quadm (p42, negative)

Percentage Accurate: 52.5% → 87.2%

Time: 14.7s

Alternatives: 11

Speedup: 19.1×

Specification

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.5% 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: 87.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -3.9 \cdot 10^{-147}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{c \cdot \left(a \cdot 4\right)}{b - t_0}}{a}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;-0.5 \cdot \frac{b + t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* a (* c 4.0))))))
   (if (<= b -4.2e+104)
     (/ (- c) b)
     (if (<= b -3.9e-147)
       (* -0.5 (/ (/ (* c (* a 4.0)) (- b t_0)) a))
       (if (<= b 2.1e+25) (* -0.5 (/ (+ b t_0) a)) (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (a * (c * 4.0))));
	double tmp;
	if (b <= -4.2e+104) {
		tmp = -c / b;
	} else if (b <= -3.9e-147) {
		tmp = -0.5 * (((c * (a * 4.0)) / (b - t_0)) / a);
	} else if (b <= 2.1e+25) {
		tmp = -0.5 * ((b + t_0) / a);
	} 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 = sqrt(((b * b) - (a * (c * 4.0d0))))
    if (b <= (-4.2d+104)) then
        tmp = -c / b
    else if (b <= (-3.9d-147)) then
        tmp = (-0.5d0) * (((c * (a * 4.0d0)) / (b - t_0)) / a)
    else if (b <= 2.1d+25) then
        tmp = (-0.5d0) * ((b + t_0) / a)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (a * (c * 4.0))));
	double tmp;
	if (b <= -4.2e+104) {
		tmp = -c / b;
	} else if (b <= -3.9e-147) {
		tmp = -0.5 * (((c * (a * 4.0)) / (b - t_0)) / a);
	} else if (b <= 2.1e+25) {
		tmp = -0.5 * ((b + t_0) / a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (a * (c * 4.0))))
	tmp = 0
	if b <= -4.2e+104:
		tmp = -c / b
	elif b <= -3.9e-147:
		tmp = -0.5 * (((c * (a * 4.0)) / (b - t_0)) / a)
	elif b <= 2.1e+25:
		tmp = -0.5 * ((b + t_0) / a)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 4.0))))
	tmp = 0.0
	if (b <= -4.2e+104)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= -3.9e-147)
		tmp = Float64(-0.5 * Float64(Float64(Float64(c * Float64(a * 4.0)) / Float64(b - t_0)) / a));
	elseif (b <= 2.1e+25)
		tmp = Float64(-0.5 * Float64(Float64(b + t_0) / a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - (a * (c * 4.0))));
	tmp = 0.0;
	if (b <= -4.2e+104)
		tmp = -c / b;
	elseif (b <= -3.9e-147)
		tmp = -0.5 * (((c * (a * 4.0)) / (b - t_0)) / a);
	elseif (b <= 2.1e+25)
		tmp = -0.5 * ((b + t_0) / a);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -4.2e+104], N[((-c) / b), $MachinePrecision], If[LessEqual[b, -3.9e-147], N[(-0.5 * N[(N[(N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision] / N[(b - t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.1e+25], N[(-0.5 * N[(N[(b + t$95$0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.1999999999999997e104

   1. Initial program 3.8%

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

   2. Taylor expanded in b around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   4. Simplified 100.0%

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

   if -4.1999999999999997e104 < b < -3.8999999999999998e-147

   1. Initial program 42.6%

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

   3. Simplified 42.6%

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

      1. fma-udef 42.6%

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

      2. associate-*r* 42.6%

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

      3. metadata-eval 42.6%

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

      4. distribute-rgt-neg-in 42.6%

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

      5. *-commutative 42.6%

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

      6. +-commutative 42.6%

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

      7. sub-neg 42.6%

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

      8. *-commutative 42.6%

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

      9. associate-*l* 42.6%

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

   5. Applied egg-rr 42.6%

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

      1. flip-+ 42.2%

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

      2. add-sqr-sqrt 42.1%

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

   7. Applied egg-rr 42.1%

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

   8. Taylor expanded in b around 0 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-*l* 72.7%

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

   10. Simplified 72.7%

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

   if -3.8999999999999998e-147 < b < 2.0999999999999999e25

   1. Initial program 87.8%

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

   3. Simplified 87.8%

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

      1. fma-udef 87.8%

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

      2. associate-*r* 87.8%

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

      3. metadata-eval 87.8%

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

      4. distribute-rgt-neg-in 87.8%

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

      5. *-commutative 87.8%

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

      6. +-commutative 87.8%

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

      7. sub-neg 87.8%

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

      8. *-commutative 87.8%

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

      9. associate-*l* 87.8%

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

   5. Applied egg-rr 87.8%

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

   if 2.0999999999999999e25 < b

   1. Initial program 58.2%

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

   2. Taylor expanded in b around inf 97.1%

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

      1. associate-*r/ 97.1%

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

      2. mul-1-neg 97.1%

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

   4. Simplified 97.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -3.9 \cdot 10^{-147}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{c \cdot \left(a \cdot 4\right)}{b - \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}}{a}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-c}{b}\\ t_1 := -0.5 \cdot \frac{b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a}\\ \mathbf{if}\;b \leq -3 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- c) b))
        (t_1 (* -0.5 (/ (+ b (sqrt (- (* b b) (* a (* c 4.0))))) a))))
   (if (<= b -3e-45)
     t_0
     (if (<= b -6.2e-106)
       t_1
       (if (<= b -1.32e-129) t_0 (if (<= b 2.1e+25) t_1 (/ (- b) a)))))))

C

double code(double a, double b, double c) {
	double t_0 = -c / b;
	double t_1 = -0.5 * ((b + sqrt(((b * b) - (a * (c * 4.0))))) / a);
	double tmp;
	if (b <= -3e-45) {
		tmp = t_0;
	} else if (b <= -6.2e-106) {
		tmp = t_1;
	} else if (b <= -1.32e-129) {
		tmp = t_0;
	} else if (b <= 2.1e+25) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = -c / b
    t_1 = (-0.5d0) * ((b + sqrt(((b * b) - (a * (c * 4.0d0))))) / a)
    if (b <= (-3d-45)) then
        tmp = t_0
    else if (b <= (-6.2d-106)) then
        tmp = t_1
    else if (b <= (-1.32d-129)) then
        tmp = t_0
    else if (b <= 2.1d+25) then
        tmp = t_1
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = -c / b;
	double t_1 = -0.5 * ((b + Math.sqrt(((b * b) - (a * (c * 4.0))))) / a);
	double tmp;
	if (b <= -3e-45) {
		tmp = t_0;
	} else if (b <= -6.2e-106) {
		tmp = t_1;
	} else if (b <= -1.32e-129) {
		tmp = t_0;
	} else if (b <= 2.1e+25) {
		tmp = t_1;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -c / b
	t_1 = -0.5 * ((b + math.sqrt(((b * b) - (a * (c * 4.0))))) / a)
	tmp = 0
	if b <= -3e-45:
		tmp = t_0
	elif b <= -6.2e-106:
		tmp = t_1
	elif b <= -1.32e-129:
		tmp = t_0
	elif b <= 2.1e+25:
		tmp = t_1
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-c) / b)
	t_1 = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 4.0))))) / a))
	tmp = 0.0
	if (b <= -3e-45)
		tmp = t_0;
	elseif (b <= -6.2e-106)
		tmp = t_1;
	elseif (b <= -1.32e-129)
		tmp = t_0;
	elseif (b <= 2.1e+25)
		tmp = t_1;
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -c / b;
	t_1 = -0.5 * ((b + sqrt(((b * b) - (a * (c * 4.0))))) / a);
	tmp = 0.0;
	if (b <= -3e-45)
		tmp = t_0;
	elseif (b <= -6.2e-106)
		tmp = t_1;
	elseif (b <= -1.32e-129)
		tmp = t_0;
	elseif (b <= 2.1e+25)
		tmp = t_1;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-c) / b), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3e-45], t$95$0, If[LessEqual[b, -6.2e-106], t$95$1, If[LessEqual[b, -1.32e-129], t$95$0, If[LessEqual[b, 2.1e+25], t$95$1, N[((-b) / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.00000000000000011e-45 or -6.19999999999999971e-106 < b < -1.31999999999999992e-129

   1. Initial program 17.2%

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

   2. Taylor expanded in b around -inf 86.3%

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

      1. associate-*r/ 86.3%

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

      2. neg-mul-1 86.3%

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

   4. Simplified 86.3%

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

   if -3.00000000000000011e-45 < b < -6.19999999999999971e-106 or -1.31999999999999992e-129 < b < 2.0999999999999999e25

   1. Initial program 84.4%

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

   3. Simplified 84.4%

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

      1. fma-udef 84.4%

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

      2. associate-*r* 84.4%

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

      3. metadata-eval 84.4%

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

      4. distribute-rgt-neg-in 84.4%

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

      5. *-commutative 84.4%

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

      6. +-commutative 84.4%

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

      7. sub-neg 84.4%

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

      8. *-commutative 84.4%

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

      9. associate-*l* 84.4%

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

   5. Applied egg-rr 84.4%

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

   if 2.0999999999999999e25 < b

   1. Initial program 58.2%

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

   2. Taylor expanded in b around inf 97.1%

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

      1. associate-*r/ 97.1%

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

      2. mul-1-neg 97.1%

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

   4. Simplified 97.1%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-45}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a}\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{+25}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-c}{b}\\ t_1 := -0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\right)\\ \mathbf{if}\;b \leq -2.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- c) b))
        (t_1 (* -0.5 (+ (/ b a) (/ (sqrt (* c (* a -4.0))) a)))))
   (if (<= b -2.6e-45)
     t_0
     (if (<= b -6.2e-106)
       t_1
       (if (<= b -1.32e-129)
         t_0
         (if (<= b 9.5e-53) t_1 (- (/ c b) (/ b a))))))))

C

double code(double a, double b, double c) {
	double t_0 = -c / b;
	double t_1 = -0.5 * ((b / a) + (sqrt((c * (a * -4.0))) / a));
	double tmp;
	if (b <= -2.6e-45) {
		tmp = t_0;
	} else if (b <= -6.2e-106) {
		tmp = t_1;
	} else if (b <= -1.32e-129) {
		tmp = t_0;
	} else if (b <= 9.5e-53) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = -c / b
    t_1 = (-0.5d0) * ((b / a) + (sqrt((c * (a * (-4.0d0)))) / a))
    if (b <= (-2.6d-45)) then
        tmp = t_0
    else if (b <= (-6.2d-106)) then
        tmp = t_1
    else if (b <= (-1.32d-129)) then
        tmp = t_0
    else if (b <= 9.5d-53) then
        tmp = t_1
    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 = -c / b;
	double t_1 = -0.5 * ((b / a) + (Math.sqrt((c * (a * -4.0))) / a));
	double tmp;
	if (b <= -2.6e-45) {
		tmp = t_0;
	} else if (b <= -6.2e-106) {
		tmp = t_1;
	} else if (b <= -1.32e-129) {
		tmp = t_0;
	} else if (b <= 9.5e-53) {
		tmp = t_1;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -c / b
	t_1 = -0.5 * ((b / a) + (math.sqrt((c * (a * -4.0))) / a))
	tmp = 0
	if b <= -2.6e-45:
		tmp = t_0
	elif b <= -6.2e-106:
		tmp = t_1
	elif b <= -1.32e-129:
		tmp = t_0
	elif b <= 9.5e-53:
		tmp = t_1
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-c) / b)
	t_1 = Float64(-0.5 * Float64(Float64(b / a) + Float64(sqrt(Float64(c * Float64(a * -4.0))) / a)))
	tmp = 0.0
	if (b <= -2.6e-45)
		tmp = t_0;
	elseif (b <= -6.2e-106)
		tmp = t_1;
	elseif (b <= -1.32e-129)
		tmp = t_0;
	elseif (b <= 9.5e-53)
		tmp = t_1;
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -c / b;
	t_1 = -0.5 * ((b / a) + (sqrt((c * (a * -4.0))) / a));
	tmp = 0.0;
	if (b <= -2.6e-45)
		tmp = t_0;
	elseif (b <= -6.2e-106)
		tmp = t_1;
	elseif (b <= -1.32e-129)
		tmp = t_0;
	elseif (b <= 9.5e-53)
		tmp = t_1;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-c) / b), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[(N[(b / a), $MachinePrecision] + N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.6e-45], t$95$0, If[LessEqual[b, -6.2e-106], t$95$1, If[LessEqual[b, -1.32e-129], t$95$0, If[LessEqual[b, 9.5e-53], t$95$1, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.59999999999999987e-45 or -6.19999999999999971e-106 < b < -1.31999999999999992e-129

   1. Initial program 17.2%

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

   2. Taylor expanded in b around -inf 86.3%

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

      1. associate-*r/ 86.3%

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

      2. neg-mul-1 86.3%

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

   4. Simplified 86.3%

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

   if -2.59999999999999987e-45 < b < -6.19999999999999971e-106 or -1.31999999999999992e-129 < b < 9.5000000000000008e-53

   1. Initial program 81.5%

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

   3. Simplified 81.5%

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

      1. fma-udef 81.5%

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

      2. associate-*r* 81.5%

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

      3. metadata-eval 81.5%

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

      4. distribute-rgt-neg-in 81.5%

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

      5. *-commutative 81.5%

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

      6. +-commutative 81.5%

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

      7. sub-neg 81.5%

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

      8. prod-diff 81.3%

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

      9. *-commutative 81.3%

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

      10. fma-neg 81.3%

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

      11. prod-diff 81.3%

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

      12. *-commutative 81.3%

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

      13. fma-neg 81.3%

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

      14. associate-+l+ 81.3%

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

   5. Applied egg-rr 81.3%

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

      1. fma-udef 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-*l* 81.3%

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

      4. +-commutative 81.3%

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

      5. fma-def 81.3%

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

      6. *-commutative 81.3%

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

      7. count-2 81.3%

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

      8. fma-def 81.3%

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

      9. associate-*r* 81.3%

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

      10. *-commutative 81.3%

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

   7. Simplified 81.3%

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

   8. Taylor expanded in b around 0 77.2%

      \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{b}{a} + \sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right) + -4 \cdot \left(c \cdot a\right)} \cdot \frac{1}{a}\right)} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 77.2%

         \[\leadsto -0.5 \cdot \color{blue}{\left(1 \cdot \left(\frac{b}{a} + \sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right) + -4 \cdot \left(c \cdot a\right)} \cdot \frac{1}{a}\right)\right)} \]

      2. +-commutative 77.2%

         \[\leadsto -0.5 \cdot \left(1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right) + -4 \cdot \left(c \cdot a\right)} \cdot \frac{1}{a} + \frac{b}{a}\right)}\right) \]

      3. un-div-inv 77.2%

         \[\leadsto -0.5 \cdot \left(1 \cdot \left(\color{blue}{\frac{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right) + -4 \cdot \left(c \cdot a\right)}}{a}} + \frac{b}{a}\right)\right) \]

      4. fma-def 77.2%

         \[\leadsto -0.5 \cdot \left(1 \cdot \left(\frac{\sqrt{\color{blue}{\mathsf{fma}\left(2, -4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right), -4 \cdot \left(c \cdot a\right)\right)}}}{a} + \frac{b}{a}\right)\right) \]

      5. distribute-rgt-out 77.2%

         \[\leadsto -0.5 \cdot \left(1 \cdot \left(\frac{\sqrt{\mathsf{fma}\left(2, \color{blue}{\left(c \cdot a\right) \cdot \left(-4 + 4\right)}, -4 \cdot \left(c \cdot a\right)\right)}}{a} + \frac{b}{a}\right)\right) \]

      6. *-commutative 77.2%

         \[\leadsto -0.5 \cdot \left(1 \cdot \left(\frac{\sqrt{\mathsf{fma}\left(2, \color{blue}{\left(a \cdot c\right)} \cdot \left(-4 + 4\right), -4 \cdot \left(c \cdot a\right)\right)}}{a} + \frac{b}{a}\right)\right) \]

      7. metadata-eval 77.2%

         \[\leadsto -0.5 \cdot \left(1 \cdot \left(\frac{\sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot \color{blue}{0}, -4 \cdot \left(c \cdot a\right)\right)}}{a} + \frac{b}{a}\right)\right) \]

      8. *-commutative 77.2%

         \[\leadsto -0.5 \cdot \left(1 \cdot \left(\frac{\sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot 0, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} + \frac{b}{a}\right)\right) \]

   10. Applied egg-rr 77.2%

       \[\leadsto -0.5 \cdot \color{blue}{\left(1 \cdot \left(\frac{\sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot 0, -4 \cdot \left(a \cdot c\right)\right)}}{a} + \frac{b}{a}\right)\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 77.2%

          \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{\sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot 0, -4 \cdot \left(a \cdot c\right)\right)}}{a} + \frac{b}{a}\right)} \]

       2. +-commutative 77.2%

          \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot 0, -4 \cdot \left(a \cdot c\right)\right)}}{a}\right)} \]

       3. fma-udef 77.2%

          \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{\color{blue}{2 \cdot \left(\left(a \cdot c\right) \cdot 0\right) + -4 \cdot \left(a \cdot c\right)}}}{a}\right) \]

       4. mul0-rgt 77.4%

          \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{2 \cdot \color{blue}{0} + -4 \cdot \left(a \cdot c\right)}}{a}\right) \]

       5. metadata-eval 77.4%

          \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{\color{blue}{0} + -4 \cdot \left(a \cdot c\right)}}{a}\right) \]

       6. +-lft-identity 77.4%

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

       7. associate-*r* 77.4%

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

       8. *-commutative 77.4%

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

       9. *-commutative 77.4%

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

   12. Simplified 77.4%

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

   if 9.5000000000000008e-53 < b

   1. Initial program 66.1%

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

   2. Taylor expanded in b around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. unsub-neg 93.4%

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

   4. Simplified 93.4%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\right)\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-53}:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -2.3 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-106}:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} + t_0 \cdot \frac{1}{a}\right)\\ \mathbf{elif}\;b \leq -1.16 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-51}:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} + \frac{t_0}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* c (* a -4.0)))) (t_1 (/ (- c) b)))
   (if (<= b -2.3e-43)
     t_1
     (if (<= b -6.5e-106)
       (* -0.5 (+ (/ b a) (* t_0 (/ 1.0 a))))
       (if (<= b -1.16e-129)
         t_1
         (if (<= b 4.4e-51)
           (* -0.5 (+ (/ b a) (/ t_0 a)))
           (- (/ c b) (/ b a))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((c * (a * -4.0)));
	double t_1 = -c / b;
	double tmp;
	if (b <= -2.3e-43) {
		tmp = t_1;
	} else if (b <= -6.5e-106) {
		tmp = -0.5 * ((b / a) + (t_0 * (1.0 / a)));
	} else if (b <= -1.16e-129) {
		tmp = t_1;
	} else if (b <= 4.4e-51) {
		tmp = -0.5 * ((b / a) + (t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((c * (a * (-4.0d0))))
    t_1 = -c / b
    if (b <= (-2.3d-43)) then
        tmp = t_1
    else if (b <= (-6.5d-106)) then
        tmp = (-0.5d0) * ((b / a) + (t_0 * (1.0d0 / a)))
    else if (b <= (-1.16d-129)) then
        tmp = t_1
    else if (b <= 4.4d-51) then
        tmp = (-0.5d0) * ((b / a) + (t_0 / 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 t_0 = Math.sqrt((c * (a * -4.0)));
	double t_1 = -c / b;
	double tmp;
	if (b <= -2.3e-43) {
		tmp = t_1;
	} else if (b <= -6.5e-106) {
		tmp = -0.5 * ((b / a) + (t_0 * (1.0 / a)));
	} else if (b <= -1.16e-129) {
		tmp = t_1;
	} else if (b <= 4.4e-51) {
		tmp = -0.5 * ((b / a) + (t_0 / a));
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((c * (a * -4.0)))
	t_1 = -c / b
	tmp = 0
	if b <= -2.3e-43:
		tmp = t_1
	elif b <= -6.5e-106:
		tmp = -0.5 * ((b / a) + (t_0 * (1.0 / a)))
	elif b <= -1.16e-129:
		tmp = t_1
	elif b <= 4.4e-51:
		tmp = -0.5 * ((b / a) + (t_0 / a))
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(c * Float64(a * -4.0)))
	t_1 = Float64(Float64(-c) / b)
	tmp = 0.0
	if (b <= -2.3e-43)
		tmp = t_1;
	elseif (b <= -6.5e-106)
		tmp = Float64(-0.5 * Float64(Float64(b / a) + Float64(t_0 * Float64(1.0 / a))));
	elseif (b <= -1.16e-129)
		tmp = t_1;
	elseif (b <= 4.4e-51)
		tmp = Float64(-0.5 * Float64(Float64(b / a) + Float64(t_0 / a)));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt((c * (a * -4.0)));
	t_1 = -c / b;
	tmp = 0.0;
	if (b <= -2.3e-43)
		tmp = t_1;
	elseif (b <= -6.5e-106)
		tmp = -0.5 * ((b / a) + (t_0 * (1.0 / a)));
	elseif (b <= -1.16e-129)
		tmp = t_1;
	elseif (b <= 4.4e-51)
		tmp = -0.5 * ((b / a) + (t_0 / a));
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-c) / b), $MachinePrecision]}, If[LessEqual[b, -2.3e-43], t$95$1, If[LessEqual[b, -6.5e-106], N[(-0.5 * N[(N[(b / a), $MachinePrecision] + N[(t$95$0 * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.16e-129], t$95$1, If[LessEqual[b, 4.4e-51], N[(-0.5 * N[(N[(b / a), $MachinePrecision] + N[(t$95$0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.2999999999999999e-43 or -6.4999999999999997e-106 < b < -1.16e-129

   1. Initial program 17.2%

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

   2. Taylor expanded in b around -inf 86.3%

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

      1. associate-*r/ 86.3%

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

      2. neg-mul-1 86.3%

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

   4. Simplified 86.3%

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

   if -2.2999999999999999e-43 < b < -6.4999999999999997e-106

   1. Initial program 62.7%

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

   3. Simplified 62.7%

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

      1. fma-udef 62.7%

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

      2. associate-*r* 62.7%

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

      3. metadata-eval 62.7%

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

      4. distribute-rgt-neg-in 62.7%

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

      5. *-commutative 62.7%

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

      6. +-commutative 62.7%

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

      7. sub-neg 62.7%

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

      8. prod-diff 62.3%

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

      9. *-commutative 62.3%

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

      10. fma-neg 62.3%

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

      11. prod-diff 62.3%

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

      12. *-commutative 62.3%

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

      13. fma-neg 62.3%

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

      14. associate-+l+ 62.3%

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

   5. Applied egg-rr 62.3%

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

      1. fma-udef 62.3%

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

      2. *-commutative 62.3%

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

      3. associate-*l* 62.3%

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

      4. +-commutative 62.3%

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

      5. fma-def 62.3%

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

      6. *-commutative 62.3%

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

      7. count-2 62.3%

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

      8. fma-def 62.5%

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

      9. associate-*r* 62.5%

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

      10. *-commutative 62.5%

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

   7. Simplified 62.5%

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

   8. Taylor expanded in b around 0 60.5%

      \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{b}{a} + \sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right) + -4 \cdot \left(c \cdot a\right)} \cdot \frac{1}{a}\right)} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 60.5%

         \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \color{blue}{\left(1 \cdot \sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right) + -4 \cdot \left(c \cdot a\right)}\right)} \cdot \frac{1}{a}\right) \]

      2. fma-def 60.5%

         \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \left(1 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(2, -4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right), -4 \cdot \left(c \cdot a\right)\right)}}\right) \cdot \frac{1}{a}\right) \]

      3. distribute-rgt-out 60.5%

         \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \left(1 \cdot \sqrt{\mathsf{fma}\left(2, \color{blue}{\left(c \cdot a\right) \cdot \left(-4 + 4\right)}, -4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \frac{1}{a}\right) \]

      4. *-commutative 60.5%

         \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \left(1 \cdot \sqrt{\mathsf{fma}\left(2, \color{blue}{\left(a \cdot c\right)} \cdot \left(-4 + 4\right), -4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \frac{1}{a}\right) \]

      5. metadata-eval 60.5%

         \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \left(1 \cdot \sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot \color{blue}{0}, -4 \cdot \left(c \cdot a\right)\right)}\right) \cdot \frac{1}{a}\right) \]

      6. *-commutative 60.5%

         \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \left(1 \cdot \sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot 0, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}\right) \cdot \frac{1}{a}\right) \]

   10. Applied egg-rr 60.5%

       \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \color{blue}{\left(1 \cdot \sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot 0, -4 \cdot \left(a \cdot c\right)\right)}\right)} \cdot \frac{1}{a}\right) \]
   11. Step-by-step derivation

       1. *-lft-identity 60.5%

          \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \color{blue}{\sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot 0, -4 \cdot \left(a \cdot c\right)\right)}} \cdot \frac{1}{a}\right) \]

       2. fma-udef 60.5%

          \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \sqrt{\color{blue}{2 \cdot \left(\left(a \cdot c\right) \cdot 0\right) + -4 \cdot \left(a \cdot c\right)}} \cdot \frac{1}{a}\right) \]

       3. mul0-rgt 60.9%

          \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \sqrt{2 \cdot \color{blue}{0} + -4 \cdot \left(a \cdot c\right)} \cdot \frac{1}{a}\right) \]

       4. metadata-eval 60.9%

          \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \sqrt{\color{blue}{0} + -4 \cdot \left(a \cdot c\right)} \cdot \frac{1}{a}\right) \]

       5. +-lft-identity 60.9%

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

       6. associate-*r* 60.9%

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

       7. *-commutative 60.9%

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

       8. *-commutative 60.9%

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

   12. Simplified 60.9%

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

   if -1.16e-129 < b < 4.4e-51

   1. Initial program 85.1%

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

   3. Simplified 85.1%

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

      1. fma-udef 85.1%

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

      2. associate-*r* 85.1%

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

      3. metadata-eval 85.1%

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

      4. distribute-rgt-neg-in 85.1%

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

      5. *-commutative 85.1%

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

      6. +-commutative 85.1%

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

      7. sub-neg 85.1%

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

      8. prod-diff 84.9%

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

      9. *-commutative 84.9%

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

      10. fma-neg 84.9%

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

      11. prod-diff 84.9%

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

      12. *-commutative 84.9%

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

      13. fma-neg 84.9%

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

      14. associate-+l+ 84.9%

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

   5. Applied egg-rr 84.9%

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

      1. fma-udef 84.9%

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

      2. *-commutative 84.9%

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

      3. associate-*l* 84.9%

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

      4. +-commutative 84.9%

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

      5. fma-def 84.9%

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

      6. *-commutative 84.9%

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

      7. count-2 84.9%

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

      8. fma-def 84.9%

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

      9. associate-*r* 84.9%

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

      10. *-commutative 84.9%

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

   7. Simplified 84.9%

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

   8. Taylor expanded in b around 0 80.4%

      \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{b}{a} + \sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right) + -4 \cdot \left(c \cdot a\right)} \cdot \frac{1}{a}\right)} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 80.4%

         \[\leadsto -0.5 \cdot \color{blue}{\left(1 \cdot \left(\frac{b}{a} + \sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right) + -4 \cdot \left(c \cdot a\right)} \cdot \frac{1}{a}\right)\right)} \]

      2. +-commutative 80.4%

         \[\leadsto -0.5 \cdot \left(1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right) + -4 \cdot \left(c \cdot a\right)} \cdot \frac{1}{a} + \frac{b}{a}\right)}\right) \]

      3. un-div-inv 80.4%

         \[\leadsto -0.5 \cdot \left(1 \cdot \left(\color{blue}{\frac{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right) + -4 \cdot \left(c \cdot a\right)}}{a}} + \frac{b}{a}\right)\right) \]

      4. fma-def 80.4%

         \[\leadsto -0.5 \cdot \left(1 \cdot \left(\frac{\sqrt{\color{blue}{\mathsf{fma}\left(2, -4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right), -4 \cdot \left(c \cdot a\right)\right)}}}{a} + \frac{b}{a}\right)\right) \]

      5. distribute-rgt-out 80.4%

         \[\leadsto -0.5 \cdot \left(1 \cdot \left(\frac{\sqrt{\mathsf{fma}\left(2, \color{blue}{\left(c \cdot a\right) \cdot \left(-4 + 4\right)}, -4 \cdot \left(c \cdot a\right)\right)}}{a} + \frac{b}{a}\right)\right) \]

      6. *-commutative 80.4%

         \[\leadsto -0.5 \cdot \left(1 \cdot \left(\frac{\sqrt{\mathsf{fma}\left(2, \color{blue}{\left(a \cdot c\right)} \cdot \left(-4 + 4\right), -4 \cdot \left(c \cdot a\right)\right)}}{a} + \frac{b}{a}\right)\right) \]

      7. metadata-eval 80.4%

         \[\leadsto -0.5 \cdot \left(1 \cdot \left(\frac{\sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot \color{blue}{0}, -4 \cdot \left(c \cdot a\right)\right)}}{a} + \frac{b}{a}\right)\right) \]

      8. *-commutative 80.4%

         \[\leadsto -0.5 \cdot \left(1 \cdot \left(\frac{\sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot 0, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)}}{a} + \frac{b}{a}\right)\right) \]

   10. Applied egg-rr 80.4%

       \[\leadsto -0.5 \cdot \color{blue}{\left(1 \cdot \left(\frac{\sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot 0, -4 \cdot \left(a \cdot c\right)\right)}}{a} + \frac{b}{a}\right)\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 80.4%

          \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{\sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot 0, -4 \cdot \left(a \cdot c\right)\right)}}{a} + \frac{b}{a}\right)} \]

       2. +-commutative 80.4%

          \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot 0, -4 \cdot \left(a \cdot c\right)\right)}}{a}\right)} \]

       3. fma-udef 80.4%

          \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{\color{blue}{2 \cdot \left(\left(a \cdot c\right) \cdot 0\right) + -4 \cdot \left(a \cdot c\right)}}}{a}\right) \]

       4. mul0-rgt 80.5%

          \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{2 \cdot \color{blue}{0} + -4 \cdot \left(a \cdot c\right)}}{a}\right) \]

       5. metadata-eval 80.5%

          \[\leadsto -0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{\color{blue}{0} + -4 \cdot \left(a \cdot c\right)}}{a}\right) \]

       6. +-lft-identity 80.5%

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

       7. associate-*r* 80.5%

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

       8. *-commutative 80.5%

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

       9. *-commutative 80.5%

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

   12. Simplified 80.5%

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

   if 4.4e-51 < b

   1. Initial program 66.1%

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

   2. Taylor expanded in b around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. unsub-neg 93.4%

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

   4. Simplified 93.4%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-43}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-106}:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} + \sqrt{c \cdot \left(a \cdot -4\right)} \cdot \frac{1}{a}\right)\\ \mathbf{elif}\;b \leq -1.16 \cdot 10^{-129}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-51}:\\ \;\;\;\;-0.5 \cdot \left(\frac{b}{a} + \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-c}{b}\\ t_1 := -0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.08 \cdot 10^{-131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- c) b)) (t_1 (* -0.5 (/ (+ b (sqrt (* -4.0 (* c a)))) a))))
   (if (<= b -3.5e-45)
     t_0
     (if (<= b -6.2e-106)
       t_1
       (if (<= b -1.08e-131)
         t_0
         (if (<= b 2.1e-53) t_1 (- (/ c b) (/ b a))))))))

C

double code(double a, double b, double c) {
	double t_0 = -c / b;
	double t_1 = -0.5 * ((b + sqrt((-4.0 * (c * a)))) / a);
	double tmp;
	if (b <= -3.5e-45) {
		tmp = t_0;
	} else if (b <= -6.2e-106) {
		tmp = t_1;
	} else if (b <= -1.08e-131) {
		tmp = t_0;
	} else if (b <= 2.1e-53) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = -c / b
    t_1 = (-0.5d0) * ((b + sqrt(((-4.0d0) * (c * a)))) / a)
    if (b <= (-3.5d-45)) then
        tmp = t_0
    else if (b <= (-6.2d-106)) then
        tmp = t_1
    else if (b <= (-1.08d-131)) then
        tmp = t_0
    else if (b <= 2.1d-53) then
        tmp = t_1
    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 = -c / b;
	double t_1 = -0.5 * ((b + Math.sqrt((-4.0 * (c * a)))) / a);
	double tmp;
	if (b <= -3.5e-45) {
		tmp = t_0;
	} else if (b <= -6.2e-106) {
		tmp = t_1;
	} else if (b <= -1.08e-131) {
		tmp = t_0;
	} else if (b <= 2.1e-53) {
		tmp = t_1;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = -c / b
	t_1 = -0.5 * ((b + math.sqrt((-4.0 * (c * a)))) / a)
	tmp = 0
	if b <= -3.5e-45:
		tmp = t_0
	elif b <= -6.2e-106:
		tmp = t_1
	elif b <= -1.08e-131:
		tmp = t_0
	elif b <= 2.1e-53:
		tmp = t_1
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-c) / b)
	t_1 = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(-4.0 * Float64(c * a)))) / a))
	tmp = 0.0
	if (b <= -3.5e-45)
		tmp = t_0;
	elseif (b <= -6.2e-106)
		tmp = t_1;
	elseif (b <= -1.08e-131)
		tmp = t_0;
	elseif (b <= 2.1e-53)
		tmp = t_1;
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = -c / b;
	t_1 = -0.5 * ((b + sqrt((-4.0 * (c * a)))) / a);
	tmp = 0.0;
	if (b <= -3.5e-45)
		tmp = t_0;
	elseif (b <= -6.2e-106)
		tmp = t_1;
	elseif (b <= -1.08e-131)
		tmp = t_0;
	elseif (b <= 2.1e-53)
		tmp = t_1;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[((-c) / b), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[(N[(b + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.5e-45], t$95$0, If[LessEqual[b, -6.2e-106], t$95$1, If[LessEqual[b, -1.08e-131], t$95$0, If[LessEqual[b, 2.1e-53], t$95$1, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.5e-45 or -6.19999999999999971e-106 < b < -1.07999999999999996e-131

   1. Initial program 17.2%

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

   2. Taylor expanded in b around -inf 86.3%

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

      1. associate-*r/ 86.3%

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

      2. neg-mul-1 86.3%

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

   4. Simplified 86.3%

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

   if -3.5e-45 < b < -6.19999999999999971e-106 or -1.07999999999999996e-131 < b < 2.09999999999999977e-53

   1. Initial program 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in a around inf 77.4%

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

      1. *-commutative 77.4%

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

   6. Simplified 77.4%

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

   if 2.09999999999999977e-53 < b

   1. Initial program 66.1%

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

   2. Taylor expanded in b around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. unsub-neg 93.4%

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

   4. Simplified 93.4%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{elif}\;b \leq -1.08 \cdot 10^{-131}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-53}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-54}:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.32e-129)
   (/ (- c) b)
   (if (<= b 1.65e-54)
     (* -0.5 (/ (sqrt (* c (* a -4.0))) a))
     (- (/ c b) (/ b a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.32e-129) {
		tmp = -c / b;
	} else if (b <= 1.65e-54) {
		tmp = -0.5 * (sqrt((c * (a * -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 <= (-1.32d-129)) then
        tmp = -c / b
    else if (b <= 1.65d-54) then
        tmp = (-0.5d0) * (sqrt((c * (a * (-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 <= -1.32e-129) {
		tmp = -c / b;
	} else if (b <= 1.65e-54) {
		tmp = -0.5 * (Math.sqrt((c * (a * -4.0))) / a);
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.32e-129:
		tmp = -c / b
	elif b <= 1.65e-54:
		tmp = -0.5 * (math.sqrt((c * (a * -4.0))) / a)
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.32e-129)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.65e-54)
		tmp = Float64(-0.5 * Float64(sqrt(Float64(c * Float64(a * -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 <= -1.32e-129)
		tmp = -c / b;
	elseif (b <= 1.65e-54)
		tmp = -0.5 * (sqrt((c * (a * -4.0))) / a);
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.32e-129], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.65e-54], N[(-0.5 * N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.31999999999999992e-129

   1. Initial program 22.7%

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

   2. Taylor expanded in b around -inf 79.3%

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

      1. associate-*r/ 79.3%

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

      2. neg-mul-1 79.3%

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

   4. Simplified 79.3%

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

   if -1.31999999999999992e-129 < b < 1.64999999999999996e-54

   1. Initial program 85.1%

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

   3. Simplified 85.1%

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

      1. fma-udef 85.1%

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

      2. associate-*r* 85.1%

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

      3. metadata-eval 85.1%

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

      4. distribute-rgt-neg-in 85.1%

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

      5. *-commutative 85.1%

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

      6. +-commutative 85.1%

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

      7. sub-neg 85.1%

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

      8. prod-diff 84.9%

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

      9. *-commutative 84.9%

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

      10. fma-neg 84.9%

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

      11. prod-diff 84.9%

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

      12. *-commutative 84.9%

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

      13. fma-neg 84.9%

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

      14. associate-+l+ 84.9%

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

   5. Applied egg-rr 84.9%

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

      1. fma-udef 84.9%

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

      2. *-commutative 84.9%

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

      3. associate-*l* 84.9%

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

      4. +-commutative 84.9%

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

      5. fma-def 84.9%

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

      6. *-commutative 84.9%

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

      7. count-2 84.9%

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

      8. fma-def 84.9%

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

      9. associate-*r* 84.9%

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

      10. *-commutative 84.9%

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

   7. Simplified 84.9%

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

   8. Taylor expanded in b around 0 80.4%

      \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{b}{a} + \sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right) + -4 \cdot \left(c \cdot a\right)} \cdot \frac{1}{a}\right)} \]

   9. Taylor expanded in b around 0 78.6%

      \[\leadsto -0.5 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right) + -4 \cdot \left(c \cdot a\right)} \cdot \frac{1}{a}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 78.6%

          \[\leadsto -0.5 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(-4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right)\right) + -4 \cdot \left(c \cdot a\right)} \cdot 1}{a}} \]

       2. fma-def 78.6%

          \[\leadsto -0.5 \cdot \frac{\sqrt{\color{blue}{\mathsf{fma}\left(2, -4 \cdot \left(c \cdot a\right) + 4 \cdot \left(c \cdot a\right), -4 \cdot \left(c \cdot a\right)\right)}} \cdot 1}{a} \]

       3. distribute-rgt-out 78.6%

          \[\leadsto -0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(2, \color{blue}{\left(c \cdot a\right) \cdot \left(-4 + 4\right)}, -4 \cdot \left(c \cdot a\right)\right)} \cdot 1}{a} \]

       4. *-commutative 78.6%

          \[\leadsto -0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(2, \color{blue}{\left(a \cdot c\right)} \cdot \left(-4 + 4\right), -4 \cdot \left(c \cdot a\right)\right)} \cdot 1}{a} \]

       5. metadata-eval 78.6%

          \[\leadsto -0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot \color{blue}{0}, -4 \cdot \left(c \cdot a\right)\right)} \cdot 1}{a} \]

       6. *-commutative 78.6%

          \[\leadsto -0.5 \cdot \frac{\sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot 0, -4 \cdot \color{blue}{\left(a \cdot c\right)}\right)} \cdot 1}{a} \]

       7. *-rgt-identity 78.6%

          \[\leadsto -0.5 \cdot \frac{\color{blue}{\sqrt{\mathsf{fma}\left(2, \left(a \cdot c\right) \cdot 0, -4 \cdot \left(a \cdot c\right)\right)}}}{a} \]

       8. fma-udef 78.6%

          \[\leadsto -0.5 \cdot \frac{\sqrt{\color{blue}{2 \cdot \left(\left(a \cdot c\right) \cdot 0\right) + -4 \cdot \left(a \cdot c\right)}}}{a} \]

       9. mul0-rgt 78.8%

          \[\leadsto -0.5 \cdot \frac{\sqrt{2 \cdot \color{blue}{0} + -4 \cdot \left(a \cdot c\right)}}{a} \]

       10. metadata-eval 78.8%

           \[\leadsto -0.5 \cdot \frac{\sqrt{\color{blue}{0} + -4 \cdot \left(a \cdot c\right)}}{a} \]

       11. +-lft-identity 78.8%

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

       12. associate-*r* 78.8%

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

       13. *-commutative 78.8%

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

       14. *-commutative 78.8%

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

   11. Simplified 78.8%

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

   if 1.64999999999999996e-54 < b

   1. Initial program 66.1%

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

   2. Taylor expanded in b around inf 93.4%

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

      1. mul-1-neg 93.4%

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

      2. unsub-neg 93.4%

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

   4. Simplified 93.4%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-129}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-54}:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{c \cdot \left(a \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 7: 67.7% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (- c) b) (- (/ c b) (/ b a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-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 <= (-2d-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 <= -2e-310) {
		tmp = -c / b;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = -c / b
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-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 <= -2e-310)
		tmp = -c / b;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[((-c) / b), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 34.4%

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

   2. Taylor expanded in b around -inf 65.9%

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

      1. associate-*r/ 65.9%

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

      2. neg-mul-1 65.9%

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

   4. Simplified 65.9%

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

   if -1.999999999999994e-310 < b

   1. Initial program 72.4%

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

   2. Taylor expanded in b around inf 67.2%

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

      1. mul-1-neg 67.2%

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

      2. unsub-neg 67.2%

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

   4. Simplified 67.2%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 8: 43.5% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -9.8e+51) (/ c b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.8e+51) {
		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 <= (-9.8d+51)) 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 <= -9.8e+51) {
		tmp = c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.8e+51:
		tmp = c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.8e+51)
		tmp = 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 <= -9.8e+51)
		tmp = c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.8e+51], N[(c / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.79999999999999967e51

   1. Initial program 14.0%

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

   2. Taylor expanded in b around inf 2.3%

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

      1. +-commutative 2.3%

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

      2. *-commutative 2.3%

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

      3. fma-def 2.3%

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

      4. *-commutative 2.3%

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

      5. associate-/l* 2.3%

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

   4. Simplified 2.3%

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

   5. Taylor expanded in b around 0 34.3%

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

   if -9.79999999999999967e51 < b

   1. Initial program 66.9%

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

   2. Taylor expanded in b around inf 45.1%

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

      1. associate-*r/ 45.1%

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

      2. mul-1-neg 45.1%

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

   4. Simplified 45.1%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 9: 67.6% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-261}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.5e-261) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.5e-261) {
		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 <= (-5.5d-261)) 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 <= -5.5e-261) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.5e-261:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.5e-261)
		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 <= -5.5e-261)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.5e-261], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.50000000000000042e-261

   1. Initial program 30.7%

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

   2. Taylor expanded in b around -inf 69.3%

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

      1. associate-*r/ 69.3%

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

      2. neg-mul-1 69.3%

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

   4. Simplified 69.3%

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

   if -5.50000000000000042e-261 < b

   1. Initial program 73.9%

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

   2. Taylor expanded in b around inf 63.6%

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

      1. associate-*r/ 63.6%

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

      2. mul-1-neg 63.6%

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

   4. Simplified 63.6%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-261}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 10: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

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

   1. clear-num 52.5%

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

   2. inv-pow 52.5%

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

3. Applied egg-rr 34.1%

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

   1. unpow-1 34.1%

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

   2. fma-udef 34.0%

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

   3. *-commutative 34.0%

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

   4. associate-*l* 34.4%

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

   5. *-commutative 34.4%

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

   6. fma-def 34.4%

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

5. Simplified 34.4%

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

6. Taylor expanded in b around -inf 2.6%

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

7. Final simplification 2.6%

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

Alternative 11: 10.8% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

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

2. Taylor expanded in b around inf 31.4%

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

   1. +-commutative 31.4%

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

   2. *-commutative 31.4%

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

   3. fma-def 31.4%

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

   4. *-commutative 31.4%

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

   5. associate-/l* 33.3%

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

4. Simplified 33.3%

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

5. Taylor expanded in b around 0 11.5%

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

6. Final simplification 11.5%

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

Developer target: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 a))))
     (/ (- (- b) t_0) (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * 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 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - (4.0 * (a * c))));
	double tmp;
	if (b < 0.0) {
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-b - t_0) / (2.0 * a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - (4.0 * (a * c))))
	tmp = 0
	if b < 0.0:
		tmp = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a))));
	else
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - (4.0 * (a * c))));
	tmp = 0.0;
	if (b < 0.0)
		tmp = c / (a * ((-b + t_0) / (2.0 * a)));
	else
		tmp = (-b - t_0) / (2.0 * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023230 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

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