Quadratic roots, full range

Percentage Accurate: 51.4% → 85.1%

Time: 10.3s

Alternatives: 9

Speedup: 19.1×

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: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(b + \left(b - 2 \cdot \frac{c}{\frac{b}{a}}\right)\right) \cdot -0.5}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+153)
   (/ (* (+ b (- b (* 2.0 (/ c (/ b a))))) -0.5) a)
   (if (<= b 2.8e-65)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))
     (- (/ (- c) b) (/ c (/ (/ (pow b 3.0) a) c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+153) {
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a;
	} else if (b <= 2.8e-65) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (-c / b) - (c / ((pow(b, 3.0) / a) / c));
	}
	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 <= (-5d+153)) then
        tmp = ((b + (b - (2.0d0 * (c / (b / a))))) * (-0.5d0)) / a
    else if (b <= 2.8d-65) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (2.0d0 * a)
    else
        tmp = (-c / b) - (c / (((b ** 3.0d0) / a) / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+153) {
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a;
	} else if (b <= 2.8e-65) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	} else {
		tmp = (-c / b) - (c / ((Math.pow(b, 3.0) / a) / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+153:
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a
	elif b <= 2.8e-65:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)
	else:
		tmp = (-c / b) - (c / ((math.pow(b, 3.0) / a) / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+153)
		tmp = Float64(Float64(Float64(b + Float64(b - Float64(2.0 * Float64(c / Float64(b / a))))) * -0.5) / a);
	elseif (b <= 2.8e-65)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(c / Float64(Float64((b ^ 3.0) / a) / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+153)
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a;
	elseif (b <= 2.8e-65)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a);
	else
		tmp = (-c / b) - (c / (((b ^ 3.0) / a) / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+153], N[(N[(N[(b + N[(b - N[(2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.8e-65], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(c / N[(N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.00000000000000018e153

   1. Initial program 26.4%

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

      1. neg-sub0 26.4%

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

      2. associate-+l- 26.4%

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

      3. sub0-neg 26.4%

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

      4. neg-mul-1 26.4%

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

      5. associate-*l/ 26.4%

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

      6. *-commutative 26.4%

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

      7. associate-/r* 26.4%

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

      8. /-rgt-identity 26.4%

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

      9. metadata-eval 26.4%

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

   3. Simplified 26.7%

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

   4. Taylor expanded in b around -inf 80.2%

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

      1. neg-mul-1 80.2%

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

      2. unsub-neg 80.2%

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

      3. *-commutative 80.2%

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

      4. associate-/l* 97.4%

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

      5. associate-/r/ 97.4%

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

   6. Simplified 97.4%

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

      1. associate-*r/ 97.7%

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

      2. associate--r- 97.7%

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

      3. *-commutative 97.7%

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

      4. associate-*l/ 80.4%

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

      5. associate-/l* 97.7%

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

   8. Applied egg-rr 97.7%

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

   if -5.00000000000000018e153 < b < 2.8e-65

   1. Initial program 83.0%

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

   if 2.8e-65 < b

   1. Initial program 13.8%

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

      1. neg-sub0 13.8%

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

      2. associate-+l- 13.8%

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

      3. sub0-neg 13.8%

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

      4. neg-mul-1 13.8%

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

      5. associate-*l/ 13.8%

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

      6. *-commutative 13.8%

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

      7. associate-/r* 13.8%

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

      8. /-rgt-identity 13.8%

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

      9. metadata-eval 13.8%

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

   3. Simplified 13.9%

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

   4. Taylor expanded in b around inf 77.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. +-commutative 77.5%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      2. mul-1-neg 77.5%

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

      3. unsub-neg 77.5%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      4. associate-*r/ 77.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      5. neg-mul-1 77.5%

         \[\leadsto \frac{\color{blue}{-c}}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      6. associate-/l* 80.7%

         \[\leadsto \frac{-c}{b} - \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}} \]

      7. unpow2 80.7%

         \[\leadsto \frac{-c}{b} - \frac{\color{blue}{c \cdot c}}{\frac{{b}^{3}}{a}} \]

      8. associate-/l* 93.4%

         \[\leadsto \frac{-c}{b} - \color{blue}{\frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}} \]

   6. Simplified 93.4%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(b + \left(b - 2 \cdot \frac{c}{\frac{b}{a}}\right)\right) \cdot -0.5}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}\\ \end{array} \]

Alternative 2: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+83}:\\ \;\;\;\;\frac{\left(b + \left(b - 2 \cdot \frac{c}{\frac{b}{a}}\right)\right) \cdot -0.5}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-67}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e+83)
   (/ (* (+ b (- b (* 2.0 (/ c (/ b a))))) -0.5) a)
   (if (<= b 6.6e-67)
     (* (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (/ 0.5 a))
     (- (/ (- c) b) (/ c (/ (/ (pow b 3.0) a) c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e+83) {
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a;
	} else if (b <= 6.6e-67) {
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	} else {
		tmp = (-c / b) - (c / ((pow(b, 3.0) / a) / c));
	}
	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 <= (-4.6d+83)) then
        tmp = ((b + (b - (2.0d0 * (c / (b / a))))) * (-0.5d0)) / a
    else if (b <= 6.6d-67) then
        tmp = (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b) * (0.5d0 / a)
    else
        tmp = (-c / b) - (c / (((b ** 3.0d0) / a) / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e+83) {
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a;
	} else if (b <= 6.6e-67) {
		tmp = (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	} else {
		tmp = (-c / b) - (c / ((Math.pow(b, 3.0) / a) / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.6e+83:
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a
	elif b <= 6.6e-67:
		tmp = (math.sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a)
	else:
		tmp = (-c / b) - (c / ((math.pow(b, 3.0) / a) / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.6e+83)
		tmp = Float64(Float64(Float64(b + Float64(b - Float64(2.0 * Float64(c / Float64(b / a))))) * -0.5) / a);
	elseif (b <= 6.6e-67)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(c / Float64(Float64((b ^ 3.0) / a) / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.6e+83)
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a;
	elseif (b <= 6.6e-67)
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	else
		tmp = (-c / b) - (c / (((b ^ 3.0) / a) / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.6e+83], N[(N[(N[(b + N[(b - N[(2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 6.6e-67], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(c / N[(N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.5999999999999999e83

   1. Initial program 45.5%

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

      1. neg-sub0 45.5%

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

      2. associate-+l- 45.5%

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

      3. sub0-neg 45.5%

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

      4. neg-mul-1 45.5%

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

      5. associate-*l/ 45.4%

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

      6. *-commutative 45.4%

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

      7. associate-/r* 45.4%

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

      8. /-rgt-identity 45.4%

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

      9. metadata-eval 45.4%

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

   3. Simplified 45.6%

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

   4. Taylor expanded in b around -inf 85.2%

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

      1. neg-mul-1 85.2%

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

      2. unsub-neg 85.2%

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

      3. *-commutative 85.2%

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

      4. associate-/l* 98.0%

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

      5. associate-/r/ 98.0%

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

   6. Simplified 98.0%

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

      1. associate-*r/ 98.3%

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

      2. associate--r- 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*l/ 85.5%

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

      5. associate-/l* 98.3%

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

   8. Applied egg-rr 98.3%

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

   if -4.5999999999999999e83 < b < 6.6000000000000003e-67

   1. Initial program 80.8%

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

      1. /-rgt-identity 80.8%

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

      2. metadata-eval 80.8%

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

      3. associate-/l* 80.8%

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

      4. associate-*r/ 80.6%

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

      5. +-commutative 80.6%

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

      6. unsub-neg 80.6%

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

      7. fma-neg 80.6%

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

      8. associate-*l* 80.6%

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

      9. *-commutative 80.6%

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

      10. distribute-rgt-neg-in 80.6%

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

      11. metadata-eval 80.6%

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

      12. associate-/r* 80.6%

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

      13. metadata-eval 80.6%

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

      14. metadata-eval 80.6%

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

   3. Simplified 80.6%

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

      1. fma-udef 80.6%

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

      2. associate-*l* 79.7%

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

   5. Applied egg-rr 79.7%

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

   if 6.6000000000000003e-67 < b

   1. Initial program 13.8%

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

      1. neg-sub0 13.8%

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

      2. associate-+l- 13.8%

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

      3. sub0-neg 13.8%

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

      4. neg-mul-1 13.8%

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

      5. associate-*l/ 13.8%

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

      6. *-commutative 13.8%

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

      7. associate-/r* 13.8%

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

      8. /-rgt-identity 13.8%

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

      9. metadata-eval 13.8%

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

   3. Simplified 13.9%

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

   4. Taylor expanded in b around inf 77.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. +-commutative 77.5%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      2. mul-1-neg 77.5%

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

      3. unsub-neg 77.5%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      4. associate-*r/ 77.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      5. neg-mul-1 77.5%

         \[\leadsto \frac{\color{blue}{-c}}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      6. associate-/l* 80.7%

         \[\leadsto \frac{-c}{b} - \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}} \]

      7. unpow2 80.7%

         \[\leadsto \frac{-c}{b} - \frac{\color{blue}{c \cdot c}}{\frac{{b}^{3}}{a}} \]

      8. associate-/l* 93.4%

         \[\leadsto \frac{-c}{b} - \color{blue}{\frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}} \]

   6. Simplified 93.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{+83}:\\ \;\;\;\;\frac{\left(b + \left(b - 2 \cdot \frac{c}{\frac{b}{a}}\right)\right) \cdot -0.5}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-67}:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}\\ \end{array} \]

Alternative 3: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(b + \left(b - 2 \cdot \frac{c}{\frac{b}{a}}\right)\right) \cdot -0.5}{a}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{-68}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.35e-57)
   (/ (* (+ b (- b (* 2.0 (/ c (/ b a))))) -0.5) a)
   (if (<= b 1.32e-68)
     (* (- b (sqrt (* c (* a -4.0)))) (/ -0.5 a))
     (- (/ (- c) b) (/ c (/ (/ (pow b 3.0) a) c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-57) {
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a;
	} else if (b <= 1.32e-68) {
		tmp = (b - sqrt((c * (a * -4.0)))) * (-0.5 / a);
	} else {
		tmp = (-c / b) - (c / ((pow(b, 3.0) / a) / c));
	}
	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 <= (-1.35d-57)) then
        tmp = ((b + (b - (2.0d0 * (c / (b / a))))) * (-0.5d0)) / a
    else if (b <= 1.32d-68) then
        tmp = (b - sqrt((c * (a * (-4.0d0))))) * ((-0.5d0) / a)
    else
        tmp = (-c / b) - (c / (((b ** 3.0d0) / a) / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.35e-57) {
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a;
	} else if (b <= 1.32e-68) {
		tmp = (b - Math.sqrt((c * (a * -4.0)))) * (-0.5 / a);
	} else {
		tmp = (-c / b) - (c / ((Math.pow(b, 3.0) / a) / c));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.35e-57:
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a
	elif b <= 1.32e-68:
		tmp = (b - math.sqrt((c * (a * -4.0)))) * (-0.5 / a)
	else:
		tmp = (-c / b) - (c / ((math.pow(b, 3.0) / a) / c))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.35e-57)
		tmp = Float64(Float64(Float64(b + Float64(b - Float64(2.0 * Float64(c / Float64(b / a))))) * -0.5) / a);
	elseif (b <= 1.32e-68)
		tmp = Float64(Float64(b - sqrt(Float64(c * Float64(a * -4.0)))) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(c / Float64(Float64((b ^ 3.0) / a) / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.35e-57)
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a;
	elseif (b <= 1.32e-68)
		tmp = (b - sqrt((c * (a * -4.0)))) * (-0.5 / a);
	else
		tmp = (-c / b) - (c / (((b ^ 3.0) / a) / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.35e-57], N[(N[(N[(b + N[(b - N[(2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.32e-68], N[(N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(c / N[(N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.3500000000000001e-57

   1. Initial program 64.2%

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

      1. neg-sub0 64.2%

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

      2. associate-+l- 64.2%

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

      3. sub0-neg 64.2%

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

      4. neg-mul-1 64.2%

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

      5. associate-*l/ 64.0%

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

      6. *-commutative 64.0%

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

      7. associate-/r* 64.0%

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

      8. /-rgt-identity 64.0%

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

      9. metadata-eval 64.0%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in b around -inf 83.6%

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

      1. neg-mul-1 83.6%

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

      2. unsub-neg 83.6%

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

      3. *-commutative 83.6%

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

      4. associate-/l* 91.7%

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

      5. associate-/r/ 91.7%

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

   6. Simplified 91.7%

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

      1. associate-*r/ 92.0%

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

      2. associate--r- 92.0%

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

      3. *-commutative 92.0%

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

      4. associate-*l/ 83.9%

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

      5. associate-/l* 92.0%

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

   8. Applied egg-rr 92.0%

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

   if -1.3500000000000001e-57 < b < 1.32e-68

   1. Initial program 74.3%

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

      1. neg-sub0 74.3%

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

      2. associate-+l- 74.3%

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

      3. sub0-neg 74.3%

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

      4. neg-mul-1 74.3%

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

      5. associate-*l/ 74.2%

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

      6. *-commutative 74.2%

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

      7. associate-/r* 74.2%

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

      8. /-rgt-identity 74.2%

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

      9. metadata-eval 74.2%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in a around inf 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-*r* 72.6%

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

   6. Simplified 72.6%

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

   if 1.32e-68 < b

   1. Initial program 13.8%

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

      1. neg-sub0 13.8%

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

      2. associate-+l- 13.8%

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

      3. sub0-neg 13.8%

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

      4. neg-mul-1 13.8%

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

      5. associate-*l/ 13.8%

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

      6. *-commutative 13.8%

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

      7. associate-/r* 13.8%

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

      8. /-rgt-identity 13.8%

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

      9. metadata-eval 13.8%

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

   3. Simplified 13.9%

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

   4. Taylor expanded in b around inf 77.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]
   5. Step-by-step derivation

      1. +-commutative 77.5%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      2. mul-1-neg 77.5%

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

      3. unsub-neg 77.5%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

      4. associate-*r/ 77.5%

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      5. neg-mul-1 77.5%

         \[\leadsto \frac{\color{blue}{-c}}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]

      6. associate-/l* 80.7%

         \[\leadsto \frac{-c}{b} - \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}} \]

      7. unpow2 80.7%

         \[\leadsto \frac{-c}{b} - \frac{\color{blue}{c \cdot c}}{\frac{{b}^{3}}{a}} \]

      8. associate-/l* 93.4%

         \[\leadsto \frac{-c}{b} - \color{blue}{\frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}} \]

   6. Simplified 93.4%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{-57}:\\ \;\;\;\;\frac{\left(b + \left(b - 2 \cdot \frac{c}{\frac{b}{a}}\right)\right) \cdot -0.5}{a}\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{-68}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{c}{\frac{\frac{{b}^{3}}{a}}{c}}\\ \end{array} \]

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-56}:\\ \;\;\;\;\frac{\left(b + \left(b - 2 \cdot \frac{c}{\frac{b}{a}}\right)\right) \cdot -0.5}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-94}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-56)
   (/ (* (+ b (- b (* 2.0 (/ c (/ b a))))) -0.5) a)
   (if (<= b 1.55e-94)
     (* (- b (sqrt (* c (* a -4.0)))) (/ -0.5 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-56) {
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a;
	} else if (b <= 1.55e-94) {
		tmp = (b - sqrt((c * (a * -4.0)))) * (-0.5 / 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 <= (-3d-56)) then
        tmp = ((b + (b - (2.0d0 * (c / (b / a))))) * (-0.5d0)) / a
    else if (b <= 1.55d-94) then
        tmp = (b - sqrt((c * (a * (-4.0d0))))) * ((-0.5d0) / 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 <= -3e-56) {
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a;
	} else if (b <= 1.55e-94) {
		tmp = (b - Math.sqrt((c * (a * -4.0)))) * (-0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e-56:
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a
	elif b <= 1.55e-94:
		tmp = (b - math.sqrt((c * (a * -4.0)))) * (-0.5 / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-56)
		tmp = Float64(Float64(Float64(b + Float64(b - Float64(2.0 * Float64(c / Float64(b / a))))) * -0.5) / a);
	elseif (b <= 1.55e-94)
		tmp = Float64(Float64(b - sqrt(Float64(c * Float64(a * -4.0)))) * Float64(-0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e-56)
		tmp = ((b + (b - (2.0 * (c / (b / a))))) * -0.5) / a;
	elseif (b <= 1.55e-94)
		tmp = (b - sqrt((c * (a * -4.0)))) * (-0.5 / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-56], N[(N[(N[(b + N[(b - N[(2.0 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.55e-94], N[(N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.99999999999999989e-56

   1. Initial program 64.2%

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

      1. neg-sub0 64.2%

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

      2. associate-+l- 64.2%

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

      3. sub0-neg 64.2%

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

      4. neg-mul-1 64.2%

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

      5. associate-*l/ 64.0%

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

      6. *-commutative 64.0%

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

      7. associate-/r* 64.0%

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

      8. /-rgt-identity 64.0%

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

      9. metadata-eval 64.0%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in b around -inf 83.6%

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

      1. neg-mul-1 83.6%

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

      2. unsub-neg 83.6%

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

      3. *-commutative 83.6%

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

      4. associate-/l* 91.7%

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

      5. associate-/r/ 91.7%

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

   6. Simplified 91.7%

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

      1. associate-*r/ 92.0%

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

      2. associate--r- 92.0%

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

      3. *-commutative 92.0%

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

      4. associate-*l/ 83.9%

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

      5. associate-/l* 92.0%

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

   8. Applied egg-rr 92.0%

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

   if -2.99999999999999989e-56 < b < 1.5499999999999999e-94

   1. Initial program 75.5%

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

      1. neg-sub0 75.5%

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

      2. associate-+l- 75.5%

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

      3. sub0-neg 75.5%

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

      4. neg-mul-1 75.5%

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

      5. associate-*l/ 75.4%

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

      6. *-commutative 75.4%

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

      7. associate-/r* 75.4%

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

      8. /-rgt-identity 75.4%

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

      9. metadata-eval 75.4%

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

   3. Simplified 74.1%

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

   4. Taylor expanded in a around inf 73.8%

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

      1. *-commutative 73.8%

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

      2. associate-*r* 73.8%

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

   6. Simplified 73.8%

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

   if 1.5499999999999999e-94 < b

   1. Initial program 15.4%

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

      1. neg-sub0 15.4%

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

      2. associate-+l- 15.4%

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

      3. sub0-neg 15.4%

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

      4. neg-mul-1 15.4%

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

      5. associate-*l/ 15.4%

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

      6. *-commutative 15.4%

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

      7. associate-/r* 15.4%

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

      8. /-rgt-identity 15.4%

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

      9. metadata-eval 15.4%

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

   3. Simplified 15.4%

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

   4. Taylor expanded in b around inf 91.3%

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

      1. associate-*r/ 91.3%

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

      2. neg-mul-1 91.3%

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

   6. Simplified 91.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-56}:\\ \;\;\;\;\frac{\left(b + \left(b - 2 \cdot \frac{c}{\frac{b}{a}}\right)\right) \cdot -0.5}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-94}:\\ \;\;\;\;\left(b - \sqrt{c \cdot \left(a \cdot -4\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5: 68.0% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 68.1%

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

      1. neg-sub0 68.1%

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

      2. associate-+l- 68.1%

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

      3. sub0-neg 68.1%

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

      4. neg-mul-1 68.1%

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

      5. associate-*l/ 67.9%

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

      6. *-commutative 67.9%

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

      7. associate-/r* 67.9%

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

      8. /-rgt-identity 67.9%

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

      9. metadata-eval 67.9%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in b around -inf 65.5%

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

      1. mul-1-neg 65.5%

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

      2. unsub-neg 65.5%

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

   6. Simplified 65.5%

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

   if -1.999999999999994e-310 < b

   1. Initial program 30.3%

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

      1. neg-sub0 30.3%

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

      2. associate-+l- 30.3%

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

      3. sub0-neg 30.3%

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

      4. neg-mul-1 30.3%

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

      5. associate-*l/ 30.2%

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

      6. *-commutative 30.2%

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

      7. associate-/r* 30.2%

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

      8. /-rgt-identity 30.2%

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

      9. metadata-eval 30.2%

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

   3. Simplified 29.5%

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

   4. Taylor expanded in b around inf 73.1%

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

      1. associate-*r/ 73.1%

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

      2. neg-mul-1 73.1%

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

   6. Simplified 73.1%

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

4. Final simplification 69.4%

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

Alternative 6: 43.5% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2400000:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 2400000.0) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2400000.0) {
		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 <= 2400000.0d0) 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 <= 2400000.0) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2400000.0:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2400000.0)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2400000.0)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2400000.0], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.4e6

   1. Initial program 65.3%

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

      1. neg-sub0 65.3%

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

      2. associate-+l- 65.3%

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

      3. sub0-neg 65.3%

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

      4. neg-mul-1 65.3%

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

      5. associate-*l/ 65.1%

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

      6. *-commutative 65.1%

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

      7. associate-/r* 65.1%

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

      8. /-rgt-identity 65.1%

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

      9. metadata-eval 65.1%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in b around -inf 47.2%

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

      1. associate-*r/ 47.2%

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

      2. neg-mul-1 47.2%

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

   6. Simplified 47.2%

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

   if 2.4e6 < b

   1. Initial program 13.1%

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

      1. neg-sub0 13.1%

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

      2. associate-+l- 13.1%

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

      3. sub0-neg 13.1%

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

      4. neg-mul-1 13.1%

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

      5. associate-*l/ 13.1%

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

      6. *-commutative 13.1%

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

      7. associate-/r* 13.1%

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

      8. /-rgt-identity 13.1%

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

      9. metadata-eval 13.1%

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

   3. Simplified 13.1%

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

   4. Taylor expanded in b around -inf 2.9%

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

      1. neg-mul-1 2.9%

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

      2. unsub-neg 2.9%

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

      3. *-commutative 2.9%

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

      4. associate-/l* 2.9%

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

      5. associate-/r/ 2.9%

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

   6. Simplified 2.9%

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

   7. Taylor expanded in b around 0 32.5%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2400000:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 7: 67.8% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.42 \cdot 10^{-288}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.42e-288) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.42e-288) {
		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 <= 1.42d-288) 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 <= 1.42e-288) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.42e-288:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.42e-288)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.42e-288)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.42e-288], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.42e-288

   1. Initial program 67.8%

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

      1. neg-sub0 67.8%

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

      2. associate-+l- 67.8%

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

      3. sub0-neg 67.8%

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

      4. neg-mul-1 67.8%

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

      5. associate-*l/ 67.7%

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

      6. *-commutative 67.7%

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

      7. associate-/r* 67.7%

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

      8. /-rgt-identity 67.7%

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

      9. metadata-eval 67.7%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in b around -inf 63.9%

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

      1. associate-*r/ 63.9%

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

      2. neg-mul-1 63.9%

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

   6. Simplified 63.9%

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

   if 1.42e-288 < b

   1. Initial program 30.0%

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

      1. neg-sub0 30.0%

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

      2. associate-+l- 30.0%

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

      3. sub0-neg 30.0%

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

      4. neg-mul-1 30.0%

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

      5. associate-*l/ 29.9%

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

      6. *-commutative 29.9%

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

      7. associate-/r* 29.9%

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

      8. /-rgt-identity 29.9%

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

      9. metadata-eval 29.9%

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

   3. Simplified 29.2%

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

   4. Taylor expanded in b around inf 74.1%

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

      1. associate-*r/ 74.1%

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

      2. neg-mul-1 74.1%

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

   6. Simplified 74.1%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.42 \cdot 10^{-288}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

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 48.7%

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

   1. flip-+ 29.3%

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

   2. pow2 29.3%

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

   3. add-sqr-sqrt 29.3%

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

   4. *-commutative 29.3%

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

   5. *-commutative 29.3%

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

   6. *-commutative 29.3%

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

   7. *-commutative 29.3%

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

3. Applied egg-rr 29.3%

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

   1. unpow2 29.3%

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

   2. add-sqr-sqrt 29.3%

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

   3. flip-+ 48.7%

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

   4. +-commutative 48.7%

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

   5. add-sqr-sqrt 33.2%

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

   6. sqrt-prod 45.6%

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

   7. unpow2 45.6%

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

   8. neg-mul-1 45.6%

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

   9. unpow-prod-down 45.6%

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

   10. metadata-eval 45.6%

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

   11. pow2 45.6%

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

   12. *-un-lft-identity 45.6%

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

   13. sqrt-prod 12.5%

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

   14. add-sqr-sqrt 30.8%

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

5. Applied egg-rr 30.8%

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

6. Taylor expanded in b around inf 2.6%

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

7. Final simplification 2.6%

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

Alternative 9: 11.2% 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 48.7%

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

   1. neg-sub0 48.7%

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

   2. associate-+l- 48.7%

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

   3. sub0-neg 48.7%

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

   4. neg-mul-1 48.7%

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

   5. associate-*l/ 48.6%

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

   6. *-commutative 48.6%

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

   7. associate-/r* 48.6%

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

   8. /-rgt-identity 48.6%

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

   9. metadata-eval 48.6%

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

3. Simplified 48.3%

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

4. Taylor expanded in b around -inf 30.4%

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

   1. neg-mul-1 30.4%

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

   2. unsub-neg 30.4%

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

   3. *-commutative 30.4%

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

   4. associate-/l* 33.1%

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

   5. associate-/r/ 33.1%

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

6. Simplified 33.1%

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

7. Taylor expanded in b around 0 12.3%

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

8. Final simplification 12.3%

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

Reproduce

herbie shell --seed 2023229 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))