quadm (p42, negative)

Percentage Accurate: 60.0% → 83.4%

Time: 16.5s

Alternatives: 10

Speedup: 0.9×

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: 60.0% 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: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.85 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{a}{b} \cdot \left(-c\right)}{a}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.85e+48)
   (/ (* (/ a b) (- c)) a)
   (if (<= b 2.15e+151)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* a 2.0))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.85e+48) {
		tmp = ((a / b) * -c) / a;
	} else if (b <= 2.15e+151) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	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 <= (-2.85d+48)) then
        tmp = ((a / b) * -c) / a
    else if (b <= 2.15d+151) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (a * 2.0d0)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.85e+48) {
		tmp = ((a / b) * -c) / a;
	} else if (b <= 2.15e+151) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.85e+48:
		tmp = ((a / b) * -c) / a
	elif b <= 2.15e+151:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.85e+48)
		tmp = Float64(Float64(Float64(a / b) * Float64(-c)) / a);
	elseif (b <= 2.15e+151)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.85e+48)
		tmp = ((a / b) * -c) / a;
	elseif (b <= 2.15e+151)
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (a * 2.0);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.85e+48], N[(N[(N[(a / b), $MachinePrecision] * (-c)), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.15e+151], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.84999999999999984e48

   1. Initial program 21.1%

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

      1. *-commutative 21.1%

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

      2. sqr-neg 21.1%

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

      3. *-commutative 21.1%

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

      4. sqr-neg 21.1%

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

      5. associate-*r* 21.1%

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

      6. *-commutative 21.1%

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

   3. Simplified 21.1%

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

   5. Taylor expanded in b around -inf 68.3%

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

      1. associate-/l* 71.9%

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

   7. Simplified 71.9%

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

      1. *-commutative 71.9%

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

      2. times-frac 71.9%

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

      3. associate-/r/ 81.8%

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

      4. metadata-eval 81.8%

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

   9. Applied egg-rr 81.8%

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

   if -2.84999999999999984e48 < b < 2.14999999999999991e151

   1. Initial program 77.3%

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

   if 2.14999999999999991e151 < b

   1. Initial program 43.5%

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

      1. *-commutative 43.5%

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

      2. sqr-neg 43.5%

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

      3. *-commutative 43.5%

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

      4. sqr-neg 43.5%

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

      5. associate-*r* 43.5%

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

      6. *-commutative 43.5%

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

   3. Simplified 43.5%

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

   5. Taylor expanded in b around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.85 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{a}{b} \cdot \left(-c\right)}{a}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{a}{b} \cdot \left(-c\right)}{a}\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-56}:\\ \;\;\;\;-0.5 \cdot \frac{0}{a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-27}:\\ \;\;\;\;-0.5 \cdot \frac{b + {\left(\left(a \cdot c\right) \cdot -4\right)}^{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.22e+48)
   (/ (* (/ a b) (- c)) a)
   (if (<= b -9e-56)
     (* -0.5 (/ 0.0 a))
     (if (<= b 4.4e-27)
       (* -0.5 (/ (+ b (pow (* (* a c) -4.0) 0.5)) a))
       (- (/ c b) (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.22e+48) {
		tmp = ((a / b) * -c) / a;
	} else if (b <= -9e-56) {
		tmp = -0.5 * (0.0 / a);
	} else if (b <= 4.4e-27) {
		tmp = -0.5 * ((b + pow(((a * c) * -4.0), 0.5)) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	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 <= (-1.22d+48)) then
        tmp = ((a / b) * -c) / a
    else if (b <= (-9d-56)) then
        tmp = (-0.5d0) * (0.0d0 / a)
    else if (b <= 4.4d-27) then
        tmp = (-0.5d0) * ((b + (((a * c) * (-4.0d0)) ** 0.5d0)) / a)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.22e+48) {
		tmp = ((a / b) * -c) / a;
	} else if (b <= -9e-56) {
		tmp = -0.5 * (0.0 / a);
	} else if (b <= 4.4e-27) {
		tmp = -0.5 * ((b + Math.pow(((a * c) * -4.0), 0.5)) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.22e+48:
		tmp = ((a / b) * -c) / a
	elif b <= -9e-56:
		tmp = -0.5 * (0.0 / a)
	elif b <= 4.4e-27:
		tmp = -0.5 * ((b + math.pow(((a * c) * -4.0), 0.5)) / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.22e+48)
		tmp = Float64(Float64(Float64(a / b) * Float64(-c)) / a);
	elseif (b <= -9e-56)
		tmp = Float64(-0.5 * Float64(0.0 / a));
	elseif (b <= 4.4e-27)
		tmp = Float64(-0.5 * Float64(Float64(b + (Float64(Float64(a * c) * -4.0) ^ 0.5)) / a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.22e+48)
		tmp = ((a / b) * -c) / a;
	elseif (b <= -9e-56)
		tmp = -0.5 * (0.0 / a);
	elseif (b <= 4.4e-27)
		tmp = -0.5 * ((b + (((a * c) * -4.0) ^ 0.5)) / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.22e+48], N[(N[(N[(a / b), $MachinePrecision] * (-c)), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, -9e-56], N[(-0.5 * N[(0.0 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.4e-27], N[(-0.5 * N[(N[(b + N[Power[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.22000000000000004e48

   1. Initial program 21.1%

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

      1. *-commutative 21.1%

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

      2. sqr-neg 21.1%

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

      3. *-commutative 21.1%

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

      4. sqr-neg 21.1%

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

      5. associate-*r* 21.1%

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

      6. *-commutative 21.1%

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

   3. Simplified 21.1%

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

   5. Taylor expanded in b around -inf 68.3%

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

      1. associate-/l* 71.9%

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

   7. Simplified 71.9%

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

      1. *-commutative 71.9%

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

      2. times-frac 71.9%

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

      3. associate-/r/ 81.8%

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

      4. metadata-eval 81.8%

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

   9. Applied egg-rr 81.8%

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

   if -1.22000000000000004e48 < b < -9.0000000000000001e-56

   1. Initial program 71.2%

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

      1. sub-neg 71.2%

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

      2. distribute-neg-out 71.2%

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

      3. neg-mul-1 71.2%

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

      4. times-frac 71.2%

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

      5. metadata-eval 71.2%

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

      6. sub-neg 71.2%

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

      7. +-commutative 71.2%

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

      8. *-commutative 71.2%

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

      9. distribute-lft-neg-in 71.2%

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

      10. distribute-rgt-neg-out 71.2%

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

      11. associate-*l* 71.2%

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

      12. fma-def 71.2%

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

      13. distribute-lft-neg-in 71.2%

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

      14. distribute-rgt-neg-in 71.2%

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

      15. metadata-eval 71.2%

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

   3. Simplified 71.2%

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

      1. add-sqr-sqrt 42.5%

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

      2. pow2 42.5%

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

      3. pow1/2 42.5%

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

      4. sqrt-pow1 42.5%

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

      5. pow2 42.5%

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

      6. metadata-eval 42.5%

         \[\leadsto -0.5 \cdot \frac{b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   6. Applied egg-rr 42.5%

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}{a} \]

   7. Taylor expanded in b around -inf 64.3%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{b + -1 \cdot b}}{a} \]
   8. Step-by-step derivation

      1. distribute-rgt1-in 64.3%

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

      2. metadata-eval 64.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot b}{a} \]

      3. mul0-lft 64.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{a} \]

   9. Simplified 64.3%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{a} \]

   if -9.0000000000000001e-56 < b < 4.39999999999999974e-27

   1. Initial program 71.6%

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

      1. sub-neg 71.6%

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

      2. distribute-neg-out 71.6%

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

      3. neg-mul-1 71.6%

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

      4. times-frac 71.6%

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

      5. metadata-eval 71.6%

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

      6. sub-neg 71.6%

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

      7. +-commutative 71.6%

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

      8. *-commutative 71.6%

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

      9. distribute-lft-neg-in 71.6%

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

      10. distribute-rgt-neg-out 71.6%

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

      11. associate-*l* 71.6%

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

      12. fma-def 71.6%

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

      13. distribute-lft-neg-in 71.6%

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

      14. distribute-rgt-neg-in 71.6%

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

      15. metadata-eval 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in a around inf 63.9%

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

      1. *-commutative 63.9%

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

      2. associate-*r* 63.9%

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

   7. Simplified 63.9%

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

      1. pow1/2 63.9%

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

      2. associate-*r* 63.9%

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

   9. Applied egg-rr 63.9%

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

   if 4.39999999999999974e-27 < b

   1. Initial program 70.3%

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

      1. *-commutative 70.3%

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

      2. sqr-neg 70.3%

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

      3. *-commutative 70.3%

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

      4. sqr-neg 70.3%

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

      5. associate-*r* 70.3%

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

      6. *-commutative 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in b around inf 89.4%

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

      1. +-commutative 89.4%

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

      2. mul-1-neg 89.4%

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

      3. unsub-neg 89.4%

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

   7. Simplified 89.4%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{a}{b} \cdot \left(-c\right)}{a}\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-56}:\\ \;\;\;\;-0.5 \cdot \frac{0}{a}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-27}:\\ \;\;\;\;-0.5 \cdot \frac{b + {\left(\left(a \cdot c\right) \cdot -4\right)}^{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{a}{b} \cdot \left(-c\right)}{a}\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-56}:\\ \;\;\;\;-0.5 \cdot \frac{0}{a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-20}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e+47)
   (/ (* (/ a b) (- c)) a)
   (if (<= b -9e-56)
     (* -0.5 (/ 0.0 a))
     (if (<= b 2.2e-20)
       (* -0.5 (/ (+ b (sqrt (* a (* c -4.0)))) a))
       (- (/ c b) (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e+47) {
		tmp = ((a / b) * -c) / a;
	} else if (b <= -9e-56) {
		tmp = -0.5 * (0.0 / a);
	} else if (b <= 2.2e-20) {
		tmp = -0.5 * ((b + sqrt((a * (c * -4.0)))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	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 <= (-2.2d+47)) then
        tmp = ((a / b) * -c) / a
    else if (b <= (-9d-56)) then
        tmp = (-0.5d0) * (0.0d0 / a)
    else if (b <= 2.2d-20) then
        tmp = (-0.5d0) * ((b + sqrt((a * (c * (-4.0d0))))) / a)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e+47) {
		tmp = ((a / b) * -c) / a;
	} else if (b <= -9e-56) {
		tmp = -0.5 * (0.0 / a);
	} else if (b <= 2.2e-20) {
		tmp = -0.5 * ((b + Math.sqrt((a * (c * -4.0)))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.2e+47:
		tmp = ((a / b) * -c) / a
	elif b <= -9e-56:
		tmp = -0.5 * (0.0 / a)
	elif b <= 2.2e-20:
		tmp = -0.5 * ((b + math.sqrt((a * (c * -4.0)))) / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e+47)
		tmp = Float64(Float64(Float64(a / b) * Float64(-c)) / a);
	elseif (b <= -9e-56)
		tmp = Float64(-0.5 * Float64(0.0 / a));
	elseif (b <= 2.2e-20)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(a * Float64(c * -4.0)))) / a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.2e+47)
		tmp = ((a / b) * -c) / a;
	elseif (b <= -9e-56)
		tmp = -0.5 * (0.0 / a);
	elseif (b <= 2.2e-20)
		tmp = -0.5 * ((b + sqrt((a * (c * -4.0)))) / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e+47], N[(N[(N[(a / b), $MachinePrecision] * (-c)), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, -9e-56], N[(-0.5 * N[(0.0 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.2e-20], N[(-0.5 * N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.1999999999999999e47

   1. Initial program 21.1%

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

      1. *-commutative 21.1%

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

      2. sqr-neg 21.1%

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

      3. *-commutative 21.1%

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

      4. sqr-neg 21.1%

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

      5. associate-*r* 21.1%

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

      6. *-commutative 21.1%

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

   3. Simplified 21.1%

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

   5. Taylor expanded in b around -inf 68.3%

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

      1. associate-/l* 71.9%

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

   7. Simplified 71.9%

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

      1. *-commutative 71.9%

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

      2. times-frac 71.9%

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

      3. associate-/r/ 81.8%

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

      4. metadata-eval 81.8%

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

   9. Applied egg-rr 81.8%

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

   if -2.1999999999999999e47 < b < -9.0000000000000001e-56

   1. Initial program 71.2%

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

      1. sub-neg 71.2%

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

      2. distribute-neg-out 71.2%

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

      3. neg-mul-1 71.2%

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

      4. times-frac 71.2%

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

      5. metadata-eval 71.2%

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

      6. sub-neg 71.2%

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

      7. +-commutative 71.2%

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

      8. *-commutative 71.2%

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

      9. distribute-lft-neg-in 71.2%

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

      10. distribute-rgt-neg-out 71.2%

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

      11. associate-*l* 71.2%

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

      12. fma-def 71.2%

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

      13. distribute-lft-neg-in 71.2%

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

      14. distribute-rgt-neg-in 71.2%

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

      15. metadata-eval 71.2%

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

   3. Simplified 71.2%

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

      1. add-sqr-sqrt 42.5%

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

      2. pow2 42.5%

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

      3. pow1/2 42.5%

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

      4. sqrt-pow1 42.5%

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

      5. pow2 42.5%

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

      6. metadata-eval 42.5%

         \[\leadsto -0.5 \cdot \frac{b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   6. Applied egg-rr 42.5%

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}{a} \]

   7. Taylor expanded in b around -inf 64.3%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{b + -1 \cdot b}}{a} \]
   8. Step-by-step derivation

      1. distribute-rgt1-in 64.3%

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

      2. metadata-eval 64.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot b}{a} \]

      3. mul0-lft 64.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{a} \]

   9. Simplified 64.3%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{a} \]

   if -9.0000000000000001e-56 < b < 2.19999999999999991e-20

   1. Initial program 71.6%

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

      1. sub-neg 71.6%

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

      2. distribute-neg-out 71.6%

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

      3. neg-mul-1 71.6%

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

      4. times-frac 71.6%

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

      5. metadata-eval 71.6%

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

      6. sub-neg 71.6%

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

      7. +-commutative 71.6%

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

      8. *-commutative 71.6%

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

      9. distribute-lft-neg-in 71.6%

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

      10. distribute-rgt-neg-out 71.6%

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

      11. associate-*l* 71.6%

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

      12. fma-def 71.6%

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

      13. distribute-lft-neg-in 71.6%

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

      14. distribute-rgt-neg-in 71.6%

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

      15. metadata-eval 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in a around inf 63.9%

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

      1. *-commutative 63.9%

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

      2. associate-*r* 63.9%

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

   7. Simplified 63.9%

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

   if 2.19999999999999991e-20 < b

   1. Initial program 70.3%

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

      1. *-commutative 70.3%

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

      2. sqr-neg 70.3%

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

      3. *-commutative 70.3%

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

      4. sqr-neg 70.3%

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

      5. associate-*r* 70.3%

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

      6. *-commutative 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in b around inf 89.4%

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

      1. +-commutative 89.4%

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

      2. mul-1-neg 89.4%

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

      3. unsub-neg 89.4%

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

   7. Simplified 89.4%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{a}{b} \cdot \left(-c\right)}{a}\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-56}:\\ \;\;\;\;-0.5 \cdot \frac{0}{a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-20}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{0}{a}\\ \mathbf{if}\;b \leq -3.7 \cdot 10^{+227}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -2.35 \cdot 10^{+63}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-308}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ 0.0 a))))
   (if (<= b -3.7e+227)
     t_0
     (if (<= b -2.35e+63)
       (/ (- c) b)
       (if (<= b 2e-308) t_0 (- (/ c b) (/ b a)))))))

C

double code(double a, double b, double c) {
	double t_0 = -0.5 * (0.0 / a);
	double tmp;
	if (b <= -3.7e+227) {
		tmp = t_0;
	} else if (b <= -2.35e+63) {
		tmp = -c / b;
	} else if (b <= 2e-308) {
		tmp = t_0;
	} else {
		tmp = (c / b) - (b / a);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (0.0d0 / a)
    if (b <= (-3.7d+227)) then
        tmp = t_0
    else if (b <= (-2.35d+63)) then
        tmp = -c / b
    else if (b <= 2d-308) then
        tmp = t_0
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -0.5 * (0.0 / a);
	double tmp;
	if (b <= -3.7e+227) {
		tmp = t_0;
	} else if (b <= -2.35e+63) {
		tmp = -c / b;
	} else if (b <= 2e-308) {
		tmp = t_0;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -0.5 * (0.0 / a)
	tmp = 0
	if b <= -3.7e+227:
		tmp = t_0
	elif b <= -2.35e+63:
		tmp = -c / b
	elif b <= 2e-308:
		tmp = t_0
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(-0.5 * Float64(0.0 / a))
	tmp = 0.0
	if (b <= -3.7e+227)
		tmp = t_0;
	elseif (b <= -2.35e+63)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 2e-308)
		tmp = t_0;
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -0.5 * (0.0 / a);
	tmp = 0.0;
	if (b <= -3.7e+227)
		tmp = t_0;
	elseif (b <= -2.35e+63)
		tmp = -c / b;
	elseif (b <= 2e-308)
		tmp = t_0;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-0.5 * N[(0.0 / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.7e+227], t$95$0, If[LessEqual[b, -2.35e+63], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 2e-308], t$95$0, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.6999999999999999e227 or -2.3500000000000001e63 < b < 1.9999999999999998e-308

   1. Initial program 54.1%

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

      1. sub-neg 54.1%

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

      2. distribute-neg-out 54.1%

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

      3. neg-mul-1 54.1%

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

      4. times-frac 54.1%

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

      5. metadata-eval 54.1%

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

      6. sub-neg 54.1%

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

      7. +-commutative 54.1%

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

      8. *-commutative 54.1%

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

      9. distribute-lft-neg-in 54.1%

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

      10. distribute-rgt-neg-out 54.1%

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

      11. associate-*l* 54.1%

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

      12. fma-def 54.1%

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

      13. distribute-lft-neg-in 54.1%

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

      14. distribute-rgt-neg-in 54.1%

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

      15. metadata-eval 54.1%

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

   3. Simplified 54.1%

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

      1. add-sqr-sqrt 44.7%

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

      2. pow2 44.7%

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

      3. pow1/2 44.7%

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

      4. sqrt-pow1 44.7%

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

      5. pow2 44.7%

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

      6. metadata-eval 44.7%

         \[\leadsto -0.5 \cdot \frac{b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   6. Applied egg-rr 44.7%

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}{a} \]

   7. Taylor expanded in b around -inf 44.8%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{b + -1 \cdot b}}{a} \]
   8. Step-by-step derivation

      1. distribute-rgt1-in 44.8%

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

      2. metadata-eval 44.8%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot b}{a} \]

      3. mul0-lft 44.8%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{a} \]

   9. Simplified 44.8%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{a} \]

   if -3.6999999999999999e227 < b < -2.3500000000000001e63

   1. Initial program 32.8%

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

      1. *-commutative 32.8%

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

      2. sqr-neg 32.8%

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

      3. *-commutative 32.8%

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

      4. sqr-neg 32.8%

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

      5. associate-*r* 32.8%

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

      6. *-commutative 32.8%

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

   3. Simplified 32.8%

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

   5. Taylor expanded in b around -inf 65.8%

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

      1. mul-1-neg 65.8%

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

   7. Simplified 65.8%

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

   if 1.9999999999999998e-308 < b

   1. Initial program 70.6%

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

      1. *-commutative 70.6%

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

      2. sqr-neg 70.6%

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

      3. *-commutative 70.6%

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

      4. sqr-neg 70.6%

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

      5. associate-*r* 70.6%

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

      6. *-commutative 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in b around inf 66.9%

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

      1. +-commutative 66.9%

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

      2. mul-1-neg 66.9%

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

      3. unsub-neg 66.9%

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

   7. Simplified 66.9%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{+227}:\\ \;\;\;\;-0.5 \cdot \frac{0}{a}\\ \mathbf{elif}\;b \leq -2.35 \cdot 10^{+63}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-308}:\\ \;\;\;\;-0.5 \cdot \frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{a}{b} \cdot \left(-c\right)}{a}\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{-38}:\\ \;\;\;\;-0.5 \cdot \frac{0}{a}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.1e+46)
   (/ (* (/ a b) (- c)) a)
   (if (<= b -1.4e-38)
     (* -0.5 (/ 0.0 a))
     (if (<= b -5e-310) (/ (- c) b) (- (/ c b) (/ b a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.1e+46) {
		tmp = ((a / b) * -c) / a;
	} else if (b <= -1.4e-38) {
		tmp = -0.5 * (0.0 / a);
	} else if (b <= -5e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	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 <= (-4.1d+46)) then
        tmp = ((a / b) * -c) / a
    else if (b <= (-1.4d-38)) then
        tmp = (-0.5d0) * (0.0d0 / a)
    else if (b <= (-5d-310)) then
        tmp = -c / b
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.1e+46) {
		tmp = ((a / b) * -c) / a;
	} else if (b <= -1.4e-38) {
		tmp = -0.5 * (0.0 / a);
	} else if (b <= -5e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.1e+46:
		tmp = ((a / b) * -c) / a
	elif b <= -1.4e-38:
		tmp = -0.5 * (0.0 / a)
	elif b <= -5e-310:
		tmp = -c / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.1e+46)
		tmp = Float64(Float64(Float64(a / b) * Float64(-c)) / a);
	elseif (b <= -1.4e-38)
		tmp = Float64(-0.5 * Float64(0.0 / a));
	elseif (b <= -5e-310)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.1e+46)
		tmp = ((a / b) * -c) / a;
	elseif (b <= -1.4e-38)
		tmp = -0.5 * (0.0 / a);
	elseif (b <= -5e-310)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.1e+46], N[(N[(N[(a / b), $MachinePrecision] * (-c)), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, -1.4e-38], N[(-0.5 * N[(0.0 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -5e-310], N[((-c) / b), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.1e46

   1. Initial program 21.1%

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

      1. *-commutative 21.1%

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

      2. sqr-neg 21.1%

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

      3. *-commutative 21.1%

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

      4. sqr-neg 21.1%

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

      5. associate-*r* 21.1%

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

      6. *-commutative 21.1%

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

   3. Simplified 21.1%

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

   5. Taylor expanded in b around -inf 68.3%

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

      1. associate-/l* 71.9%

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

   7. Simplified 71.9%

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

      1. *-commutative 71.9%

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

      2. times-frac 71.9%

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

      3. associate-/r/ 81.8%

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

      4. metadata-eval 81.8%

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

   9. Applied egg-rr 81.8%

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

   if -4.1e46 < b < -1.4e-38

   1. Initial program 78.9%

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

      1. sub-neg 78.9%

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

      2. distribute-neg-out 78.9%

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

      3. neg-mul-1 78.9%

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

      4. times-frac 78.9%

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

      5. metadata-eval 78.9%

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

      6. sub-neg 78.9%

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

      7. +-commutative 78.9%

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

      8. *-commutative 78.9%

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

      9. distribute-lft-neg-in 78.9%

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

      10. distribute-rgt-neg-out 78.9%

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

      11. associate-*l* 78.9%

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

      12. fma-def 78.9%

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

      13. distribute-lft-neg-in 78.9%

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

      14. distribute-rgt-neg-in 78.9%

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

      15. metadata-eval 78.9%

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

   3. Simplified 78.9%

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

      1. add-sqr-sqrt 49.4%

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

      2. pow2 49.4%

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

      3. pow1/2 49.4%

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

      4. sqrt-pow1 49.4%

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

      5. pow2 49.4%

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

      6. metadata-eval 49.4%

         \[\leadsto -0.5 \cdot \frac{b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   6. Applied egg-rr 49.4%

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}{a} \]

   7. Taylor expanded in b around -inf 70.7%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{b + -1 \cdot b}}{a} \]
   8. Step-by-step derivation

      1. distribute-rgt1-in 70.7%

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

      2. metadata-eval 70.7%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot b}{a} \]

      3. mul0-lft 70.7%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{a} \]

   9. Simplified 70.7%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{a} \]

   if -1.4e-38 < b < -4.999999999999985e-310

   1. Initial program 68.1%

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

      1. *-commutative 68.1%

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

      2. sqr-neg 68.1%

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

      3. *-commutative 68.1%

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

      4. sqr-neg 68.1%

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

      5. associate-*r* 68.1%

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

      6. *-commutative 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in b around -inf 17.0%

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

      1. mul-1-neg 17.0%

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

   7. Simplified 17.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 70.6%

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

      1. *-commutative 70.6%

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

      2. sqr-neg 70.6%

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

      3. *-commutative 70.6%

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

      4. sqr-neg 70.6%

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

      5. associate-*r* 70.6%

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

      6. *-commutative 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in b around inf 66.9%

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

      1. +-commutative 66.9%

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

      2. mul-1-neg 66.9%

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

      3. unsub-neg 66.9%

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

   7. Simplified 66.9%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{a}{b} \cdot \left(-c\right)}{a}\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{-38}:\\ \;\;\;\;-0.5 \cdot \frac{0}{a}\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{0}{a}\\ \mathbf{if}\;b \leq -1.9 \cdot 10^{+234}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -3.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -2.42 \cdot 10^{-222}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ 0.0 a))))
   (if (<= b -1.9e+234)
     t_0
     (if (<= b -3.2e+62) (/ (- c) b) (if (<= b -2.42e-222) t_0 (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double t_0 = -0.5 * (0.0 / a);
	double tmp;
	if (b <= -1.9e+234) {
		tmp = t_0;
	} else if (b <= -3.2e+62) {
		tmp = -c / b;
	} else if (b <= -2.42e-222) {
		tmp = t_0;
	} else {
		tmp = -b / a;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (0.0d0 / a)
    if (b <= (-1.9d+234)) then
        tmp = t_0
    else if (b <= (-3.2d+62)) then
        tmp = -c / b
    else if (b <= (-2.42d-222)) then
        tmp = t_0
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -0.5 * (0.0 / a);
	double tmp;
	if (b <= -1.9e+234) {
		tmp = t_0;
	} else if (b <= -3.2e+62) {
		tmp = -c / b;
	} else if (b <= -2.42e-222) {
		tmp = t_0;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -0.5 * (0.0 / a)
	tmp = 0
	if b <= -1.9e+234:
		tmp = t_0
	elif b <= -3.2e+62:
		tmp = -c / b
	elif b <= -2.42e-222:
		tmp = t_0
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(-0.5 * Float64(0.0 / a))
	tmp = 0.0
	if (b <= -1.9e+234)
		tmp = t_0;
	elseif (b <= -3.2e+62)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= -2.42e-222)
		tmp = t_0;
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -0.5 * (0.0 / a);
	tmp = 0.0;
	if (b <= -1.9e+234)
		tmp = t_0;
	elseif (b <= -3.2e+62)
		tmp = -c / b;
	elseif (b <= -2.42e-222)
		tmp = t_0;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-0.5 * N[(0.0 / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.9e+234], t$95$0, If[LessEqual[b, -3.2e+62], N[((-c) / b), $MachinePrecision], If[LessEqual[b, -2.42e-222], t$95$0, N[((-b) / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.9e234 or -3.19999999999999984e62 < b < -2.4200000000000001e-222

   1. Initial program 52.1%

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

      1. sub-neg 52.1%

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

      2. distribute-neg-out 52.1%

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

      3. neg-mul-1 52.1%

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

      4. times-frac 52.1%

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

      5. metadata-eval 52.1%

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

      6. sub-neg 52.1%

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

      7. +-commutative 52.1%

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

      8. *-commutative 52.1%

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

      9. distribute-lft-neg-in 52.1%

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

      10. distribute-rgt-neg-out 52.1%

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

      11. associate-*l* 52.1%

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

      12. fma-def 52.1%

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

      13. distribute-lft-neg-in 52.1%

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

      14. distribute-rgt-neg-in 52.1%

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

      15. metadata-eval 52.1%

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

   3. Simplified 52.1%

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

      1. add-sqr-sqrt 41.6%

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

      2. pow2 41.6%

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

      3. pow1/2 41.6%

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

      4. sqrt-pow1 41.6%

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

      5. pow2 41.6%

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

      6. metadata-eval 41.6%

         \[\leadsto -0.5 \cdot \frac{b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   6. Applied egg-rr 41.6%

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}{a} \]

   7. Taylor expanded in b around -inf 49.8%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{b + -1 \cdot b}}{a} \]
   8. Step-by-step derivation

      1. distribute-rgt1-in 49.8%

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

      2. metadata-eval 49.8%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot b}{a} \]

      3. mul0-lft 49.8%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{a} \]

   9. Simplified 49.8%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{a} \]

   if -1.9e234 < b < -3.19999999999999984e62

   1. Initial program 32.8%

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

      1. *-commutative 32.8%

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

      2. sqr-neg 32.8%

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

      3. *-commutative 32.8%

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

      4. sqr-neg 32.8%

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

      5. associate-*r* 32.8%

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

      6. *-commutative 32.8%

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

   3. Simplified 32.8%

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

   5. Taylor expanded in b around -inf 65.8%

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

      1. mul-1-neg 65.8%

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

   7. Simplified 65.8%

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

   if -2.4200000000000001e-222 < b

   1. Initial program 70.7%

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

      1. *-commutative 70.7%

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

      2. sqr-neg 70.7%

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

      3. *-commutative 70.7%

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

      4. sqr-neg 70.7%

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

      5. associate-*r* 70.7%

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

      6. *-commutative 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in b around inf 61.8%

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

      1. associate-*r/ 61.8%

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

      2. mul-1-neg 61.8%

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

   7. Simplified 61.8%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{+234}:\\ \;\;\;\;-0.5 \cdot \frac{0}{a}\\ \mathbf{elif}\;b \leq -3.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -2.42 \cdot 10^{-222}:\\ \;\;\;\;-0.5 \cdot \frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{a}{b}}{a} \cdot \left(-c\right)\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.5e+44)
   (* (/ (/ a b) a) (- c))
   (if (<= b -5e-310) (* -0.5 (/ 0.0 a)) (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e+44) {
		tmp = ((a / b) / a) * -c;
	} else if (b <= -5e-310) {
		tmp = -0.5 * (0.0 / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	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 <= (-7.5d+44)) then
        tmp = ((a / b) / a) * -c
    else if (b <= (-5d-310)) then
        tmp = (-0.5d0) * (0.0d0 / a)
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e+44) {
		tmp = ((a / b) / a) * -c;
	} else if (b <= -5e-310) {
		tmp = -0.5 * (0.0 / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.5e+44:
		tmp = ((a / b) / a) * -c
	elif b <= -5e-310:
		tmp = -0.5 * (0.0 / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.5e+44)
		tmp = Float64(Float64(Float64(a / b) / a) * Float64(-c));
	elseif (b <= -5e-310)
		tmp = Float64(-0.5 * Float64(0.0 / a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.5e+44)
		tmp = ((a / b) / a) * -c;
	elseif (b <= -5e-310)
		tmp = -0.5 * (0.0 / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.5e+44], N[(N[(N[(a / b), $MachinePrecision] / a), $MachinePrecision] * (-c)), $MachinePrecision], If[LessEqual[b, -5e-310], N[(-0.5 * N[(0.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.50000000000000027e44

   1. Initial program 21.1%

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

      1. *-commutative 21.1%

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

      2. sqr-neg 21.1%

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

      3. *-commutative 21.1%

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

      4. sqr-neg 21.1%

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

      5. associate-*r* 21.1%

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

      6. *-commutative 21.1%

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

   3. Simplified 21.1%

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

   5. Taylor expanded in b around -inf 68.3%

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

      1. associate-/l* 71.9%

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

   7. Simplified 71.9%

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

      1. expm1-log1p-u 64.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-2 \cdot \frac{a}{\frac{b}{c}}}{a \cdot 2}\right)\right)} \]

      2. expm1-udef 57.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-2 \cdot \frac{a}{\frac{b}{c}}}{a \cdot 2}\right)} - 1} \]

      3. *-commutative 57.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-2 \cdot \frac{a}{\frac{b}{c}}}{\color{blue}{2 \cdot a}}\right)} - 1 \]

      4. times-frac 57.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-2}{2} \cdot \frac{\frac{a}{\frac{b}{c}}}{a}}\right)} - 1 \]

      5. metadata-eval 57.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-1} \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)} - 1 \]

      6. associate-/r/ 64.3%

         \[\leadsto e^{\mathsf{log1p}\left(-1 \cdot \frac{\color{blue}{\frac{a}{b} \cdot c}}{a}\right)} - 1 \]

   9. Applied egg-rr 64.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-1 \cdot \frac{\frac{a}{b} \cdot c}{a}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 74.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 \cdot \frac{\frac{a}{b} \cdot c}{a}\right)\right)} \]

       2. expm1-log1p 81.8%

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

       3. mul-1-neg 81.8%

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

       4. associate-/l* 65.4%

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

       5. distribute-neg-frac 65.4%

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

   11. Simplified 65.4%

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

       1. distribute-frac-neg 65.4%

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

       2. add-sqr-sqrt 44.0%

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

       3. sqrt-unprod 58.0%

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

       4. sqr-neg 58.0%

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

       5. sqrt-unprod 26.8%

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

       6. add-sqr-sqrt 38.4%

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

       7. associate-/r/ 47.0%

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

       8. add-sqr-sqrt 35.4%

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

       9. sqrt-unprod 68.3%

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

       10. sqr-neg 68.3%

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

       11. sqrt-unprod 54.2%

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

       12. add-sqr-sqrt 80.5%

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

   13. Applied egg-rr 80.5%

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

   if -7.50000000000000027e44 < b < -4.999999999999985e-310

   1. Initial program 71.7%

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

      1. sub-neg 71.7%

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

      2. distribute-neg-out 71.7%

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

      3. neg-mul-1 71.7%

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

      4. times-frac 71.7%

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

      5. metadata-eval 71.7%

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

      6. sub-neg 71.7%

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

      7. +-commutative 71.7%

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

      8. *-commutative 71.7%

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

      9. distribute-lft-neg-in 71.7%

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

      10. distribute-rgt-neg-out 71.7%

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

      11. associate-*l* 71.7%

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

      12. fma-def 71.7%

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

      13. distribute-lft-neg-in 71.7%

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

      14. distribute-rgt-neg-in 71.7%

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

      15. metadata-eval 71.7%

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

   3. Simplified 71.7%

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

      1. add-sqr-sqrt 60.3%

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

      2. pow2 60.3%

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

      3. pow1/2 60.3%

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

      4. sqrt-pow1 60.3%

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

      5. pow2 60.3%

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

      6. metadata-eval 60.3%

         \[\leadsto -0.5 \cdot \frac{b + {\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   6. Applied egg-rr 60.3%

      \[\leadsto -0.5 \cdot \frac{b + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}{a} \]

   7. Taylor expanded in b around -inf 30.3%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{b + -1 \cdot b}}{a} \]
   8. Step-by-step derivation

      1. distribute-rgt1-in 30.3%

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

      2. metadata-eval 30.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0} \cdot b}{a} \]

      3. mul0-lft 30.3%

         \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{a} \]

   9. Simplified 30.3%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{0}}{a} \]

   if -4.999999999999985e-310 < b

   1. Initial program 70.6%

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

      1. *-commutative 70.6%

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

      2. sqr-neg 70.6%

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

      3. *-commutative 70.6%

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

      4. sqr-neg 70.6%

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

      5. associate-*r* 70.6%

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

      6. *-commutative 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in b around inf 66.9%

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

      1. +-commutative 66.9%

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

      2. mul-1-neg 66.9%

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

      3. unsub-neg 66.9%

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

   7. Simplified 66.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{a}{b}}{a} \cdot \left(-c\right)\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{-300}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.4e-300) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.4e-300) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	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-300)) then
        tmp = -c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.4e-300) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.4e-300:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.4e-300)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.4e-300)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.4e-300], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.4000000000000003e-300

   1. Initial program 47.8%

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

      1. *-commutative 47.8%

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

      2. sqr-neg 47.8%

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

      3. *-commutative 47.8%

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

      4. sqr-neg 47.8%

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

      5. associate-*r* 47.8%

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

      6. *-commutative 47.8%

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

   3. Simplified 47.8%

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

   5. Taylor expanded in b around -inf 38.6%

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

      1. mul-1-neg 38.6%

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

   7. Simplified 38.6%

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

   if -7.4000000000000003e-300 < b

   1. Initial program 71.1%

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

      1. *-commutative 71.1%

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

      2. sqr-neg 71.1%

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

      3. *-commutative 71.1%

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

      4. sqr-neg 71.1%

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

      5. associate-*r* 71.1%

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

      6. *-commutative 71.1%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in b around inf 65.4%

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

      1. associate-*r/ 65.4%

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

      2. mul-1-neg 65.4%

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

   7. Simplified 65.4%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{-300}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 22.1% accurate, 29.0× 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(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 59.7%

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

   1. *-commutative 59.7%

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

   2. sqr-neg 59.7%

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

   3. *-commutative 59.7%

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

   4. sqr-neg 59.7%

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

   5. associate-*r* 59.7%

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

   6. *-commutative 59.7%

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

3. Simplified 59.7%

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

5. Taylor expanded in b around -inf 20.1%

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

   1. mul-1-neg 20.1%

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

7. Simplified 20.1%

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

8. Final simplification 20.1%

   \[\leadsto \frac{-c}{b} \]
9. Add Preprocessing

Alternative 10: 11.0% 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 59.7%

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

   1. *-commutative 59.7%

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

   2. sqr-neg 59.7%

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

   3. *-commutative 59.7%

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

   4. sqr-neg 59.7%

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

   5. associate-*r* 59.7%

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

   6. *-commutative 59.7%

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

3. Simplified 59.7%

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

5. Taylor expanded in b around inf 34.8%

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

   1. +-commutative 34.8%

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

   2. mul-1-neg 34.8%

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

   3. unsub-neg 34.8%

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

7. Simplified 34.8%

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

8. Taylor expanded in c around inf 9.1%

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

9. Final simplification 9.1%

   \[\leadsto \frac{c}{b} \]
10. Add Preprocessing

Developer target: 86.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{t\_2 - \frac{b}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{2} + t\_2}{-a}\\ \end{array} \end{array} \]

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) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

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 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	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 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	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 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	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(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	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 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	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[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024031 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (if (< b 0.0) (/ c (- (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))) (/ (+ (/ 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)))))) (- a)))

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