Quadratic roots, full range

Percentage Accurate: 51.9% → 85.2%

Time: 12.4s

Alternatives: 8

Speedup: 12.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e+150)
   (/ b (- a))
   (if (<= b 9.8e-108)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e+150) {
		tmp = b / -a;
	} else if (b <= 9.8e-108) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = 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 <= (-2.3d+150)) then
        tmp = b / -a
    else if (b <= 9.8d-108) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e+150) {
		tmp = b / -a;
	} else if (b <= 9.8e-108) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.3e+150:
		tmp = b / -a
	elif b <= 9.8e-108:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e+150)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 9.8e-108)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.3e+150)
		tmp = b / -a;
	elseif (b <= 9.8e-108)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.3e+150], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 9.8e-108], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.30000000000000001e150

   1. Initial program 44.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. distribute-neg-frac2 97.9%

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

   5. Simplified 97.9%

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

   if -2.30000000000000001e150 < b < 9.7999999999999996e-108

   1. Initial program 84.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if 9.7999999999999996e-108 < b

   1. Initial program 22.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 84.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. mul-1-neg 84.2%

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

      2. distribute-neg-frac2 84.2%

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

   5. Simplified 84.2%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-63}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.75e-63)
   (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))
   (if (<= b 2e-114) (/ (- (sqrt (* -4.0 (* a c))) b) (* a 2.0)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.75e-63) {
		tmp = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 2e-114) {
		tmp = (sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.75d-63)) then
        tmp = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
    else if (b <= 2d-114) then
        tmp = (sqrt(((-4.0d0) * (a * c))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.75e-63) {
		tmp = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 2e-114) {
		tmp = (Math.sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.75e-63:
		tmp = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
	elif b <= 2e-114:
		tmp = (math.sqrt((-4.0 * (a * c))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.75e-63)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	elseif (b <= 2e-114)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(a * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.75e-63)
		tmp = b * ((c / (b ^ 2.0)) + (-1.0 / a));
	elseif (b <= 2e-114)
		tmp = (sqrt((-4.0 * (a * c))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.75e-63], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-114], N[(N[(N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.75000000000000002e-63

   1. Initial program 70.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 84.7%

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

      1. mul-1-neg 84.7%

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

      2. distribute-rgt-neg-in 84.7%

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

      3. +-commutative 84.7%

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

      4. mul-1-neg 84.7%

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

      5. unsub-neg 84.7%

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

   5. Simplified 84.7%

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

   if -1.75000000000000002e-63 < b < 2.0000000000000001e-114

   1. Initial program 80.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 77.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]

   if 2.0000000000000001e-114 < b

   1. Initial program 23.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 82.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. mul-1-neg 82.7%

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

      2. distribute-neg-frac2 82.7%

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

   5. Simplified 82.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{-63}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{-62}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.4e-62)
   (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))
   (if (<= b 2.8e-114)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.4e-62) {
		tmp = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 2.8e-114) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.4d-62)) then
        tmp = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
    else if (b <= 2.8d-114) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.4e-62) {
		tmp = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 2.8e-114) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.4e-62:
		tmp = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
	elif b <= 2.8e-114:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.4e-62)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	elseif (b <= 2.8e-114)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.4e-62)
		tmp = b * ((c / (b ^ 2.0)) + (-1.0 / a));
	elseif (b <= 2.8e-114)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.4e-62], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.8e-114], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.3999999999999996e-62

   1. Initial program 70.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 84.7%

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

      1. mul-1-neg 84.7%

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

      2. distribute-rgt-neg-in 84.7%

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

      3. +-commutative 84.7%

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

      4. mul-1-neg 84.7%

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

      5. unsub-neg 84.7%

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

   5. Simplified 84.7%

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

   if -7.3999999999999996e-62 < b < 2.8000000000000001e-114

   1. Initial program 80.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. prod-diff 79.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -c \cdot \left(4 \cdot a\right)\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}}{2 \cdot a} \]

      2. *-commutative 79.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(4 \cdot a\right) \cdot c}\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}{2 \cdot a} \]

      3. fma-define 79.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)\right)} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}{2 \cdot a} \]

      4. associate-+l+ 79.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b + \left(\left(-\left(4 \cdot a\right) \cdot c\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}}{2 \cdot a} \]

      5. fma-define 79.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(-\left(4 \cdot a\right) \cdot c\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}}{2 \cdot a} \]

      6. *-commutative 79.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-\color{blue}{c \cdot \left(4 \cdot a\right)}\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{2 \cdot a} \]

      7. distribute-rgt-neg-in 79.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{2 \cdot a} \]

      8. fma-define 79.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\mathsf{fma}\left(c, -4 \cdot a, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}\right)}}{2 \cdot a} \]

      9. *-commutative 79.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -\color{blue}{a \cdot 4}, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)\right)}}{2 \cdot a} \]

      10. distribute-rgt-neg-in 79.7%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, \color{blue}{a \cdot \left(-4\right)}, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)\right)}}{2 \cdot a} \]

      11. metadata-eval 79.7%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot \color{blue}{-4}, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)\right)}}{2 \cdot a} \]

      12. *-commutative 79.7%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(-c, 4 \cdot a, \color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right)\right)}}{2 \cdot a} \]

      13. fma-undefine 79.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{\left(-c\right) \cdot \left(4 \cdot a\right) + \left(4 \cdot a\right) \cdot c}\right)\right)}}{2 \cdot a} \]

      14. distribute-lft-neg-in 79.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{\left(-c \cdot \left(4 \cdot a\right)\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{2 \cdot a} \]

      15. distribute-rgt-neg-in 79.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{c \cdot \left(-4 \cdot a\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{2 \cdot a} \]

      16. fma-define 79.7%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{\mathsf{fma}\left(c, -4 \cdot a, \left(4 \cdot a\right) \cdot c\right)}\right)\right)}}{2 \cdot a} \]

   4. Applied egg-rr 79.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, \left(4 \cdot a\right) \cdot c\right)\right)\right)}}}{2 \cdot a} \]

   5. Taylor expanded in b around 0 76.9%

      \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} + -1 \cdot b}}{2 \cdot a} \]
   6. Step-by-step derivation

      1. mul-1-neg 76.9%

         \[\leadsto \frac{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} + \color{blue}{\left(-b\right)}}{2 \cdot a} \]

      2. unsub-neg 76.9%

         \[\leadsto \frac{\color{blue}{\sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a} \]

      3. associate-*r* 76.9%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot a\right) \cdot c} + 4 \cdot \left(a \cdot c\right)} - b}{2 \cdot a} \]

      4. associate-*r* 76.9%

         \[\leadsto \frac{\sqrt{\left(-8 \cdot a\right) \cdot c + \color{blue}{\left(4 \cdot a\right) \cdot c}} - b}{2 \cdot a} \]

      5. distribute-rgt-in 77.3%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-8 \cdot a + 4 \cdot a\right)}} - b}{2 \cdot a} \]

      6. distribute-rgt-out 77.3%

         \[\leadsto \frac{\sqrt{c \cdot \color{blue}{\left(a \cdot \left(-8 + 4\right)\right)}} - b}{2 \cdot a} \]

      7. metadata-eval 77.3%

         \[\leadsto \frac{\sqrt{c \cdot \left(a \cdot \color{blue}{-4}\right)} - b}{2 \cdot a} \]

      8. *-commutative 77.3%

         \[\leadsto \frac{\sqrt{c \cdot \color{blue}{\left(-4 \cdot a\right)}} - b}{2 \cdot a} \]

   7. Simplified 77.3%

      \[\leadsto \frac{\color{blue}{\sqrt{c \cdot \left(-4 \cdot a\right)} - b}}{2 \cdot a} \]

   if 2.8000000000000001e-114 < b

   1. Initial program 23.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 82.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. mul-1-neg 82.7%

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

      2. distribute-neg-frac2 82.7%

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

   5. Simplified 82.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{-62}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-63}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{c \cdot \left(a \cdot -4\right)} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e-63)
   (* b (+ (/ c (pow b 2.0)) (/ -1.0 a)))
   (if (<= b 2.75e-114) (* (sqrt (* c (* a -4.0))) (/ 0.5 a)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-63) {
		tmp = b * ((c / pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 2.75e-114) {
		tmp = sqrt((c * (a * -4.0))) * (0.5 / a);
	} else {
		tmp = 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 <= (-2.8d-63)) then
        tmp = b * ((c / (b ** 2.0d0)) + ((-1.0d0) / a))
    else if (b <= 2.75d-114) then
        tmp = sqrt((c * (a * (-4.0d0)))) * (0.5d0 / a)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-63) {
		tmp = b * ((c / Math.pow(b, 2.0)) + (-1.0 / a));
	} else if (b <= 2.75e-114) {
		tmp = Math.sqrt((c * (a * -4.0))) * (0.5 / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e-63:
		tmp = b * ((c / math.pow(b, 2.0)) + (-1.0 / a))
	elif b <= 2.75e-114:
		tmp = math.sqrt((c * (a * -4.0))) * (0.5 / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e-63)
		tmp = Float64(b * Float64(Float64(c / (b ^ 2.0)) + Float64(-1.0 / a)));
	elseif (b <= 2.75e-114)
		tmp = Float64(sqrt(Float64(c * Float64(a * -4.0))) * Float64(0.5 / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.8e-63)
		tmp = b * ((c / (b ^ 2.0)) + (-1.0 / a));
	elseif (b <= 2.75e-114)
		tmp = sqrt((c * (a * -4.0))) * (0.5 / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e-63], N[(b * N[(N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.75e-114], N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8000000000000002e-63

   1. Initial program 70.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 84.7%

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

      1. mul-1-neg 84.7%

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

      2. distribute-rgt-neg-in 84.7%

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

      3. +-commutative 84.7%

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

      4. mul-1-neg 84.7%

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

      5. unsub-neg 84.7%

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

   5. Simplified 84.7%

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

   if -2.8000000000000002e-63 < b < 2.75000000000000005e-114

   1. Initial program 80.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. prod-diff 79.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -c \cdot \left(4 \cdot a\right)\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}}{2 \cdot a} \]

      2. *-commutative 79.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(4 \cdot a\right) \cdot c}\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}{2 \cdot a} \]

      3. fma-define 79.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)\right)} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}{2 \cdot a} \]

      4. associate-+l+ 79.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b + \left(\left(-\left(4 \cdot a\right) \cdot c\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}}{2 \cdot a} \]

      5. fma-define 79.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(-\left(4 \cdot a\right) \cdot c\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}}{2 \cdot a} \]

      6. *-commutative 79.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-\color{blue}{c \cdot \left(4 \cdot a\right)}\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{2 \cdot a} \]

      7. distribute-rgt-neg-in 79.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{2 \cdot a} \]

      8. fma-define 79.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\mathsf{fma}\left(c, -4 \cdot a, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}\right)}}{2 \cdot a} \]

      9. *-commutative 79.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -\color{blue}{a \cdot 4}, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)\right)}}{2 \cdot a} \]

      10. distribute-rgt-neg-in 79.7%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, \color{blue}{a \cdot \left(-4\right)}, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)\right)}}{2 \cdot a} \]

      11. metadata-eval 79.7%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot \color{blue}{-4}, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)\right)}}{2 \cdot a} \]

      12. *-commutative 79.7%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(-c, 4 \cdot a, \color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right)\right)}}{2 \cdot a} \]

      13. fma-undefine 79.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{\left(-c\right) \cdot \left(4 \cdot a\right) + \left(4 \cdot a\right) \cdot c}\right)\right)}}{2 \cdot a} \]

      14. distribute-lft-neg-in 79.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{\left(-c \cdot \left(4 \cdot a\right)\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{2 \cdot a} \]

      15. distribute-rgt-neg-in 79.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{c \cdot \left(-4 \cdot a\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{2 \cdot a} \]

      16. fma-define 79.7%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{\mathsf{fma}\left(c, -4 \cdot a, \left(4 \cdot a\right) \cdot c\right)}\right)\right)}}{2 \cdot a} \]

   4. Applied egg-rr 79.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, \left(4 \cdot a\right) \cdot c\right)\right)\right)}}}{2 \cdot a} \]

   5. Taylor expanded in b around 0 76.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{a} \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 76.6%

         \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{a}\right) \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}} \]

      2. associate-*r/ 76.6%

         \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{a}} \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} \]

      3. metadata-eval 76.6%

         \[\leadsto \frac{\color{blue}{0.5}}{a} \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} \]

      4. associate-*r* 76.5%

         \[\leadsto \frac{0.5}{a} \cdot \sqrt{\color{blue}{\left(-8 \cdot a\right) \cdot c} + 4 \cdot \left(a \cdot c\right)} \]

      5. associate-*r* 76.5%

         \[\leadsto \frac{0.5}{a} \cdot \sqrt{\left(-8 \cdot a\right) \cdot c + \color{blue}{\left(4 \cdot a\right) \cdot c}} \]

      6. distribute-rgt-in 77.0%

         \[\leadsto \frac{0.5}{a} \cdot \sqrt{\color{blue}{c \cdot \left(-8 \cdot a + 4 \cdot a\right)}} \]

      7. distribute-rgt-out 77.0%

         \[\leadsto \frac{0.5}{a} \cdot \sqrt{c \cdot \color{blue}{\left(a \cdot \left(-8 + 4\right)\right)}} \]

      8. metadata-eval 77.0%

         \[\leadsto \frac{0.5}{a} \cdot \sqrt{c \cdot \left(a \cdot \color{blue}{-4}\right)} \]

      9. *-commutative 77.0%

         \[\leadsto \frac{0.5}{a} \cdot \sqrt{c \cdot \color{blue}{\left(-4 \cdot a\right)}} \]

   7. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \sqrt{c \cdot \left(-4 \cdot a\right)}} \]

   if 2.75000000000000005e-114 < b

   1. Initial program 23.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 82.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. mul-1-neg 82.7%

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

      2. distribute-neg-frac2 82.7%

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

   5. Simplified 82.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-63}:\\ \;\;\;\;b \cdot \left(\frac{c}{{b}^{2}} + \frac{-1}{a}\right)\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{c \cdot \left(a \cdot -4\right)} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-77}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{c \cdot \left(a \cdot -4\right)} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.7e-77)
   (/ b (- a))
   (if (<= b 2.8e-114) (* (sqrt (* c (* a -4.0))) (/ 0.5 a)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.7e-77) {
		tmp = b / -a;
	} else if (b <= 2.8e-114) {
		tmp = sqrt((c * (a * -4.0))) * (0.5 / a);
	} else {
		tmp = 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-77)) then
        tmp = b / -a
    else if (b <= 2.8d-114) then
        tmp = sqrt((c * (a * (-4.0d0)))) * (0.5d0 / a)
    else
        tmp = 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-77) {
		tmp = b / -a;
	} else if (b <= 2.8e-114) {
		tmp = Math.sqrt((c * (a * -4.0))) * (0.5 / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.7e-77:
		tmp = b / -a
	elif b <= 2.8e-114:
		tmp = math.sqrt((c * (a * -4.0))) * (0.5 / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.7e-77)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 2.8e-114)
		tmp = Float64(sqrt(Float64(c * Float64(a * -4.0))) * Float64(0.5 / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.7e-77)
		tmp = b / -a;
	elseif (b <= 2.8e-114)
		tmp = sqrt((c * (a * -4.0))) * (0.5 / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.7e-77], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 2.8e-114], N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.7e-77

   1. Initial program 69.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 83.4%

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

      1. mul-1-neg 83.4%

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

      2. distribute-neg-frac2 83.4%

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

   5. Simplified 83.4%

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

   if -2.7e-77 < b < 2.8000000000000001e-114

   1. Initial program 81.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. prod-diff 81.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -c \cdot \left(4 \cdot a\right)\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}}{2 \cdot a} \]

      2. *-commutative 81.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(4 \cdot a\right) \cdot c}\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}{2 \cdot a} \]

      3. fma-define 81.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)\right)} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}{2 \cdot a} \]

      4. associate-+l+ 81.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b + \left(\left(-\left(4 \cdot a\right) \cdot c\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}}{2 \cdot a} \]

      5. fma-define 81.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(-\left(4 \cdot a\right) \cdot c\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}}{2 \cdot a} \]

      6. *-commutative 81.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-\color{blue}{c \cdot \left(4 \cdot a\right)}\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{2 \cdot a} \]

      7. distribute-rgt-neg-in 81.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{2 \cdot a} \]

      8. fma-define 81.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\mathsf{fma}\left(c, -4 \cdot a, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}\right)}}{2 \cdot a} \]

      9. *-commutative 81.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -\color{blue}{a \cdot 4}, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)\right)}}{2 \cdot a} \]

      10. distribute-rgt-neg-in 81.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, \color{blue}{a \cdot \left(-4\right)}, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)\right)}}{2 \cdot a} \]

      11. metadata-eval 81.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot \color{blue}{-4}, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)\right)}}{2 \cdot a} \]

      12. *-commutative 81.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(-c, 4 \cdot a, \color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right)\right)}}{2 \cdot a} \]

      13. fma-undefine 81.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{\left(-c\right) \cdot \left(4 \cdot a\right) + \left(4 \cdot a\right) \cdot c}\right)\right)}}{2 \cdot a} \]

      14. distribute-lft-neg-in 81.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{\left(-c \cdot \left(4 \cdot a\right)\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{2 \cdot a} \]

      15. distribute-rgt-neg-in 81.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{c \cdot \left(-4 \cdot a\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{2 \cdot a} \]

      16. fma-define 81.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{\mathsf{fma}\left(c, -4 \cdot a, \left(4 \cdot a\right) \cdot c\right)}\right)\right)}}{2 \cdot a} \]

   4. Applied egg-rr 81.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, \left(4 \cdot a\right) \cdot c\right)\right)\right)}}}{2 \cdot a} \]

   5. Taylor expanded in b around 0 77.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{a} \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 77.7%

         \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{a}\right) \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)}} \]

      2. associate-*r/ 77.7%

         \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{a}} \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} \]

      3. metadata-eval 77.7%

         \[\leadsto \frac{\color{blue}{0.5}}{a} \cdot \sqrt{-8 \cdot \left(a \cdot c\right) + 4 \cdot \left(a \cdot c\right)} \]

      4. associate-*r* 77.7%

         \[\leadsto \frac{0.5}{a} \cdot \sqrt{\color{blue}{\left(-8 \cdot a\right) \cdot c} + 4 \cdot \left(a \cdot c\right)} \]

      5. associate-*r* 77.7%

         \[\leadsto \frac{0.5}{a} \cdot \sqrt{\left(-8 \cdot a\right) \cdot c + \color{blue}{\left(4 \cdot a\right) \cdot c}} \]

      6. distribute-rgt-in 78.1%

         \[\leadsto \frac{0.5}{a} \cdot \sqrt{\color{blue}{c \cdot \left(-8 \cdot a + 4 \cdot a\right)}} \]

      7. distribute-rgt-out 78.1%

         \[\leadsto \frac{0.5}{a} \cdot \sqrt{c \cdot \color{blue}{\left(a \cdot \left(-8 + 4\right)\right)}} \]

      8. metadata-eval 78.1%

         \[\leadsto \frac{0.5}{a} \cdot \sqrt{c \cdot \left(a \cdot \color{blue}{-4}\right)} \]

      9. *-commutative 78.1%

         \[\leadsto \frac{0.5}{a} \cdot \sqrt{c \cdot \color{blue}{\left(-4 \cdot a\right)}} \]

   7. Simplified 78.1%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \sqrt{c \cdot \left(-4 \cdot a\right)}} \]

   if 2.8000000000000001e-114 < b

   1. Initial program 23.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 82.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. mul-1-neg 82.7%

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

      2. distribute-neg-frac2 82.7%

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

   5. Simplified 82.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-77}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{c \cdot \left(a \cdot -4\right)} \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-68}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{c \cdot -4}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-68)
   (/ b (- a))
   (if (<= b 1.55e-108) (* 0.5 (sqrt (/ (* c -4.0) a))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-68) {
		tmp = b / -a;
	} else if (b <= 1.55e-108) {
		tmp = 0.5 * sqrt(((c * -4.0) / a));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3d-68)) then
        tmp = b / -a
    else if (b <= 1.55d-108) then
        tmp = 0.5d0 * sqrt(((c * (-4.0d0)) / a))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-68) {
		tmp = b / -a;
	} else if (b <= 1.55e-108) {
		tmp = 0.5 * Math.sqrt(((c * -4.0) / a));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e-68:
		tmp = b / -a
	elif b <= 1.55e-108:
		tmp = 0.5 * math.sqrt(((c * -4.0) / a))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-68)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.55e-108)
		tmp = Float64(0.5 * sqrt(Float64(Float64(c * -4.0) / a)));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e-68)
		tmp = b / -a;
	elseif (b <= 1.55e-108)
		tmp = 0.5 * sqrt(((c * -4.0) / a));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-68], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.55e-108], N[(0.5 * N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] / a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3e-68

   1. Initial program 70.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 84.3%

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

      1. mul-1-neg 84.3%

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

      2. distribute-neg-frac2 84.3%

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

   5. Simplified 84.3%

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

   if -3e-68 < b < 1.55000000000000007e-108

   1. Initial program 79.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. prod-diff 78.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -c \cdot \left(4 \cdot a\right)\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}}{2 \cdot a} \]

      2. *-commutative 78.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{\left(4 \cdot a\right) \cdot c}\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}{2 \cdot a} \]

      3. fma-define 78.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)\right)} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)}}{2 \cdot a} \]

      4. associate-+l+ 78.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b + \left(\left(-\left(4 \cdot a\right) \cdot c\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}}{2 \cdot a} \]

      5. fma-define 78.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(-\left(4 \cdot a\right) \cdot c\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}}{2 \cdot a} \]

      6. *-commutative 78.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-\color{blue}{c \cdot \left(4 \cdot a\right)}\right) + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{2 \cdot a} \]

      7. distribute-rgt-neg-in 78.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-4 \cdot a\right)} + \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}}{2 \cdot a} \]

      8. fma-define 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\mathsf{fma}\left(c, -4 \cdot a, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)}\right)}}{2 \cdot a} \]

      9. *-commutative 78.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, -\color{blue}{a \cdot 4}, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)\right)}}{2 \cdot a} \]

      10. distribute-rgt-neg-in 78.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, \color{blue}{a \cdot \left(-4\right)}, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)\right)}}{2 \cdot a} \]

      11. metadata-eval 78.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot \color{blue}{-4}, \mathsf{fma}\left(-c, 4 \cdot a, c \cdot \left(4 \cdot a\right)\right)\right)\right)}}{2 \cdot a} \]

      12. *-commutative 78.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(-c, 4 \cdot a, \color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right)\right)}}{2 \cdot a} \]

      13. fma-undefine 78.9%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{\left(-c\right) \cdot \left(4 \cdot a\right) + \left(4 \cdot a\right) \cdot c}\right)\right)}}{2 \cdot a} \]

      14. distribute-lft-neg-in 78.9%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{\left(-c \cdot \left(4 \cdot a\right)\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{2 \cdot a} \]

      15. distribute-rgt-neg-in 78.9%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{c \cdot \left(-4 \cdot a\right)} + \left(4 \cdot a\right) \cdot c\right)\right)}}{2 \cdot a} \]

      16. fma-define 78.8%

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \color{blue}{\mathsf{fma}\left(c, -4 \cdot a, \left(4 \cdot a\right) \cdot c\right)}\right)\right)}}{2 \cdot a} \]

   4. Applied egg-rr 78.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, \left(4 \cdot a\right) \cdot c\right)\right)\right)}}}{2 \cdot a} \]

   5. Taylor expanded in a around inf 44.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{-8 \cdot c + 4 \cdot c}{a}}} \]
   6. Step-by-step derivation

      1. distribute-rgt-out 44.3%

         \[\leadsto 0.5 \cdot \sqrt{\frac{\color{blue}{c \cdot \left(-8 + 4\right)}}{a}} \]

      2. metadata-eval 44.3%

         \[\leadsto 0.5 \cdot \sqrt{\frac{c \cdot \color{blue}{-4}}{a}} \]

   7. Simplified 44.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{c \cdot -4}{a}}} \]

   if 1.55000000000000007e-108 < b

   1. Initial program 22.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 84.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. mul-1-neg 84.2%

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

      2. distribute-neg-frac2 84.2%

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

   5. Simplified 84.2%

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

Alternative 7: 67.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.4 \cdot 10^{-305}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.4e-305) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.4e-305) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 7.4d-305) then
        tmp = b / -a
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.4e-305) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.4e-305:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.4e-305)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7.4e-305)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.4e-305], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.39999999999999954e-305

   1. Initial program 73.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 60.9%

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

      1. mul-1-neg 60.9%

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

      2. distribute-neg-frac2 60.9%

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

   5. Simplified 60.9%

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

   if 7.39999999999999954e-305 < b

   1. Initial program 35.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 66.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. mul-1-neg 66.7%

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

      2. distribute-neg-frac2 66.7%

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

   5. Simplified 66.7%

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

Alternative 8: 35.2% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around -inf 33.2%

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

   1. mul-1-neg 33.2%

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

   2. distribute-neg-frac2 33.2%

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

5. Simplified 33.2%

   \[\leadsto \color{blue}{\frac{b}{-a}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))