quadp (p42, positive)

Percentage Accurate: 52.6% → 84.9%

Time: 17.5s

Alternatives: 10

Speedup: 19.1×

Specification

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.6% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+158}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.95 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-1 - \frac{c}{\frac{b}{\frac{a}{b}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+158)
   (- (/ c b) (/ b a))
   (if (<= b 2.95e-121)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (* (/ c b) (- -1.0 (/ c (/ b (/ a b))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+158) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.95e-121) {
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = (c / b) * (-1.0 - (c / (b / (a / 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 <= (-2d+158)) then
        tmp = (c / b) - (b / a)
    else if (b <= 2.95d-121) then
        tmp = (sqrt(((b * b) - (4.0d0 * (c * a)))) - b) / (a * 2.0d0)
    else
        tmp = (c / b) * ((-1.0d0) - (c / (b / (a / b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+158) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.95e-121) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	} else {
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e+158:
		tmp = (c / b) - (b / a)
	elif b <= 2.95e-121:
		tmp = (math.sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0)
	else:
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+158)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 2.95e-121)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) * Float64(-1.0 - Float64(c / Float64(b / Float64(a / b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e+158)
		tmp = (c / b) - (b / a);
	elseif (b <= 2.95e-121)
		tmp = (sqrt(((b * b) - (4.0 * (c * a)))) - b) / (a * 2.0);
	else
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+158], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.95e-121], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * N[(-1.0 - N[(c / N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.99999999999999991e158

   1. Initial program 52.3%

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

   2. Taylor expanded in b around -inf 98.2%

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

      1. +-commutative 98.2%

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

      2. mul-1-neg 98.2%

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

      3. unsub-neg 98.2%

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

   4. Simplified 98.2%

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

   if -1.99999999999999991e158 < b < 2.94999999999999998e-121

   1. Initial program 88.0%

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

   if 2.94999999999999998e-121 < b

   1. Initial program 17.1%

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

   2. Taylor expanded in b around inf 72.8%

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

      1. distribute-lft-out 72.8%

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

      2. unpow2 72.8%

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

   4. Simplified 72.8%

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

      1. +-commutative 72.8%

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

      2. associate-*r* 75.7%

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

      3. *-commutative 75.7%

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

      4. unpow3 75.7%

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

      5. times-frac 83.6%

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

      6. *-lft-identity 83.6%

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

      7. distribute-rgt-out 83.6%

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

      8. associate-/l* 87.7%

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

      9. associate-/l* 87.6%

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

   6. Applied egg-rr 87.6%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{c}{b} \cdot \left(\frac{c}{\frac{b}{\frac{a}{b}}} + 1\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+158}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.95 \cdot 10^{-121}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-1 - \frac{c}{\frac{b}{\frac{a}{b}}}\right)\\ \end{array} \]

Alternative 2: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{-123}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-1 - \frac{c}{\frac{b}{\frac{a}{b}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e+143)
   (- (/ c b) (/ b a))
   (if (<= b 1.42e-123)
     (* (/ 0.5 a) (- (sqrt (+ (* b b) (* a (* c -4.0)))) b))
     (* (/ c b) (- -1.0 (/ c (/ b (/ a b))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e+143) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.42e-123) {
		tmp = (0.5 / a) * (sqrt(((b * b) + (a * (c * -4.0)))) - b);
	} else {
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-9.5d+143)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.42d-123) then
        tmp = (0.5d0 / a) * (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b)
    else
        tmp = (c / b) * ((-1.0d0) - (c / (b / (a / b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e+143) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.42e-123) {
		tmp = (0.5 / a) * (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b);
	} else {
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.5e+143:
		tmp = (c / b) - (b / a)
	elif b <= 1.42e-123:
		tmp = (0.5 / a) * (math.sqrt(((b * b) + (a * (c * -4.0)))) - b)
	else:
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.5e+143)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.42e-123)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b));
	else
		tmp = Float64(Float64(c / b) * Float64(-1.0 - Float64(c / Float64(b / Float64(a / b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9.5e+143)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.42e-123)
		tmp = (0.5 / a) * (sqrt(((b * b) + (a * (c * -4.0)))) - b);
	else
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.5e+143], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.42e-123], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * N[(-1.0 - N[(c / N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.50000000000000066e143

   1. Initial program 57.3%

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

   2. Taylor expanded in b around -inf 98.4%

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

      1. +-commutative 98.4%

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

      2. mul-1-neg 98.4%

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

      3. unsub-neg 98.4%

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

   4. Simplified 98.4%

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

   if -9.50000000000000066e143 < b < 1.42000000000000008e-123

   1. Initial program 87.3%

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

      1. flip-- 48.8%

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

      2. clear-num 48.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{1}{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

      3. sqrt-div 48.3%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

      4. metadata-eval 48.3%

         \[\leadsto \frac{\left(-b\right) + \frac{\color{blue}{1}}{\sqrt{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}{2 \cdot a} \]

      5. clear-num 48.3%

         \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + 4 \cdot \left(a \cdot c\right)}}}}}}{2 \cdot a} \]

      6. flip-- 86.7%

         \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}{2 \cdot a} \]

      7. fma-neg 86.7%

         \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

      8. distribute-lft-neg-in 86.7%

         \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)}}}}{2 \cdot a} \]

      9. *-commutative 86.7%

         \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)}}}}{2 \cdot a} \]

      10. associate-*l* 86.7%

          \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot \left(-4\right)\right)}\right)}}}}{2 \cdot a} \]

      11. metadata-eval 86.7%

          \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot \color{blue}{-4}\right)\right)}}}}{2 \cdot a} \]

   3. Applied egg-rr 86.7%

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

      1. sqrt-div 87.2%

         \[\leadsto \frac{\left(-b\right) + \frac{1}{\color{blue}{\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}}}{2 \cdot a} \]

      2. metadata-eval 87.2%

         \[\leadsto \frac{\left(-b\right) + \frac{1}{\frac{\color{blue}{1}}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}}{2 \cdot a} \]

      3. remove-double-div 87.2%

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

      4. associate-*r* 87.2%

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

      5. *-commutative 87.2%

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

      6. metadata-eval 87.2%

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

      7. distribute-lft-neg-in 87.2%

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

      8. fma-neg 87.3%

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

      9. clear-num 87.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}} \]

      10. associate-/r/ 87.1%

          \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 87.1%

          \[\leadsto \color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 87.1%

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

      13. +-commutative 87.1%

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

      14. unsub-neg 87.1%

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

   5. Applied egg-rr 87.1%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b\right)} \]
   6. Step-by-step derivation

      1. fma-udef 87.1%

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

      2. +-commutative 87.1%

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

   7. Applied egg-rr 87.1%

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

   if 1.42000000000000008e-123 < b

   1. Initial program 17.1%

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

   2. Taylor expanded in b around inf 72.8%

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

      1. distribute-lft-out 72.8%

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

      2. unpow2 72.8%

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

   4. Simplified 72.8%

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

      1. +-commutative 72.8%

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

      2. associate-*r* 75.7%

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

      3. *-commutative 75.7%

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

      4. unpow3 75.7%

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

      5. times-frac 83.6%

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

      6. *-lft-identity 83.6%

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

      7. distribute-rgt-out 83.6%

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

      8. associate-/l* 87.7%

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

      9. associate-/l* 87.6%

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

   6. Applied egg-rr 87.6%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{c}{b} \cdot \left(\frac{c}{\frac{b}{\frac{a}{b}}} + 1\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{-123}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-1 - \frac{c}{\frac{b}{\frac{a}{b}}}\right)\\ \end{array} \]

Alternative 3: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-1 - \frac{c}{\frac{b}{\frac{a}{b}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e-86)
   (- (/ c b) (/ b a))
   (if (<= b 1.4e-155)
     (* (/ 0.5 a) (- (sqrt (* a (* c -4.0))) b))
     (* (/ c b) (- -1.0 (/ c (/ b (/ a b))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-86) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.4e-155) {
		tmp = (0.5 / a) * (sqrt((a * (c * -4.0))) - b);
	} else {
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.2d-86)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.4d-155) then
        tmp = (0.5d0 / a) * (sqrt((a * (c * (-4.0d0)))) - b)
    else
        tmp = (c / b) * ((-1.0d0) - (c / (b / (a / b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-86) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.4e-155) {
		tmp = (0.5 / a) * (Math.sqrt((a * (c * -4.0))) - b);
	} else {
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.2e-86:
		tmp = (c / b) - (b / a)
	elif b <= 1.4e-155:
		tmp = (0.5 / a) * (math.sqrt((a * (c * -4.0))) - b)
	else:
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.2e-86)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.4e-155)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(Float64(a * Float64(c * -4.0))) - b));
	else
		tmp = Float64(Float64(c / b) * Float64(-1.0 - Float64(c / Float64(b / Float64(a / b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.2e-86)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.4e-155)
		tmp = (0.5 / a) * (sqrt((a * (c * -4.0))) - b);
	else
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.2e-86], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.4e-155], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * N[(-1.0 - N[(c / N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.20000000000000006e-86

   1. Initial program 73.6%

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

   2. Taylor expanded in b around -inf 87.5%

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

      1. +-commutative 87.5%

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

      2. mul-1-neg 87.5%

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

      3. unsub-neg 87.5%

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

   4. Simplified 87.5%

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

   if -3.20000000000000006e-86 < b < 1.4e-155

   1. Initial program 82.4%

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

      1. flip-- 50.0%

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

      2. clear-num 50.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{1}{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

      3. sqrt-div 49.1%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

      4. metadata-eval 49.1%

         \[\leadsto \frac{\left(-b\right) + \frac{\color{blue}{1}}{\sqrt{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}{2 \cdot a} \]

      5. clear-num 49.1%

         \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + 4 \cdot \left(a \cdot c\right)}}}}}}{2 \cdot a} \]

      6. flip-- 81.4%

         \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}{2 \cdot a} \]

      7. fma-neg 81.4%

         \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

      8. distribute-lft-neg-in 81.4%

         \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)}}}}{2 \cdot a} \]

      9. *-commutative 81.4%

         \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)}}}}{2 \cdot a} \]

      10. associate-*l* 81.4%

          \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot \left(-4\right)\right)}\right)}}}}{2 \cdot a} \]

      11. metadata-eval 81.4%

          \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot \color{blue}{-4}\right)\right)}}}}{2 \cdot a} \]

   3. Applied egg-rr 81.4%

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

      1. sqrt-div 82.3%

         \[\leadsto \frac{\left(-b\right) + \frac{1}{\color{blue}{\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}}}{2 \cdot a} \]

      2. metadata-eval 82.3%

         \[\leadsto \frac{\left(-b\right) + \frac{1}{\frac{\color{blue}{1}}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}}{2 \cdot a} \]

      3. remove-double-div 82.4%

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

      4. associate-*r* 82.4%

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

      5. *-commutative 82.4%

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

      6. metadata-eval 82.4%

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

      7. distribute-lft-neg-in 82.4%

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

      8. fma-neg 82.4%

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

      9. clear-num 82.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}} \]

      10. associate-/r/ 82.2%

          \[\leadsto \color{blue}{\frac{1}{2 \cdot a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)} \]

      11. associate-/r* 82.2%

          \[\leadsto \color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \]

      12. metadata-eval 82.2%

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

      13. +-commutative 82.2%

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

      14. unsub-neg 82.2%

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

   5. Applied egg-rr 82.2%

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

   6. Taylor expanded in a around inf 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-*r* 78.0%

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

   8. Simplified 78.0%

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

   if 1.4e-155 < b

   1. Initial program 19.0%

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

   2. Taylor expanded in b around inf 70.0%

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

      1. distribute-lft-out 70.0%

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

      2. unpow2 70.0%

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

   4. Simplified 70.0%

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

      1. +-commutative 70.0%

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

      2. associate-*r* 72.9%

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

      3. *-commutative 72.9%

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

      4. unpow3 72.9%

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

      5. times-frac 82.3%

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

      6. *-lft-identity 82.3%

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

      7. distribute-rgt-out 82.4%

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

      8. associate-/l* 86.2%

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

      9. associate-/l* 86.2%

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

   6. Applied egg-rr 86.2%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{c}{b} \cdot \left(\frac{c}{\frac{b}{\frac{a}{b}}} + 1\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-1 - \frac{c}{\frac{b}{\frac{a}{b}}}\right)\\ \end{array} \]

Alternative 4: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-1 - \frac{c}{\frac{b}{\frac{a}{b}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-87)
   (- (/ c b) (/ b a))
   (if (<= b 1.1e-155)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (* (/ c b) (- -1.0 (/ c (/ b (/ a b))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.1e-155) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-9d-87)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.1d-155) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (c / b) * ((-1.0d0) - (c / (b / (a / b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-87) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.1e-155) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9e-87:
		tmp = (c / b) - (b / a)
	elif b <= 1.1e-155:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9e-87)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.1e-155)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c / b) * Float64(-1.0 - Float64(c / Float64(b / Float64(a / b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -9e-87)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.1e-155)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9e-87], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e-155], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * N[(-1.0 - N[(c / N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.99999999999999915e-87

   1. Initial program 73.6%

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

   2. Taylor expanded in b around -inf 87.5%

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

      1. +-commutative 87.5%

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

      2. mul-1-neg 87.5%

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

      3. unsub-neg 87.5%

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

   4. Simplified 87.5%

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

   if -8.99999999999999915e-87 < b < 1.1e-155

   1. Initial program 82.4%

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

   2. Taylor expanded in b around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*r* 78.1%

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

   4. Simplified 78.1%

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

   if 1.1e-155 < b

   1. Initial program 19.0%

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

   2. Taylor expanded in b around inf 70.0%

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

      1. distribute-lft-out 70.0%

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

      2. unpow2 70.0%

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

   4. Simplified 70.0%

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

      1. +-commutative 70.0%

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

      2. associate-*r* 72.9%

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

      3. *-commutative 72.9%

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

      4. unpow3 72.9%

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

      5. times-frac 82.3%

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

      6. *-lft-identity 82.3%

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

      7. distribute-rgt-out 82.4%

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

      8. associate-/l* 86.2%

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

      9. associate-/l* 86.2%

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

   6. Applied egg-rr 86.2%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{c}{b} \cdot \left(\frac{c}{\frac{b}{\frac{a}{b}}} + 1\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-1 - \frac{c}{\frac{b}{\frac{a}{b}}}\right)\\ \end{array} \]

Alternative 5: 66.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{-161}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-1 - \frac{c}{\frac{b}{\frac{a}{b}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.55e-161)
   (- (/ c b) (/ b a))
   (* (/ c b) (- -1.0 (/ c (/ b (/ a b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.55e-161) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = (c / b) * (-1.0 - (c / (b / (a / 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 <= 1.55d-161) then
        tmp = (c / b) - (b / a)
    else
        tmp = (c / b) * ((-1.0d0) - (c / (b / (a / b))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.55e-161) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.55e-161:
		tmp = (c / b) - (b / a)
	else:
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.55e-161)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(c / b) * Float64(-1.0 - Float64(c / Float64(b / Float64(a / b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.55e-161)
		tmp = (c / b) - (b / a);
	else
		tmp = (c / b) * (-1.0 - (c / (b / (a / b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.55e-161], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * N[(-1.0 - N[(c / N[(b / N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5499999999999999e-161

   1. Initial program 76.3%

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

   2. Taylor expanded in b around -inf 64.6%

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

      1. +-commutative 64.6%

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

      2. mul-1-neg 64.6%

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

      3. unsub-neg 64.6%

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

   4. Simplified 64.6%

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

   if 1.5499999999999999e-161 < b

   1. Initial program 19.0%

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

   2. Taylor expanded in b around inf 70.0%

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

      1. distribute-lft-out 70.0%

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

      2. unpow2 70.0%

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

   4. Simplified 70.0%

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

      1. +-commutative 70.0%

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

      2. associate-*r* 72.9%

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

      3. *-commutative 72.9%

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

      4. unpow3 72.9%

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

      5. times-frac 82.3%

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

      6. *-lft-identity 82.3%

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

      7. distribute-rgt-out 82.4%

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

      8. associate-/l* 86.2%

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

      9. associate-/l* 86.2%

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

   6. Applied egg-rr 86.2%

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{c}{b} \cdot \left(\frac{c}{\frac{b}{\frac{a}{b}}} + 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{-161}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-1 - \frac{c}{\frac{b}{\frac{a}{b}}}\right)\\ \end{array} \]

Alternative 6: 67.4% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 75.6%

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

   2. Taylor expanded in b around -inf 70.0%

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

      1. +-commutative 70.0%

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

      2. mul-1-neg 70.0%

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

      3. unsub-neg 70.0%

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

   4. Simplified 70.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 25.6%

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

   2. Taylor expanded in b around inf 77.4%

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

      1. associate-*r/ 77.4%

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

      2. neg-mul-1 77.4%

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

   4. Simplified 77.4%

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

4. Final simplification 73.4%

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

Alternative 7: 42.9% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 4.9e-5) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.9e-5) {
		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 <= 4.9d-5) 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 <= 4.9e-5) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.9e-5:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.9e-5)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.9e-5)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.9e-5], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.9e-5

   1. Initial program 70.4%

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

   2. Taylor expanded in b around -inf 55.8%

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

      1. associate-*r/ 55.8%

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

      2. neg-mul-1 55.8%

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

   4. Simplified 55.8%

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

   if 4.9e-5 < b

   1. Initial program 14.8%

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

   3. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\right)}^{-0.5} \cdot {\left(\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\right)}^{-0.5}} \]
   4. Step-by-step derivation

      1. pow-sqr 2.7%

         \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\right)}^{\left(2 \cdot -0.5\right)}} \]

      2. metadata-eval 2.7%

         \[\leadsto {\left(\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\right)}^{\color{blue}{-1}} \]

      3. unpow-1 2.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}} \]

      4. fma-def 2.7%

         \[\leadsto \frac{1}{\frac{a \cdot 2}{b + \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}}} \]

      5. +-commutative 2.7%

         \[\leadsto \frac{1}{\frac{a \cdot 2}{b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}}} \]

      6. fma-def 2.7%

         \[\leadsto \frac{1}{\frac{a \cdot 2}{b + \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}} \]

   5. Simplified 2.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}} \]

   6. Taylor expanded in b around -inf 40.4%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 8: 67.2% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-250}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9.5e-250) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9.5e-250) {
		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 <= 9.5d-250) 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 <= 9.5e-250) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9.5e-250:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9.5e-250)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 9.5e-250)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9.5e-250], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.5000000000000002e-250

   1. Initial program 76.0%

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

   2. Taylor expanded in b around -inf 66.7%

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

      1. associate-*r/ 66.7%

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

      2. neg-mul-1 66.7%

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

   4. Simplified 66.7%

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

   if 9.5000000000000002e-250 < b

   1. Initial program 22.5%

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

   2. Taylor expanded in b around inf 81.4%

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

      1. associate-*r/ 81.4%

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

      2. neg-mul-1 81.4%

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

   4. Simplified 81.4%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-250}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 9: 2.5% accurate, 38.7× 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 / 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 52.8%

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

   1. flip-- 22.0%

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

   2. clear-num 21.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{1}{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

   3. sqrt-div 21.2%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

   4. metadata-eval 21.2%

      \[\leadsto \frac{\left(-b\right) + \frac{\color{blue}{1}}{\sqrt{\frac{b \cdot b + 4 \cdot \left(a \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}}}{2 \cdot a} \]

   5. clear-num 21.4%

      \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + 4 \cdot \left(a \cdot c\right)}}}}}}{2 \cdot a} \]

   6. flip-- 51.7%

      \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\color{blue}{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}{2 \cdot a} \]

   7. fma-neg 51.7%

      \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}}}}{2 \cdot a} \]

   8. distribute-lft-neg-in 51.7%

      \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(-4\right) \cdot \left(a \cdot c\right)}\right)}}}}{2 \cdot a} \]

   9. *-commutative 51.7%

      \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}\right)}}}}{2 \cdot a} \]

   10. associate-*l* 51.7%

       \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot \left(-4\right)\right)}\right)}}}}{2 \cdot a} \]

   11. metadata-eval 51.7%

       \[\leadsto \frac{\left(-b\right) + \frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot \color{blue}{-4}\right)\right)}}}}{2 \cdot a} \]

3. Applied egg-rr 51.7%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{\frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}}}{2 \cdot a} \]

4. Taylor expanded in b around -inf 38.9%

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

   1. associate-*r/ 38.9%

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

   2. neg-mul-1 38.9%

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

   3. rem-square-sqrt 37.5%

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

   4. fabs-sqr 37.5%

      \[\leadsto \frac{\color{blue}{\left|\sqrt{-b} \cdot \sqrt{-b}\right|}}{a} \]

   5. rem-square-sqrt 39.1%

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

   6. fabs-neg 39.1%

      \[\leadsto \frac{\color{blue}{\left|b\right|}}{a} \]

   7. rem-square-sqrt 1.5%

      \[\leadsto \frac{\left|\color{blue}{\sqrt{b} \cdot \sqrt{b}}\right|}{a} \]

   8. fabs-sqr 1.5%

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

   9. rem-square-sqrt 2.3%

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

6. Simplified 2.3%

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

7. Final simplification 2.3%

   \[\leadsto \frac{b}{a} \]

Alternative 10: 10.4% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

3. Applied egg-rr 19.9%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\right)}^{-0.5} \cdot {\left(\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\right)}^{-0.5}} \]
4. Step-by-step derivation

   1. pow-sqr 32.1%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\right)}^{\left(2 \cdot -0.5\right)}} \]

   2. metadata-eval 32.1%

      \[\leadsto {\left(\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}\right)}^{\color{blue}{-1}} \]

   3. unpow-1 32.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}} \]

   4. fma-def 32.1%

      \[\leadsto \frac{1}{\frac{a \cdot 2}{b + \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}}} \]

   5. +-commutative 32.1%

      \[\leadsto \frac{1}{\frac{a \cdot 2}{b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}}} \]

   6. fma-def 32.1%

      \[\leadsto \frac{1}{\frac{a \cdot 2}{b + \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}} \]

5. Simplified 32.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}} \]

6. Taylor expanded in b around -inf 15.0%

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

7. Final simplification 15.0%

   \[\leadsto \frac{c}{b} \]

Developer target: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t_0 - t_1} \cdot \sqrt{t_0 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2023297 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (if (< b 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))) (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs (/ b 2.0)) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot (/ b 2.0) (* (sqrt (fabs a)) (sqrt (fabs c))))))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))