Cubic critical

Percentage Accurate: 51.8% → 84.6%

Time: 28.0s

Alternatives: 10

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+127}:\\ \;\;\;\;-\mathsf{fma}\left(-0.5, \frac{c}{b}, b \cdot \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.4e+127)
   (- (fma -0.5 (/ c b) (* b (/ 0.6666666666666666 a))))
   (if (<= b 2e-5)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.4e+127) {
		tmp = -fma(-0.5, (c / b), (b * (0.6666666666666666 / a)));
	} else if (b <= 2e-5) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.4e+127)
		tmp = Float64(-fma(-0.5, Float64(c / b), Float64(b * Float64(0.6666666666666666 / a))));
	elseif (b <= 2e-5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.4e+127], (-N[(-0.5 * N[(c / b), $MachinePrecision] + N[(b * N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 2e-5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.4000000000000001e127

   1. Initial program 34.7%

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

      1. sqr-neg 34.7%

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

      2. sqr-neg 34.7%

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

      3. associate-*l* 34.7%

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

   3. Simplified 34.7%

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

   5. Taylor expanded in b around -inf 92.3%

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

   6. Taylor expanded in c around 0 92.4%

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

      1. fma-define 92.4%

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

      2. associate-*r/ 92.3%

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

      3. associate-*l/ 92.6%

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

      4. *-commutative 92.6%

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

   8. Simplified 92.6%

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

   if -1.4000000000000001e127 < b < 2.00000000000000016e-5

   1. Initial program 80.8%

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

   if 2.00000000000000016e-5 < b

   1. Initial program 15.9%

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

      1. sqr-neg 15.9%

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

      2. sqr-neg 15.9%

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

      3. associate-*l* 15.9%

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

   3. Simplified 15.9%

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

   5. Taylor expanded in b around inf 90.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+127}:\\ \;\;\;\;-\mathsf{fma}\left(-0.5, \frac{c}{b}, b \cdot \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+127}:\\ \;\;\;\;-\mathsf{fma}\left(-0.5, \frac{c}{b}, b \cdot \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 0.00066:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+127)
   (- (fma -0.5 (/ c b) (* b (/ 0.6666666666666666 a))))
   (if (<= b 0.00066)
     (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+127) {
		tmp = -fma(-0.5, (c / b), (b * (0.6666666666666666 / a)));
	} else if (b <= 0.00066) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+127)
		tmp = Float64(-fma(-0.5, Float64(c / b), Float64(b * Float64(0.6666666666666666 / a))));
	elseif (b <= 0.00066)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+127], (-N[(-0.5 * N[(c / b), $MachinePrecision] + N[(b * N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 0.00066], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.99999999999999991e127

   1. Initial program 34.7%

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

      1. sqr-neg 34.7%

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

      2. sqr-neg 34.7%

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

      3. associate-*l* 34.7%

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

   3. Simplified 34.7%

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

   5. Taylor expanded in b around -inf 92.3%

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

   6. Taylor expanded in c around 0 92.4%

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

      1. fma-define 92.4%

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

      2. associate-*r/ 92.3%

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

      3. associate-*l/ 92.6%

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

      4. *-commutative 92.6%

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

   8. Simplified 92.6%

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

   if -1.99999999999999991e127 < b < 6.6e-4

   1. Initial program 80.8%

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

      1. sqr-neg 80.8%

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

      2. sqr-neg 80.8%

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

      3. associate-*l* 80.7%

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

   3. Simplified 80.7%

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

   if 6.6e-4 < b

   1. Initial program 15.9%

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

      1. sqr-neg 15.9%

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

      2. sqr-neg 15.9%

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

      3. associate-*l* 15.9%

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

   3. Simplified 15.9%

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

   5. Taylor expanded in b around inf 90.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+127}:\\ \;\;\;\;-\mathsf{fma}\left(-0.5, \frac{c}{b}, b \cdot \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 0.00066:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -180000000:\\ \;\;\;\;-\mathsf{fma}\left(-0.5, \frac{c}{b}, b \cdot \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -180000000.0)
   (- (fma -0.5 (/ c b) (* b (/ 0.6666666666666666 a))))
   (if (<= b 1.4e-5)
     (* -0.3333333333333333 (/ (- b (sqrt (* c (* a -3.0)))) a))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -180000000.0) {
		tmp = -fma(-0.5, (c / b), (b * (0.6666666666666666 / a)));
	} else if (b <= 1.4e-5) {
		tmp = -0.3333333333333333 * ((b - sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -180000000.0)
		tmp = Float64(-fma(-0.5, Float64(c / b), Float64(b * Float64(0.6666666666666666 / a))));
	elseif (b <= 1.4e-5)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(c * Float64(a * -3.0)))) / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -180000000.0], (-N[(-0.5 * N[(c / b), $MachinePrecision] + N[(b * N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 1.4e-5], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.8e8

   1. Initial program 56.7%

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

      1. sqr-neg 56.7%

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

      2. sqr-neg 56.7%

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

      3. associate-*l* 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in b around -inf 91.2%

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

   6. Taylor expanded in c around 0 91.2%

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

      1. fma-define 91.2%

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

      2. associate-*r/ 91.2%

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

      3. associate-*l/ 91.3%

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

      4. *-commutative 91.3%

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

   8. Simplified 91.3%

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

   if -1.8e8 < b < 1.39999999999999998e-5

   1. Initial program 76.7%

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

      1. sqr-neg 76.7%

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

      2. sqr-neg 76.7%

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

      3. associate-*l* 76.6%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in b around 0 66.3%

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

      1. *-un-lft-identity 66.3%

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

      2. frac-2neg 66.3%

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

      3. distribute-neg-in 66.3%

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

      4. add-sqr-sqrt 34.7%

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

      5. sqrt-unprod 66.2%

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

      6. sqr-neg 66.2%

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

      7. sqrt-unprod 31.8%

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

      8. add-sqr-sqrt 64.2%

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

      9. sub-neg 64.2%

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

      10. add-sqr-sqrt 32.4%

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

      11. sqrt-unprod 64.0%

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

      12. sqr-neg 64.0%

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

      13. sqrt-unprod 31.6%

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

      14. add-sqr-sqrt 66.3%

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

      15. *-commutative 66.3%

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

      16. associate-*r* 66.3%

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

      17. *-commutative 66.3%

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

      18. associate-*r* 66.4%

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

      19. *-commutative 66.4%

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

      20. distribute-rgt-neg-in 66.4%

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

   7. Applied egg-rr 66.4%

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

      1. associate-*r/ 66.4%

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

      2. *-commutative 66.4%

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

      3. rem-square-sqrt 0.0%

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

      4. unpow2 0.0%

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

      5. times-frac 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 66.3%

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

      8. metadata-eval 66.3%

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

      9. *-commutative 66.3%

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

      10. *-commutative 66.3%

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

   9. Simplified 66.3%

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

   if 1.39999999999999998e-5 < b

   1. Initial program 15.9%

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

      1. sqr-neg 15.9%

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

      2. sqr-neg 15.9%

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

      3. associate-*l* 15.9%

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

   3. Simplified 15.9%

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

   5. Taylor expanded in b around inf 90.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -180000000:\\ \;\;\;\;-\mathsf{fma}\left(-0.5, \frac{c}{b}, b \cdot \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -190000000:\\ \;\;\;\;-\mathsf{fma}\left(-0.5, \frac{c}{b}, b \cdot \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 0.017:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -190000000.0)
   (- (fma -0.5 (/ c b) (* b (/ 0.6666666666666666 a))))
   (if (<= b 0.017)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -190000000.0) {
		tmp = -fma(-0.5, (c / b), (b * (0.6666666666666666 / a)));
	} else if (b <= 0.017) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -190000000.0)
		tmp = Float64(-fma(-0.5, Float64(c / b), Float64(b * Float64(0.6666666666666666 / a))));
	elseif (b <= 0.017)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -190000000.0], (-N[(-0.5 * N[(c / b), $MachinePrecision] + N[(b * N[(0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 0.017], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9e8

   1. Initial program 56.7%

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

      1. sqr-neg 56.7%

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

      2. sqr-neg 56.7%

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

      3. associate-*l* 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in b around -inf 91.2%

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

   6. Taylor expanded in c around 0 91.2%

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

      1. fma-define 91.2%

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

      2. associate-*r/ 91.2%

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

      3. associate-*l/ 91.3%

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

      4. *-commutative 91.3%

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

   8. Simplified 91.3%

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

   if -1.9e8 < b < 0.017000000000000001

   1. Initial program 76.7%

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

      1. sqr-neg 76.7%

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

      2. sqr-neg 76.7%

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

      3. associate-*l* 76.6%

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

   3. Simplified 76.6%

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

   5. Taylor expanded in b around 0 66.3%

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

      1. +-commutative 66.3%

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

      2. unsub-neg 66.3%

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

      3. *-commutative 66.3%

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

      4. associate-*r* 66.3%

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

      5. *-commutative 66.3%

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

      6. associate-*r* 66.4%

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

   7. Applied egg-rr 66.4%

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

   if 0.017000000000000001 < b

   1. Initial program 15.9%

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

      1. sqr-neg 15.9%

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

      2. sqr-neg 15.9%

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

      3. associate-*l* 15.9%

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

   3. Simplified 15.9%

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

   5. Taylor expanded in b around inf 90.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -190000000:\\ \;\;\;\;-\mathsf{fma}\left(-0.5, \frac{c}{b}, b \cdot \frac{0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 0.017:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-\mathsf{fma}\left(-0.5, \frac{c}{b}, b \cdot \frac{0.6666666666666666}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 69.4%

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

      1. sqr-neg 69.4%

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

      2. sqr-neg 69.4%

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

      3. associate-*l* 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in b around -inf 60.4%

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

   6. Taylor expanded in c around 0 61.2%

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

      1. fma-define 61.2%

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

      2. associate-*r/ 61.2%

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

      3. associate-*l/ 61.3%

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

      4. *-commutative 61.3%

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

   8. Simplified 61.3%

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

   if -4.999999999999985e-310 < b

   1. Initial program 42.3%

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

      1. sqr-neg 42.3%

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

      2. sqr-neg 42.3%

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

      3. associate-*l* 42.3%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in b around inf 56.0%

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

4. Final simplification 58.9%

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

Alternative 6: 67.1% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 69.4%

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

      1. sqr-neg 69.4%

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

      2. sqr-neg 69.4%

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

      3. associate-*l* 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in b around -inf 60.4%

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

   6. Taylor expanded in c around 0 61.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 42.3%

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

      1. sqr-neg 42.3%

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

      2. sqr-neg 42.3%

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

      3. associate-*l* 42.3%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in b around inf 56.0%

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

4. Final simplification 58.8%

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

Alternative 7: 66.9% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-298}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-3}{b \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8e-298) (/ (/ 1.0 a) (/ -3.0 (* b 2.0))) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-298) {
		tmp = (1.0 / a) / (-3.0 / (b * 2.0));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 8d-298) then
        tmp = (1.0d0 / a) / ((-3.0d0) / (b * 2.0d0))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-298) {
		tmp = (1.0 / a) / (-3.0 / (b * 2.0));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8e-298:
		tmp = (1.0 / a) / (-3.0 / (b * 2.0))
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e-298)
		tmp = Float64(Float64(1.0 / a) / Float64(-3.0 / Float64(b * 2.0)));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8e-298)
		tmp = (1.0 / a) / (-3.0 / (b * 2.0));
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e-298], N[(N[(1.0 / a), $MachinePrecision] / N[(-3.0 / N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.9999999999999993e-298

   1. Initial program 68.5%

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

      1. sqr-neg 68.5%

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

      2. sqr-neg 68.5%

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

      3. associate-*l* 68.4%

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

   3. Simplified 68.4%

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

   6. Applied egg-rr 67.3%

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

      1. *-commutative 67.3%

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

      2. clear-num 67.1%

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

      3. un-div-inv 67.2%

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

   8. Applied egg-rr 67.2%

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

   9. Taylor expanded in b around -inf 59.9%

      \[\leadsto \frac{\frac{1}{a}}{\frac{-3}{\color{blue}{2 \cdot b}}} \]
   10. Step-by-step derivation

       1. *-commutative 59.9%

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

   11. Simplified 59.9%

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

   if 7.9999999999999993e-298 < b

   1. Initial program 43.0%

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

      1. sqr-neg 43.0%

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

      2. sqr-neg 43.0%

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

      3. associate-*l* 42.9%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in b around inf 56.9%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-298}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-3}{b \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-302}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.7e-302) (* b (/ -0.6666666666666666 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.7e-302) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.7d-302) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.7e-302) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.7e-302:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.7e-302)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.7e-302)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.7e-302], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.70000000000000006e-302

   1. Initial program 68.5%

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

      1. sqr-neg 68.5%

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

      2. sqr-neg 68.5%

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

      3. associate-*l* 68.4%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in b around -inf 59.8%

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

      1. *-commutative 59.8%

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

   7. Simplified 59.8%

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

      1. *-commutative 59.8%

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

      2. clear-num 59.7%

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

      3. un-div-inv 59.8%

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

   9. Applied egg-rr 59.8%

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

       1. associate-/r/ 59.8%

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

   11. Simplified 59.8%

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

   if 2.70000000000000006e-302 < b

   1. Initial program 43.0%

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

      1. sqr-neg 43.0%

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

      2. sqr-neg 43.0%

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

      3. associate-*l* 42.9%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in b around inf 56.9%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-302}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-302}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.4e-302) (/ b (* a -1.5)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-302) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.4d-302) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.4e-302) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.4e-302:
		tmp = b / (a * -1.5)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.4e-302)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.4e-302)
		tmp = b / (a * -1.5);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.4e-302], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.40000000000000022e-302

   1. Initial program 68.5%

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

      1. sqr-neg 68.5%

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

      2. sqr-neg 68.5%

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

      3. associate-*l* 68.4%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in b around -inf 59.8%

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

      1. *-commutative 59.8%

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

   7. Simplified 59.8%

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

      1. *-commutative 59.8%

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

      2. clear-num 59.7%

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

      3. un-div-inv 59.8%

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

   9. Applied egg-rr 59.8%

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

       1. associate-/r/ 59.8%

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

   11. Simplified 59.8%

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

       1. *-commutative 59.8%

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

       2. clear-num 59.8%

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

       3. un-div-inv 59.8%

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

       4. div-inv 59.9%

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

       5. metadata-eval 59.9%

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

   13. Applied egg-rr 59.9%

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

   if 2.40000000000000022e-302 < b

   1. Initial program 43.0%

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

      1. sqr-neg 43.0%

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

      2. sqr-neg 43.0%

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

      3. associate-*l* 42.9%

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

   3. Simplified 42.9%

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

   5. Taylor expanded in b around inf 56.9%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-302}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.0% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.5 (/ c b)))

C

double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.5d0) * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Python

def code(a, b, c):
	return -0.5 * (c / b)

Julia

function code(a, b, c)
	return Float64(-0.5 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.9%

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

   1. sqr-neg 56.9%

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

   2. sqr-neg 56.9%

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

   3. associate-*l* 56.9%

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

3. Simplified 56.9%

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

5. Taylor expanded in b around inf 27.0%

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

6. Final simplification 27.0%

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

Reproduce

herbie shell --seed 2024053 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))