quadp (p42, positive)

Percentage Accurate: 52.6% → 85.8%

Time: 12.8s

Alternatives: 8

Speedup: 12.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+105)
   (/ b (- a))
   (if (<= b 6.5e-81)
     (/ (- (sqrt (- (* b b) (* 4.0 (* a c)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+105) {
		tmp = b / -a;
	} else if (b <= 6.5e-81) {
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d+105)) then
        tmp = b / -a
    else if (b <= 6.5d-81) then
        tmp = (sqrt(((b * b) - (4.0d0 * (a * c)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+105) {
		tmp = b / -a;
	} else if (b <= 6.5e-81) {
		tmp = (Math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+105:
		tmp = b / -a
	elif b <= 6.5e-81:
		tmp = (math.sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+105)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 6.5e-81)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e+105)
		tmp = b / -a;
	elseif (b <= 6.5e-81)
		tmp = (sqrt(((b * b) - (4.0 * (a * c)))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+105], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 6.5e-81], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.9999999999999994e104

   1. Initial program 59.5%

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

      1. *-commutative 59.5%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in b around -inf 93.0%

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

      1. associate-*r/ 93.0%

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

      2. mul-1-neg 93.0%

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

   7. Simplified 93.0%

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

   if -9.9999999999999994e104 < b < 6.5000000000000002e-81

   1. Initial program 79.7%

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

   if 6.5000000000000002e-81 < b

   1. Initial program 19.6%

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

      1. *-commutative 19.6%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in b around inf 81.5%

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

      1. associate-*r/ 81.5%

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

      2. neg-mul-1 81.5%

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

   7. Simplified 81.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+105}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.6e-7)
   (/ b (- a))
   (if (<= b 3.7e-84)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-7) {
		tmp = b / -a;
	} else if (b <= 3.7e-84) {
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.6d-7)) then
        tmp = b / -a
    else if (b <= 3.7d-84) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-7) {
		tmp = b / -a;
	} else if (b <= 3.7e-84) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.6e-7:
		tmp = b / -a
	elif b <= 3.7e-84:
		tmp = (math.sqrt((c * (a * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.6e-7)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 3.7e-84)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.6e-7)
		tmp = b / -a;
	elseif (b <= 3.7e-84)
		tmp = (sqrt((c * (a * -4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.6e-7], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 3.7e-84], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.59999999999999999e-7

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in b around -inf 89.1%

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

      1. associate-*r/ 89.1%

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

      2. mul-1-neg 89.1%

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

   7. Simplified 89.1%

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

   if -2.59999999999999999e-7 < b < 3.6999999999999999e-84

   1. Initial program 74.2%

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

      1. *-commutative 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in b around 0 64.9%

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

      1. associate-*r* 64.9%

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

      2. *-commutative 64.9%

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

      3. *-commutative 64.9%

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

   7. Simplified 64.9%

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

   if 3.6999999999999999e-84 < b

   1. Initial program 19.6%

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

      1. *-commutative 19.6%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in b around inf 81.5%

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

      1. associate-*r/ 81.5%

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

      2. neg-mul-1 81.5%

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

   7. Simplified 81.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3.15 \cdot 10^{-80}:\\ \;\;\;\;\frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e-7)
   (/ b (- a))
   (if (<= b 3.15e-80)
     (/ (+ b (sqrt (* (* a c) -4.0))) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e-7) {
		tmp = b / -a;
	} else if (b <= 3.15e-80) {
		tmp = (b + sqrt(((a * c) * -4.0))) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.2d-7)) then
        tmp = b / -a
    else if (b <= 3.15d-80) then
        tmp = (b + sqrt(((a * c) * (-4.0d0)))) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e-7) {
		tmp = b / -a;
	} else if (b <= 3.15e-80) {
		tmp = (b + Math.sqrt(((a * c) * -4.0))) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.2e-7:
		tmp = b / -a
	elif b <= 3.15e-80:
		tmp = (b + math.sqrt(((a * c) * -4.0))) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e-7)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 3.15e-80)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(a * c) * -4.0))) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.2e-7)
		tmp = b / -a;
	elseif (b <= 3.15e-80)
		tmp = (b + sqrt(((a * c) * -4.0))) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e-7], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 3.15e-80], N[(N[(b + N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.2000000000000001e-7

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in b around -inf 89.1%

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

      1. associate-*r/ 89.1%

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

      2. mul-1-neg 89.1%

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

   7. Simplified 89.1%

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

   if -2.2000000000000001e-7 < b < 3.14999999999999983e-80

   1. Initial program 74.2%

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

      1. *-commutative 74.2%

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

   3. Simplified 74.2%

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

      1. clear-num 74.1%

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

      2. associate-/r/ 74.0%

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

      3. *-commutative 74.0%

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

      4. associate-/r* 74.0%

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

      5. metadata-eval 74.0%

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

      6. add-sqr-sqrt 45.5%

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

      7. sqrt-unprod 73.5%

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

      8. sqr-neg 73.5%

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

      9. sqrt-prod 28.4%

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

      10. add-sqr-sqrt 63.4%

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

      11. sub-neg 63.4%

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

      12. +-commutative 63.4%

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

      13. *-commutative 63.4%

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

      14. distribute-rgt-neg-in 63.4%

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

      15. fma-define 63.4%

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

      16. metadata-eval 63.4%

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

      17. pow2 63.4%

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

   6. Applied egg-rr 63.4%

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

   7. Taylor expanded in a around inf 63.1%

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

      1. associate-*r* 63.1%

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

      2. *-commutative 63.1%

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

   9. Simplified 63.1%

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

       1. associate-*l/ 63.3%

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

       2. clear-num 63.2%

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

       3. +-commutative 63.2%

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

       4. sqrt-prod 39.8%

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

       5. fma-define 39.8%

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

       6. *-commutative 39.8%

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

   11. Applied egg-rr 39.8%

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

       1. associate-/r* 39.8%

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

       2. associate-/r/ 39.8%

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

       3. div-inv 39.8%

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

       4. metadata-eval 39.8%

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

       5. fma-undefine 39.8%

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

       6. sqrt-unprod 63.1%

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

       7. associate-*r* 63.1%

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

       8. *-commutative 63.1%

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

       9. +-commutative 63.1%

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

       10. associate-*r* 63.1%

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

       11. *-commutative 63.1%

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

       12. sqrt-prod 43.0%

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

       13. *-commutative 43.0%

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

       14. sqrt-unprod 63.1%

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

       15. *-commutative 63.1%

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

   13. Applied egg-rr 63.1%

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

       1. associate-*l/ 63.3%

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

       2. *-lft-identity 63.3%

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

       3. associate-*r* 63.3%

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

       4. *-commutative 63.3%

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

       5. associate-*r* 63.3%

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

       6. *-commutative 63.3%

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

       7. *-commutative 63.3%

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

   15. Simplified 63.3%

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

   if 3.14999999999999983e-80 < b

   1. Initial program 19.6%

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

      1. *-commutative 19.6%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in b around inf 81.5%

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

      1. associate-*r/ 81.5%

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

      2. neg-mul-1 81.5%

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

   7. Simplified 81.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 3.15 \cdot 10^{-80}:\\ \;\;\;\;\frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-84}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.1e-7)
   (/ b (- a))
   (if (<= b 7e-84) (* (/ 0.5 a) (+ b (sqrt (* c (* a -4.0))))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-7) {
		tmp = b / -a;
	} else if (b <= 7e-84) {
		tmp = (0.5 / a) * (b + sqrt((c * (a * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.1d-7)) then
        tmp = b / -a
    else if (b <= 7d-84) then
        tmp = (0.5d0 / a) * (b + sqrt((c * (a * (-4.0d0)))))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-7) {
		tmp = b / -a;
	} else if (b <= 7e-84) {
		tmp = (0.5 / a) * (b + Math.sqrt((c * (a * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.1e-7:
		tmp = b / -a
	elif b <= 7e-84:
		tmp = (0.5 / a) * (b + math.sqrt((c * (a * -4.0))))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.1e-7)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 7e-84)
		tmp = Float64(Float64(0.5 / a) * Float64(b + sqrt(Float64(c * Float64(a * -4.0)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.1e-7)
		tmp = b / -a;
	elseif (b <= 7e-84)
		tmp = (0.5 / a) * (b + sqrt((c * (a * -4.0))));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.1e-7], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 7e-84], N[(N[(0.5 / a), $MachinePrecision] * N[(b + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.1e-7

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in b around -inf 89.1%

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

      1. associate-*r/ 89.1%

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

      2. mul-1-neg 89.1%

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

   7. Simplified 89.1%

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

   if -2.1e-7 < b < 7.0000000000000002e-84

   1. Initial program 74.2%

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

      1. *-commutative 74.2%

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

   3. Simplified 74.2%

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

      1. clear-num 74.1%

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

      2. associate-/r/ 74.0%

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

      3. *-commutative 74.0%

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

      4. associate-/r* 74.0%

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

      5. metadata-eval 74.0%

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

      6. add-sqr-sqrt 45.5%

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

      7. sqrt-unprod 73.5%

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

      8. sqr-neg 73.5%

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

      9. sqrt-prod 28.4%

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

      10. add-sqr-sqrt 63.4%

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

      11. sub-neg 63.4%

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

      12. +-commutative 63.4%

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

      13. *-commutative 63.4%

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

      14. distribute-rgt-neg-in 63.4%

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

      15. fma-define 63.4%

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

      16. metadata-eval 63.4%

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

      17. pow2 63.4%

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

   6. Applied egg-rr 63.4%

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

   7. Taylor expanded in a around inf 63.1%

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

      1. associate-*r* 63.1%

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

      2. *-commutative 63.1%

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

   9. Simplified 63.1%

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

   if 7.0000000000000002e-84 < b

   1. Initial program 19.6%

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

      1. *-commutative 19.6%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in b around inf 81.5%

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

      1. associate-*r/ 81.5%

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

      2. neg-mul-1 81.5%

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

   7. Simplified 81.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-84}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.9% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-264}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 8e-264) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-264) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 8d-264) then
        tmp = b / -a
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e-264) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8e-264:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e-264)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8e-264)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e-264], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.0000000000000001e-264

   1. Initial program 74.3%

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

      1. *-commutative 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in b around -inf 60.5%

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

      1. associate-*r/ 60.5%

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

      2. mul-1-neg 60.5%

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

   7. Simplified 60.5%

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

   if 8.0000000000000001e-264 < b

   1. Initial program 29.0%

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

      1. *-commutative 29.0%

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

   3. Simplified 29.0%

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

   5. Taylor expanded in b around inf 69.0%

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

      1. associate-*r/ 69.0%

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

      2. neg-mul-1 69.0%

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

   7. Simplified 69.0%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-264}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 43.9% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7300000000000:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7300000000000.0) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7300000000000.0) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 7300000000000.0d0) then
        tmp = b / -a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7300000000000.0) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7300000000000.0:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7300000000000.0)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7300000000000.0)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7300000000000.0], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.3e12

   1. Initial program 65.9%

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

      1. *-commutative 65.9%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in b around -inf 42.6%

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

      1. associate-*r/ 42.6%

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

      2. mul-1-neg 42.6%

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

   7. Simplified 42.6%

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

   if 7.3e12 < b

   1. Initial program 14.7%

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

      1. *-commutative 14.7%

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

   3. Simplified 14.7%

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

      1. clear-num 14.8%

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

      2. associate-/r/ 14.7%

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

      3. *-commutative 14.7%

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

      4. associate-/r* 14.7%

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

      5. metadata-eval 14.7%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 4.9%

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

      8. sqr-neg 4.9%

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

      9. sqrt-prod 4.9%

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

      10. add-sqr-sqrt 4.9%

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

      11. sub-neg 4.9%

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

      12. +-commutative 4.9%

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

      13. *-commutative 4.9%

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

      14. distribute-rgt-neg-in 4.9%

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

      15. fma-define 4.9%

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

      16. metadata-eval 4.9%

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

      17. pow2 4.9%

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

   6. Applied egg-rr 4.9%

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

   7. Taylor expanded in b around -inf 25.2%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7300000000000:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 10.9% 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 51.3%

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

   1. *-commutative 51.3%

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

3. Simplified 51.3%

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

   1. clear-num 51.3%

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

   2. associate-/r/ 51.2%

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

   3. *-commutative 51.2%

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

   4. associate-/r* 51.2%

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

   5. metadata-eval 51.2%

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

   6. add-sqr-sqrt 34.5%

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

   7. sqrt-unprod 47.7%

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

   8. sqr-neg 47.7%

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

   9. sqrt-prod 13.3%

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

   10. add-sqr-sqrt 32.5%

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

   11. sub-neg 32.5%

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

   12. +-commutative 32.5%

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

   13. *-commutative 32.5%

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

   14. distribute-rgt-neg-in 32.5%

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

   15. fma-define 32.5%

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

   16. metadata-eval 32.5%

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

   17. pow2 32.5%

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

6. Applied egg-rr 32.5%

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

7. Taylor expanded in b around -inf 9.4%

   \[\leadsto \color{blue}{\frac{c}{b}} \]
8. Add Preprocessing

Alternative 8: 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 51.3%

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

   1. *-commutative 51.3%

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

3. Simplified 51.3%

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

   1. clear-num 51.3%

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

   2. associate-/r/ 51.2%

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

   3. *-commutative 51.2%

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

   4. associate-/r* 51.2%

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

   5. metadata-eval 51.2%

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

   6. add-sqr-sqrt 34.5%

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

   7. sqrt-unprod 47.7%

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

   8. sqr-neg 47.7%

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

   9. sqrt-prod 13.3%

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

   10. add-sqr-sqrt 32.5%

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

   11. sub-neg 32.5%

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

   12. +-commutative 32.5%

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

   13. *-commutative 32.5%

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

   14. distribute-rgt-neg-in 32.5%

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

   15. fma-define 32.5%

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

   16. metadata-eval 32.5%

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

   17. pow2 32.5%

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

6. Applied egg-rr 32.5%

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

7. Taylor expanded in a around 0 2.8%

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

Developer Target 1: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024157 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

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