quadp (p42, positive)

Percentage Accurate: 52.5% → 84.8%

Time: 13.3s

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: 52.5% 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: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(\frac{{b}^{2}}{c} - a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.8e+29)
   (/ b (- a))
   (if (<= b 1.65e-96)
     (/ (- (sqrt (* c (- (/ (pow b 2.0) c) (* a 4.0)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.8e+29) {
		tmp = b / -a;
	} else if (b <= 1.65e-96) {
		tmp = (sqrt((c * ((pow(b, 2.0) / 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 <= (-9.8d+29)) then
        tmp = b / -a
    else if (b <= 1.65d-96) then
        tmp = (sqrt((c * (((b ** 2.0d0) / 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 <= -9.8e+29) {
		tmp = b / -a;
	} else if (b <= 1.65e-96) {
		tmp = (Math.sqrt((c * ((Math.pow(b, 2.0) / c) - (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -9.8e+29:
		tmp = b / -a
	elif b <= 1.65e-96:
		tmp = (math.sqrt((c * ((math.pow(b, 2.0) / 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 <= -9.8e+29)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.65e-96)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(Float64((b ^ 2.0) / 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 <= -9.8e+29)
		tmp = b / -a;
	elseif (b <= 1.65e-96)
		tmp = (sqrt((c * (((b ^ 2.0) / 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, -9.8e+29], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.65e-96], N[(N[(N[Sqrt[N[(c * N[(N[(N[Power[b, 2.0], $MachinePrecision] / c), $MachinePrecision] - N[(a * 4.0), $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 < -9.8000000000000003e29

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in b around -inf 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 -9.8000000000000003e29 < b < 1.64999999999999995e-96

   1. Initial program 85.9%

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

      1. *-commutative 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in c around inf 85.9%

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

   if 1.64999999999999995e-96 < b

   1. Initial program 16.9%

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

      1. *-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in b around inf 82.5%

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

      1. associate-*r/ 82.5%

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

      2. neg-mul-1 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(\frac{{b}^{2}}{c} - a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.8e+31)
   (/ b (- a))
   (if (<= b 9.8e-98)
     (/ (- (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 <= -4.8e+31) {
		tmp = b / -a;
	} else if (b <= 9.8e-98) {
		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 <= (-4.8d+31)) then
        tmp = b / -a
    else if (b <= 9.8d-98) 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 <= -4.8e+31) {
		tmp = b / -a;
	} else if (b <= 9.8e-98) {
		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 <= -4.8e+31:
		tmp = b / -a
	elif b <= 9.8e-98:
		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 <= -4.8e+31)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 9.8e-98)
		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 <= -4.8e+31)
		tmp = b / -a;
	elseif (b <= 9.8e-98)
		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, -4.8e+31], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 9.8e-98], 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 < -4.79999999999999965e31

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in b around -inf 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 -4.79999999999999965e31 < b < 9.80000000000000028e-98

   1. Initial program 85.9%

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

   if 9.80000000000000028e-98 < b

   1. Initial program 16.9%

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

      1. *-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in b around inf 82.5%

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

      1. associate-*r/ 82.5%

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

      2. neg-mul-1 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 9.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.9e-97)
   (/ b (- a))
   (if (<= b 4.2e-95)
     (/ (- (sqrt (* c (* a -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.9e-97) {
		tmp = b / -a;
	} else if (b <= 4.2e-95) {
		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 <= (-1.9d-97)) then
        tmp = b / -a
    else if (b <= 4.2d-95) 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 <= -1.9e-97) {
		tmp = b / -a;
	} else if (b <= 4.2e-95) {
		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 <= -1.9e-97:
		tmp = b / -a
	elif b <= 4.2e-95:
		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 <= -1.9e-97)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 4.2e-95)
		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 <= -1.9e-97)
		tmp = b / -a;
	elseif (b <= 4.2e-95)
		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, -1.9e-97], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 4.2e-95], 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 < -1.9e-97

   1. Initial program 66.1%

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

      1. *-commutative 66.1%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in b around -inf 87.1%

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

      1. associate-*r/ 87.1%

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

      2. mul-1-neg 87.1%

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

   7. Simplified 87.1%

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

   if -1.9e-97 < b < 4.2e-95

   1. Initial program 81.7%

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

      1. *-commutative 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in b around 0 74.1%

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

      1. associate-*r* 74.1%

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

      2. *-commutative 74.1%

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

      3. *-commutative 74.1%

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

   7. Simplified 74.1%

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

   if 4.2e-95 < b

   1. Initial program 16.9%

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

      1. *-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in b around inf 82.5%

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

      1. associate-*r/ 82.5%

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

      2. neg-mul-1 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-156}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-178}:\\ \;\;\;\;-0.5 \cdot \left(-\sqrt{c \cdot \frac{-4}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.25e-156)
   (/ b (- a))
   (if (<= b 4e-178) (* -0.5 (- (sqrt (* c (/ -4.0 a))))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.25e-156) {
		tmp = b / -a;
	} else if (b <= 4e-178) {
		tmp = -0.5 * -sqrt((c * (-4.0 / a)));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.25d-156)) then
        tmp = b / -a
    else if (b <= 4d-178) then
        tmp = (-0.5d0) * -sqrt((c * ((-4.0d0) / a)))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.25e-156) {
		tmp = b / -a;
	} else if (b <= 4e-178) {
		tmp = -0.5 * -Math.sqrt((c * (-4.0 / a)));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.25e-156:
		tmp = b / -a
	elif b <= 4e-178:
		tmp = -0.5 * -math.sqrt((c * (-4.0 / a)))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.25e-156)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 4e-178)
		tmp = Float64(-0.5 * Float64(-sqrt(Float64(c * Float64(-4.0 / a)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.25e-156)
		tmp = b / -a;
	elseif (b <= 4e-178)
		tmp = -0.5 * -sqrt((c * (-4.0 / a)));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.25e-156], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 4e-178], N[(-0.5 * (-N[Sqrt[N[(c * N[(-4.0 / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.24999999999999993e-156

   1. Initial program 69.1%

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

      1. *-commutative 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in b around -inf 82.3%

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

      1. associate-*r/ 82.3%

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

      2. mul-1-neg 82.3%

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

   7. Simplified 82.3%

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

   if -2.24999999999999993e-156 < b < 3.9999999999999998e-178

   1. Initial program 76.5%

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

      1. *-commutative 76.5%

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

   3. Simplified 76.5%

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

      1. add-cube-cbrt 75.8%

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

      2. pow3 75.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{4 \cdot \left(a \cdot c\right)}\right)}^{3}}}}{a \cdot 2} \]

      3. *-commutative 75.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{\left(a \cdot c\right) \cdot 4}}\right)}^{3}}}{a \cdot 2} \]

      4. associate-*l* 75.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - {\left(\sqrt[3]{\color{blue}{a \cdot \left(c \cdot 4\right)}}\right)}^{3}}}{a \cdot 2} \]

   6. Applied egg-rr 75.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left(\sqrt[3]{a \cdot \left(c \cdot 4\right)}\right)}^{3}}}}{a \cdot 2} \]

   7. Taylor expanded in a around -inf 0.0%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 0.0%

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

      2. associate-/l* 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 46.3%

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

   9. Simplified 46.3%

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

   if 3.9999999999999998e-178 < b

   1. Initial program 25.9%

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

      1. *-commutative 25.9%

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

   3. Simplified 25.9%

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

   5. Taylor expanded in b around inf 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

   7. Simplified 74.3%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.25 \cdot 10^{-156}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-178}:\\ \;\;\;\;-0.5 \cdot \left(-\sqrt{c \cdot \frac{-4}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		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 <= (-1d-310)) 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 <= -1e-310) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		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 <= -1e-310)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 70.4%

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

      1. *-commutative 70.4%

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

   3. Simplified 70.4%

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

   5. Taylor expanded in b around -inf 72.9%

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

      1. associate-*r/ 72.9%

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

      2. mul-1-neg 72.9%

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

   7. Simplified 72.9%

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

   if -9.999999999999969e-311 < b

   1. Initial program 33.7%

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

      1. *-commutative 33.7%

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

   3. Simplified 33.7%

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

   5. Taylor expanded in b around inf 62.9%

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

      1. associate-*r/ 62.9%

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

      2. neg-mul-1 62.9%

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

   7. Simplified 62.9%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 4.4e+86) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e+86) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 4.4d+86) then
        tmp = b / -a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4e+86) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.4e+86:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.4e+86)
		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 <= 4.4e+86)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.4e+86], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.40000000000000006e86

   1. Initial program 65.2%

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

      1. *-commutative 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in b around -inf 50.3%

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

      1. associate-*r/ 50.3%

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

      2. mul-1-neg 50.3%

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

   7. Simplified 50.3%

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

   if 4.40000000000000006e86 < b

   1. Initial program 6.1%

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

      1. *-commutative 6.1%

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

   3. Simplified 6.1%

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

      1. add-cube-cbrt 6.1%

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

      2. pow3 6.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \color{blue}{{\left(\sqrt[3]{a \cdot c}\right)}^{3}}}}{a \cdot 2} \]

   6. Applied egg-rr 6.1%

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

      1. div-inv 6.1%

         \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}}\right) \cdot \frac{1}{a \cdot 2}} \]

      2. add-sqr-sqrt 0.0%

         \[\leadsto \left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{b \cdot b - 4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}}\right) \cdot \frac{1}{a \cdot 2} \]

      3. sqrt-unprod 1.9%

         \[\leadsto \left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{b \cdot b - 4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}}\right) \cdot \frac{1}{a \cdot 2} \]

      4. sqr-neg 1.9%

         \[\leadsto \left(\sqrt{\color{blue}{b \cdot b}} + \sqrt{b \cdot b - 4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}}\right) \cdot \frac{1}{a \cdot 2} \]

      5. sqrt-prod 1.9%

         \[\leadsto \left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{b \cdot b - 4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}}\right) \cdot \frac{1}{a \cdot 2} \]

      6. add-sqr-sqrt 1.9%

         \[\leadsto \left(\color{blue}{b} + \sqrt{b \cdot b - 4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}}\right) \cdot \frac{1}{a \cdot 2} \]

      7. rem-cube-cbrt 1.9%

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

      8. cancel-sign-sub-inv 1.9%

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

      9. fma-define 1.9%

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

      10. metadata-eval 1.9%

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

   8. Applied egg-rr 1.9%

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

   9. Taylor expanded in b around -inf 31.3%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 10.7% 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 53.6%

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

   1. *-commutative 53.6%

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

3. Simplified 53.6%

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

   1. add-cube-cbrt 53.5%

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

   2. pow3 53.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \color{blue}{{\left(\sqrt[3]{a \cdot c}\right)}^{3}}}}{a \cdot 2} \]

6. Applied egg-rr 53.5%

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

   1. div-inv 53.4%

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}}\right) \cdot \frac{1}{a \cdot 2}} \]

   2. add-sqr-sqrt 38.0%

      \[\leadsto \left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{b \cdot b - 4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}}\right) \cdot \frac{1}{a \cdot 2} \]

   3. sqrt-unprod 51.4%

      \[\leadsto \left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{b \cdot b - 4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}}\right) \cdot \frac{1}{a \cdot 2} \]

   4. sqr-neg 51.4%

      \[\leadsto \left(\sqrt{\color{blue}{b \cdot b}} + \sqrt{b \cdot b - 4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}}\right) \cdot \frac{1}{a \cdot 2} \]

   5. sqrt-prod 13.4%

      \[\leadsto \left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{b \cdot b - 4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}}\right) \cdot \frac{1}{a \cdot 2} \]

   6. add-sqr-sqrt 32.6%

      \[\leadsto \left(\color{blue}{b} + \sqrt{b \cdot b - 4 \cdot {\left(\sqrt[3]{a \cdot c}\right)}^{3}}\right) \cdot \frac{1}{a \cdot 2} \]

   7. rem-cube-cbrt 32.8%

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

   8. cancel-sign-sub-inv 32.8%

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

   9. fma-define 32.8%

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

   10. metadata-eval 32.8%

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

8. Applied egg-rr 32.8%

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

9. Taylor expanded in b around -inf 8.4%

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

Alternative 8: 2.6% 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 53.6%

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

   1. *-commutative 53.6%

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

3. Simplified 53.6%

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

5. Taylor expanded in b around -inf 40.9%

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

   1. associate-*r/ 40.9%

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

   2. mul-1-neg 40.9%

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

7. Simplified 40.9%

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

   1. *-un-lft-identity 40.9%

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

   2. add-sqr-sqrt 39.4%

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

   3. sqrt-unprod 29.1%

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

   4. sqr-neg 29.1%

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

   5. sqrt-prod 1.6%

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

   6. add-sqr-sqrt 2.3%

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

9. Applied egg-rr 2.3%

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

    1. *-lft-identity 2.3%

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

11. Simplified 2.3%

    \[\leadsto \color{blue}{\frac{b}{a}} \]
12. 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 2024152 
(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)))