Quadratic roots, full range

Percentage Accurate: 51.4% → 84.7%

Time: 12.1s

Alternatives: 8

Speedup: 12.9×

Specification

Math

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

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

Initial Program: 51.4% accurate, 1.0× speedup

Math

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

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

Alternative 1: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.5e+31)
   (/ b (- a))
   (if (<= b 2.2e-50)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.5e+31) {
		tmp = b / -a;
	} else if (b <= 2.2e-50) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * 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 <= (-6.5d+31)) then
        tmp = b / -a
    else if (b <= 2.2d-50) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * 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 <= -6.5e+31) {
		tmp = b / -a;
	} else if (b <= 2.2e-50) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.5e+31:
		tmp = b / -a
	elif b <= 2.2e-50:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.5e+31)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 2.2e-50)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * 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 <= -6.5e+31)
		tmp = b / -a;
	elseif (b <= 2.2e-50)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.5e+31], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 2.2e-50], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $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 < -6.5000000000000004e31

   1. Initial program 78.7%

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

      1. *-commutative 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in b around -inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. mul-1-neg 98.7%

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

   7. Simplified 98.7%

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

   if -6.5000000000000004e31 < b < 2.1999999999999999e-50

   1. Initial program 74.2%

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

   if 2.1999999999999999e-50 < b

   1. Initial program 15.8%

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

      1. *-commutative 15.8%

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

   3. Simplified 15.8%

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

   5. Taylor expanded in b around inf 84.4%

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

      1. associate-*r/ 84.4%

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

      2. mul-1-neg 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-64}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.05e-64)
   (/ b (- a))
   (if (<= b 1.2e-49)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-64) {
		tmp = b / -a;
	} else if (b <= 1.2e-49) {
		tmp = (sqrt((a * (c * -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 <= (-2.05d-64)) then
        tmp = b / -a
    else if (b <= 1.2d-49) then
        tmp = (sqrt((a * (c * (-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 <= -2.05e-64) {
		tmp = b / -a;
	} else if (b <= 1.2e-49) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.05e-64:
		tmp = b / -a
	elif b <= 1.2e-49:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.05e-64)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 1.2e-49)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -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 <= -2.05e-64)
		tmp = b / -a;
	elseif (b <= 1.2e-49)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.05e-64], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 1.2e-49], N[(N[(N[Sqrt[N[(a * N[(c * -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 < -2.05e-64

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in b around -inf 87.0%

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

      1. associate-*r/ 87.0%

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

      2. mul-1-neg 87.0%

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

   7. Simplified 87.0%

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

   if -2.05e-64 < b < 1.19999999999999996e-49

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in b around 0 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-*r* 68.0%

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

   7. Simplified 68.0%

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

   if 1.19999999999999996e-49 < b

   1. Initial program 15.8%

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

      1. *-commutative 15.8%

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

   3. Simplified 15.8%

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

   5. Taylor expanded in b around inf 84.4%

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

      1. associate-*r/ 84.4%

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

      2. mul-1-neg 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-64}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-139}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.05e-139)
   (/ b (- a))
   (if (<= b 2.15e-51)
     (/ (+ b (sqrt (* a (* c -4.0)))) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.05e-139) {
		tmp = b / -a;
	} else if (b <= 2.15e-51) {
		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 <= (-1.05d-139)) then
        tmp = b / -a
    else if (b <= 2.15d-51) 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 <= -1.05e-139) {
		tmp = b / -a;
	} else if (b <= 2.15e-51) {
		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 <= -1.05e-139:
		tmp = b / -a
	elif b <= 2.15e-51:
		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 <= -1.05e-139)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 2.15e-51)
		tmp = Float64(Float64(b + sqrt(Float64(a * Float64(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 <= -1.05e-139)
		tmp = b / -a;
	elseif (b <= 2.15e-51)
		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, -1.05e-139], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 2.15e-51], N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05000000000000004e-139

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in b around -inf 84.3%

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

      1. associate-*r/ 84.3%

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

      2. mul-1-neg 84.3%

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

   7. Simplified 84.3%

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

   if -1.05000000000000004e-139 < b < 2.1499999999999999e-51

   1. Initial program 69.5%

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

      1. *-commutative 69.5%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in b around 0 69.4%

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

      1. *-commutative 69.4%

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

      2. associate-*r* 69.4%

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

   7. Simplified 69.4%

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

      1. +-commutative 69.4%

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

      2. *-un-lft-identity 69.4%

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

      3. fma-define 69.4%

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

      4. add-sqr-sqrt 35.6%

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

      5. sqrt-unprod 68.7%

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

      6. sqr-neg 68.7%

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

      7. sqrt-prod 33.9%

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

      8. add-sqr-sqrt 68.7%

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

   9. Applied egg-rr 68.7%

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

       1. fma-undefine 68.7%

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

       2. *-lft-identity 68.7%

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

   11. Simplified 68.7%

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

   if 2.1499999999999999e-51 < b

   1. Initial program 15.8%

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

      1. *-commutative 15.8%

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

   3. Simplified 15.8%

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

   5. Taylor expanded in b around inf 84.4%

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

      1. associate-*r/ 84.4%

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

      2. mul-1-neg 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-139}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{b + \sqrt{a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-139}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-175}:\\ \;\;\;\;-0.5 \cdot \left(-\sqrt{\frac{c \cdot -4}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.05e-139)
   (/ b (- a))
   (if (<= b 5.5e-175) (* -0.5 (- (sqrt (/ (* c -4.0) a)))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.05e-139) {
		tmp = b / -a;
	} else if (b <= 5.5e-175) {
		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 <= (-1.05d-139)) then
        tmp = b / -a
    else if (b <= 5.5d-175) 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 <= -1.05e-139) {
		tmp = b / -a;
	} else if (b <= 5.5e-175) {
		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 <= -1.05e-139:
		tmp = b / -a
	elif b <= 5.5e-175:
		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 <= -1.05e-139)
		tmp = Float64(b / Float64(-a));
	elseif (b <= 5.5e-175)
		tmp = Float64(-0.5 * Float64(-sqrt(Float64(Float64(c * -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 <= -1.05e-139)
		tmp = b / -a;
	elseif (b <= 5.5e-175)
		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, -1.05e-139], N[(b / (-a)), $MachinePrecision], If[LessEqual[b, 5.5e-175], N[(-0.5 * (-N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] / a), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.05000000000000004e-139

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in b around -inf 84.3%

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

      1. associate-*r/ 84.3%

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

      2. mul-1-neg 84.3%

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

   7. Simplified 84.3%

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

   if -1.05000000000000004e-139 < b < 5.50000000000000054e-175

   1. Initial program 77.3%

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

      1. *-commutative 77.3%

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

   3. Simplified 77.3%

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

      1. add-cube-cbrt 76.7%

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

      2. pow3 76.7%

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

   6. Applied egg-rr 76.7%

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

   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. *-commutative 0.0%

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

      2. unpow2 0.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      3. rem-square-sqrt 42.0%

         \[\leadsto -0.5 \cdot \left(\color{blue}{-1} \cdot \sqrt{\frac{c \cdot {\left(\sqrt[3]{-4}\right)}^{3}}{a}}\right) \]

      4. rem-cube-cbrt 42.4%

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

   9. Simplified 42.4%

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

   if 5.50000000000000054e-175 < b

   1. Initial program 22.4%

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

      1. *-commutative 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in b around inf 75.9%

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

      1. associate-*r/ 75.9%

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

      2. mul-1-neg 75.9%

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

   7. Simplified 75.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-139}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-175}:\\ \;\;\;\;-0.5 \cdot \left(-\sqrt{\frac{c \cdot -4}{a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in b around -inf 67.6%

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

      1. associate-*r/ 67.6%

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

      2. mul-1-neg 67.6%

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

   7. Simplified 67.6%

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

   if -1.999999999999994e-310 < b

   1. Initial program 29.2%

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

      1. *-commutative 29.2%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in b around inf 67.5%

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

      1. associate-*r/ 67.5%

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

      2. mul-1-neg 67.5%

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

   7. Simplified 67.5%

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

4. Final simplification 67.6%

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

Alternative 6: 42.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 6.2e+72) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.2e+72) {
		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 <= 6.2d+72) 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 <= 6.2e+72) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.2e+72:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.2e+72)
		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 <= 6.2e+72)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.2e+72], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.19999999999999977e72

   1. Initial program 71.6%

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

      1. *-commutative 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in b around -inf 47.9%

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

      1. associate-*r/ 47.9%

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

      2. mul-1-neg 47.9%

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

   7. Simplified 47.9%

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

   if 6.19999999999999977e72 < b

   1. Initial program 8.6%

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

      1. *-commutative 8.6%

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

   3. Simplified 8.6%

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

   5. Taylor expanded in b around inf 77.5%

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

      1. associate-*r/ 77.6%

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

      2. associate-*r* 77.6%

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

      3. *-commutative 77.6%

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

      4. associate-*l/ 69.7%

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

   7. Simplified 69.7%

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

      1. associate-/l* 71.1%

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

   9. Applied egg-rr 71.1%

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

       1. clear-num 71.0%

          \[\leadsto \frac{a \cdot -2}{b} \cdot \color{blue}{\frac{1}{\frac{a \cdot 2}{c}}} \]

       2. un-div-inv 71.0%

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

       3. associate-/l* 71.1%

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

       4. frac-2neg 71.1%

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

       5. metadata-eval 71.1%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 25.1%

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

       8. sqr-neg 25.1%

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

       9. sqrt-prod 24.5%

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

       10. add-sqr-sqrt 24.5%

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

       11. *-commutative 24.5%

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

       12. *-un-lft-identity 24.5%

           \[\leadsto \frac{a \cdot \frac{2}{b}}{\frac{2 \cdot a}{\color{blue}{1 \cdot c}}} \]

       13. times-frac 24.5%

           \[\leadsto \frac{a \cdot \frac{2}{b}}{\color{blue}{\frac{2}{1} \cdot \frac{a}{c}}} \]

       14. metadata-eval 24.5%

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

   11. Applied egg-rr 24.5%

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

       1. associate-*r/ 24.5%

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

       2. metadata-eval 24.5%

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

       3. distribute-rgt-neg-in 24.5%

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

       4. associate-/r* 24.8%

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

       5. distribute-rgt-neg-in 24.8%

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

       6. metadata-eval 24.8%

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

   13. Simplified 24.8%

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

   14. Taylor expanded in a around 0 24.6%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
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 55.4%

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

   1. *-commutative 55.4%

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

3. Simplified 55.4%

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

5. Taylor expanded in b around inf 27.0%

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

   1. associate-*r/ 27.0%

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

   2. associate-*r* 27.0%

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

   3. *-commutative 27.0%

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

   4. associate-*l/ 26.8%

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

7. Simplified 26.8%

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

   1. associate-/l* 26.4%

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

9. Applied egg-rr 26.4%

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

    1. clear-num 26.3%

       \[\leadsto \frac{a \cdot -2}{b} \cdot \color{blue}{\frac{1}{\frac{a \cdot 2}{c}}} \]

    2. un-div-inv 26.3%

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

    3. associate-/l* 26.3%

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

    4. frac-2neg 26.3%

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

    5. metadata-eval 26.3%

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

    6. add-sqr-sqrt 1.0%

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

    7. sqrt-unprod 7.8%

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

    8. sqr-neg 7.8%

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

    9. sqrt-prod 6.6%

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

    10. add-sqr-sqrt 8.2%

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

    11. *-commutative 8.2%

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

    12. *-un-lft-identity 8.2%

        \[\leadsto \frac{a \cdot \frac{2}{b}}{\frac{2 \cdot a}{\color{blue}{1 \cdot c}}} \]

    13. times-frac 8.2%

        \[\leadsto \frac{a \cdot \frac{2}{b}}{\color{blue}{\frac{2}{1} \cdot \frac{a}{c}}} \]

    14. metadata-eval 8.2%

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

11. Applied egg-rr 8.2%

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

    1. associate-*r/ 8.2%

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

    2. metadata-eval 8.2%

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

    3. distribute-rgt-neg-in 8.2%

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

    4. associate-/r* 10.9%

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

    5. distribute-rgt-neg-in 10.9%

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

    6. metadata-eval 10.9%

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

13. Simplified 10.9%

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

14. Taylor expanded in a around 0 8.5%

    \[\leadsto \color{blue}{\frac{c}{b}} \]
15. 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 55.4%

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

   1. *-commutative 55.4%

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

3. Simplified 55.4%

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

5. Taylor expanded in b around -inf 36.2%

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

   1. associate-*r/ 36.2%

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

   2. mul-1-neg 36.2%

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

7. Simplified 36.2%

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

   1. div-inv 36.1%

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

   2. add-sqr-sqrt 34.7%

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

   3. sqrt-unprod 30.0%

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

   4. sqr-neg 30.0%

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

   5. sqrt-prod 1.8%

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

   6. add-sqr-sqrt 2.6%

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

9. Applied egg-rr 2.6%

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

    1. associate-*r/ 2.6%

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

    2. *-rgt-identity 2.6%

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

11. Simplified 2.6%

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

Reproduce

herbie shell --seed 2024119 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))