666666666666 \cdot \frac{1}{a}\right) \]

      3. associate-*r/ 83.1%

         \[\leadsto b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \]

      4. metadata-eval 83.1%

         \[\leadsto b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \frac{\color{blue}{0.6666666666666666}}{a}\right) \]

   10. Simplified 83.1%

       \[\leadsto \color{blue}{b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \frac{0.6666666666666666}{a}\right)} \]

   if -8.2e7 < b < 5.29999999999999971e-67

   1. Initial program 83.4%

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

      1. sqr-neg 83.4%

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

      2. sqr-neg 83.4%

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

      3. associate-*l* 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in b around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. *-un-lft-identity 75.1%

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

      3. fma-define 75.1%

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

      4. *-commutative 75.1%

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

      5. *-commutative 75.1%

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

      6. associate-*r* 75.2%

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

      7. add-sqr-sqrt 38.1%

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

      8. sqrt-unprod 74.9%

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

      9. sqr-neg 74.9%

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

      10. sqrt-prod 37.0%

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

      11. add-sqr-sqrt 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. fma-undefine 74.0%

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

      2. *-lft-identity 74.0%

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

   9. Simplified 74.0%

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

   if 5.29999999999999971e-67 < b

   1. Initial program 13.4%

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

      1. sqr-neg 13.4%

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

      2. sqr-neg 13.4%

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

      3. associate-*l* 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in b around inf 87.5%

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

      1. *-commutative 87.5%

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

   7. Simplified 87.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -82000000:\\ \;\;\;\;b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{-67}:\\ \;\;\;\;\frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1050000000:\\ \;\;\;\;b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1050000000.0)
   (* b (- (/ (* c 0.5) (pow b 2.0)) (/ 0.6666666666666666 a)))
   (if (<= b 3.2e-67)
     (/ (- (sqrt (* (* a c) -3.0)) b) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1050000000.0) {
		tmp = b * (((c * 0.5) / pow(b, 2.0)) - (0.6666666666666666 / a));
	} else if (b <= 3.2e-67) {
		tmp = (sqrt(((a * c) * -3.0)) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-1050000000.0d0)) then
        tmp = b * (((c * 0.5d0) / (b ** 2.0d0)) - (0.6666666666666666d0 / a))
    else if (b <= 3.2d-67) then
        tmp = (sqrt(((a * c) * (-3.0d0))) - b) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1050000000.0) {
		tmp = b * (((c * 0.5) / Math.pow(b, 2.0)) - (0.6666666666666666 / a));
	} else if (b <= 3.2e-67) {
		tmp = (Math.sqrt(((a * c) * -3.0)) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1050000000.0:
		tmp = b * (((c * 0.5) / math.pow(b, 2.0)) - (0.6666666666666666 / a))
	elif b <= 3.2e-67:
		tmp = (math.sqrt(((a * c) * -3.0)) - b) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1050000000.0)
		tmp = Float64(b * Float64(Float64(Float64(c * 0.5) / (b ^ 2.0)) - Float64(0.6666666666666666 / a)));
	elseif (b <= 3.2e-67)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) * -3.0)) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1050000000.0)
		tmp = b * (((c * 0.5) / (b ^ 2.0)) - (0.6666666666666666 / a));
	elseif (b <= 3.2e-67)
		tmp = (sqrt(((a * c) * -3.0)) - b) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1050000000.0], N[(b * N[(N[(N[(c * 0.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.2e-67], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05e9

   1. Initial program 58.1%

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

      1. sqr-neg 58.1%

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

      2. sqr-neg 58.1%

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

      3. associate-*l* 58.1%

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

   3. Simplified 58.1%

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

   5. Taylor expanded in b around -inf 78.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-*r* 78.4%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\right) \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{3 \cdot a} \]

      2. mul-1-neg 78.4%

         \[\leadsto \frac{\color{blue}{\left(-b\right)} \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)}{3 \cdot a} \]

      3. associate-/l* 82.8%

         \[\leadsto \frac{\left(-b\right) \cdot \left(2 + -1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{2}}\right)}\right)}{3 \cdot a} \]

   7. Simplified 82.8%

      \[\leadsto \frac{\color{blue}{\left(-b\right) \cdot \left(2 + -1.5 \cdot \left(a \cdot \frac{c}{{b}^{2}}\right)\right)}}{3 \cdot a} \]

   8. Taylor expanded in b around inf 83.0%

      \[\leadsto \color{blue}{b \cdot \left(0.5 \cdot \frac{c}{{b}^{2}} - 0.6666666666666666 \cdot \frac{1}{a}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 83.0%

         \[\leadsto b \cdot \left(\color{blue}{\frac{0.5 \cdot c}{{b}^{2}}} - 0.6666666666666666 \cdot \frac{1}{a}\right) \]

      2. *-commutative 83.0%

         \[\leadsto b \cdot \left(\frac{\color{blue}{c \cdot 0.5}}{{b}^{2}} - 0.6666666666666666 \cdot \frac{1}{a}\right) \]

      3. associate-*r/ 83.1%

         \[\leadsto b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \]

      4. metadata-eval 83.1%

         \[\leadsto b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \frac{\color{blue}{0.6666666666666666}}{a}\right) \]

   10. Simplified 83.1%

       \[\leadsto \color{blue}{b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \frac{0.6666666666666666}{a}\right)} \]

   if -1.05e9 < b < 3.20000000000000021e-67

   1. Initial program 83.4%

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

      1. sqr-neg 83.4%

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

      2. sqr-neg 83.4%

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

      3. associate-*l* 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in b around 0 75.1%

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

   if 3.20000000000000021e-67 < b

   1. Initial program 13.4%

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

      1. sqr-neg 13.4%

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

      2. sqr-neg 13.4%

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

      3. associate-*l* 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in b around inf 87.5%

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

      1. *-commutative 87.5%

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

   7. Simplified 87.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1050000000:\\ \;\;\;\;b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot c\right) \cdot -3} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -82000000:\\ \;\;\;\;b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -82000000.0)
   (* b (- (/ (* c 0.5) (pow b 2.0)) (/ 0.6666666666666666 a)))
   (if (<= b 2.15e-66)
     (/ (- (sqrt (* c (* a -3.0))) b) (* 3.0 a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -82000000.0) {
		tmp = b * (((c * 0.5) / pow(b, 2.0)) - (0.6666666666666666 / a));
	} else if (b <= 2.15e-66) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-82000000.0d0)) then
        tmp = b * (((c * 0.5d0) / (b ** 2.0d0)) - (0.6666666666666666d0 / a))
    else if (b <= 2.15d-66) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (3.0d0 * a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -82000000.0) {
		tmp = b * (((c * 0.5) / Math.pow(b, 2.0)) - (0.6666666666666666 / a));
	} else if (b <= 2.15e-66) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -82000000.0:
		tmp = b * (((c * 0.5) / math.pow(b, 2.0)) - (0.6666666666666666 / a))
	elif b <= 2.15e-66:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (3.0 * a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -82000000.0)
		tmp = Float64(b * Float64(Float64(Float64(c * 0.5) / (b ^ 2.0)) - Float64(0.6666666666666666 / a)));
	elseif (b <= 2.15e-66)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -82000000.0)
		tmp = b * (((c * 0.5) / (b ^ 2.0)) - (0.6666666666666666 / a));
	elseif (b <= 2.15e-66)
		tmp = (sqrt((c * (a * -3.0))) - b) / (3.0 * a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -82000000.0], N[(b * N[(N[(N[(c * 0.5), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.15e-66], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.2e7

   1. Initial program 58.1%

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

      1. sqr-neg 58.1%

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

      2. sqr-neg 58.1%

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

      3. associate-*l* 58.1%

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

   3. Simplified 58.1%

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

   5. Taylor expanded in b around -inf 78.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-*r* 78.4%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\right) \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{3 \cdot a} \]

      2. mul-1-neg 78.4%

         \[\leadsto \frac{\color{blue}{\left(-b\right)} \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)}{3 \cdot a} \]

      3. associate-/l* 82.8%

         \[\leadsto \frac{\left(-b\right) \cdot \left(2 + -1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{2}}\right)}\right)}{3 \cdot a} \]

   7. Simplified 82.8%

      \[\leadsto \frac{\color{blue}{\left(-b\right) \cdot \left(2 + -1.5 \cdot \left(a \cdot \frac{c}{{b}^{2}}\right)\right)}}{3 \cdot a} \]

   8. Taylor expanded in b around inf 83.0%

      \[\leadsto \color{blue}{b \cdot \left(0.5 \cdot \frac{c}{{b}^{2}} - 0.6666666666666666 \cdot \frac{1}{a}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 83.0%

         \[\leadsto b \cdot \left(\color{blue}{\frac{0.5 \cdot c}{{b}^{2}}} - 0.6666666666666666 \cdot \frac{1}{a}\right) \]

      2. *-commutative 83.0%

         \[\leadsto b \cdot \left(\frac{\color{blue}{c \cdot 0.5}}{{b}^{2}} - 0.6666666666666666 \cdot \frac{1}{a}\right) \]

      3. associate-*r/ 83.1%

         \[\leadsto b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \color{blue}{\frac{0.6666666666666666 \cdot 1}{a}}\right) \]

      4. metadata-eval 83.1%

         \[\leadsto b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \frac{\color{blue}{0.6666666666666666}}{a}\right) \]

   10. Simplified 83.1%

       \[\leadsto \color{blue}{b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \frac{0.6666666666666666}{a}\right)} \]

   if -8.2e7 < b < 2.15000000000000007e-66

   1. Initial program 83.4%

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

      1. sqr-neg 83.4%

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

      2. sqr-neg 83.4%

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

      3. associate-*l* 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in b around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. *-commutative 75.1%

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

      3. associate-*r* 75.2%

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

   7. Simplified 75.2%

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

   if 2.15000000000000007e-66 < b

   1. Initial program 13.4%

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

      1. sqr-neg 13.4%

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

      2. sqr-neg 13.4%

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

      3. associate-*l* 13.4%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in b around inf 87.5%

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

      1. *-commutative 87.5%

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

   7. Simplified 87.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -82000000:\\ \;\;\;\;b \cdot \left(\frac{c \cdot 0.5}{{b}^{2}} - \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-4d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 70.4%

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

      1. sqr-neg 70.4%

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

      2. sqr-neg 70.4%

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

      3. associate-*l* 70.5%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in b around -inf 55.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(b \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. associate-*r* 55.6%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot b\right) \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)}}{3 \cdot a} \]

      2. mul-1-neg 55.6%

         \[\leadsto \frac{\color{blue}{\left(-b\right)} \cdot \left(2 + -1.5 \cdot \frac{a \cdot c}{{b}^{2}}\right)}{3 \cdot a} \]

      3. associate-/l* 58.4%

         \[\leadsto \frac{\left(-b\right) \cdot \left(2 + -1.5 \cdot \color{blue}{\left(a \cdot \frac{c}{{b}^{2}}\right)}\right)}{3 \cdot a} \]

   7. Simplified 58.4%

      \[\leadsto \frac{\color{blue}{\left(-b\right) \cdot \left(2 + -1.5 \cdot \left(a \cdot \frac{c}{{b}^{2}}\right)\right)}}{3 \cdot a} \]

   8. Taylor expanded in a around inf 58.7%

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

   if -3.999999999999988e-310 < b

   1. Initial program 34.2%

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

      1. sqr-neg 34.2%

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

      2. sqr-neg 34.2%

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

      3. associate-*l* 34.2%

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

   3. Simplified 34.2%

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

   5. Taylor expanded in b around inf 64.4%

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

      1. *-commutative 64.4%

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

   7. Simplified 64.4%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.7% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		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 <= (-4d-310)) 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 <= -4e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		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 <= -4e-310)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], 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 < -3.999999999999988e-310

   1. Initial program 70.4%

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

      1. sqr-neg 70.4%

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

      2. sqr-neg 70.4%

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

      3. associate-*l* 70.5%

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

   3. Simplified 70.5%

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

      1. +-commutative 70.5%

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

      2. add-sqr-sqrt 70.3%

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

      3. fma-define 70.3%

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

   6. Applied egg-rr 59.8%

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

      1. associate-*r* 59.9%

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

      2. *-commutative 59.9%

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

      3. associate-*r* 59.9%

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

   8. Simplified 59.9%

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

   9. Taylor expanded in b around -inf 57.9%

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

       1. associate-*r/ 57.9%

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

       2. *-commutative 57.9%

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

       3. associate-/l* 57.9%

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

   11. Simplified 57.9%

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

   if -3.999999999999988e-310 < b

   1. Initial program 34.2%

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

      1. sqr-neg 34.2%

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

      2. sqr-neg 34.2%

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

      3. associate-*l* 34.2%

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

   3. Simplified 34.2%

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

      1. +-commutative 34.2%

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

      2. add-sqr-sqrt 32.4%

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

      3. fma-define 31.5%

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

   6. Applied egg-rr 38.6%

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

      1. associate-*r* 38.6%

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

      2. *-commutative 38.6%

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

      3. associate-*r* 38.7%

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

   8. Simplified 38.7%

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

      1. div-inv 38.6%

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

      2. *-commutative 38.6%

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

   10. Applied egg-rr 38.6%

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

   11. Taylor expanded in b around inf 0.0%

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

       1. *-commutative 0.0%

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

       2. associate-/l* 0.0%

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

       3. associate-*r* 0.0%

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

       4. *-commutative 0.0%

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

       5. associate-*r/ 0.0%

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

       6. unpow2 0.0%

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

       7. rem-square-sqrt 64.2%

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

       8. metadata-eval 64.2%

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

   13. Simplified 64.2%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.8% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5d-310) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 4.999999999999985e-310

   1. Initial program 70.4%

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

      1. sqr-neg 70.4%

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

      2. sqr-neg 70.4%

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

      3. associate-*l* 70.5%

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

   3. Simplified 70.5%

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

      1. +-commutative 70.5%

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

      2. add-sqr-sqrt 70.3%

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

      3. fma-define 70.3%

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

   6. Applied egg-rr 59.8%

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

      1. associate-*r* 59.9%

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

      2. *-commutative 59.9%

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

      3. associate-*r* 59.9%

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

   8. Simplified 59.9%

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

   9. Taylor expanded in b around -inf 57.9%

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

       1. associate-*r/ 57.9%

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

       2. *-commutative 57.9%

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

       3. associate-/l* 57.9%

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

   11. Simplified 57.9%

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

   if 4.999999999999985e-310 < b

   1. Initial program 34.2%

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

      1. sqr-neg 34.2%

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

      2. sqr-neg 34.2%

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

      3. associate-*l* 34.2%

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

   3. Simplified 34.2%

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

   5. Taylor expanded in b around inf 64.4%

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

      1. *-commutative 64.4%

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

   7. Simplified 64.4%

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

4. Final simplification 61.4%

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

Alternative 12: 35.7% 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 51.2%

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

   1. sqr-neg 51.2%

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

   2. sqr-neg 51.2%

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

   3. associate-*l* 51.2%

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

3. Simplified 51.2%

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

   1. +-commutative 51.2%

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

   2. add-sqr-sqrt 50.2%

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

   3. fma-define 49.7%

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

6. Applied egg-rr 48.5%

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

   1. associate-*r* 48.6%

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

   2. *-commutative 48.6%

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

   3. associate-*r* 48.6%

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

8. Simplified 48.6%

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

9. Taylor expanded in b around -inf 28.5%

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

    1. associate-*r/ 28.5%

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

    2. *-commutative 28.5%

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

    3. associate-/l* 28.5%

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

11. Simplified 28.5%

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

12. Final simplification 28.5%

    \[\leadsto b \cdot \frac{-0.6666666666666666}{a} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024078 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))