Quadratic roots, medium range

Percentage Accurate: 32.0% → 99.4%

Time: 13.3s

Alternatives: 5

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 32.0% 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(4 \cdot a\right)\\ \frac{\frac{t\_0}{\left(-b\right) - \sqrt{{b}^{2} - t\_0}}}{a \cdot 2} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* 4.0 a))))
   (/ (/ t_0 (- (- b) (sqrt (- (pow b 2.0) t_0)))) (* a 2.0))))

C

double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	return (t_0 / (-b - sqrt((pow(b, 2.0) - t_0)))) / (a * 2.0);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = c * (4.0d0 * a)
    code = (t_0 / (-b - sqrt(((b ** 2.0d0) - t_0)))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (4.0 * a);
	return (t_0 / (-b - Math.sqrt((Math.pow(b, 2.0) - t_0)))) / (a * 2.0);
}

Python

def code(a, b, c):
	t_0 = c * (4.0 * a)
	return (t_0 / (-b - math.sqrt((math.pow(b, 2.0) - t_0)))) / (a * 2.0)

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(4.0 * a))
	return Float64(Float64(t_0 / Float64(Float64(-b) - sqrt(Float64((b ^ 2.0) - t_0)))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = c * (4.0 * a);
	tmp = (t_0 / (-b - sqrt(((b ^ 2.0) - t_0)))) / (a * 2.0);
end

Wolfram

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

Derivation

1. Initial program 30.7%

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

   1. *-commutative 30.7%

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

3. Simplified 30.7%

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

   1. add-sqr-sqrt 30.8%

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

   2. distribute-rgt-neg-in 30.8%

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

6. Applied egg-rr 30.8%

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

   1. flip-+ 31.0%

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

   2. pow2 31.0%

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

   3. distribute-rgt-neg-out 31.0%

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

   4. add-sqr-sqrt 30.8%

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

   5. add-sqr-sqrt 31.6%

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

   6. pow2 31.6%

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

   7. *-commutative 31.6%

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

   8. distribute-rgt-neg-out 31.6%

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

   9. add-sqr-sqrt 31.6%

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

   10. pow2 31.6%

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

   11. *-commutative 31.6%

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

8. Applied egg-rr 31.6%

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

   1. associate--r- 99.4%

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

   2. unpow2 99.4%

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

   3. unpow2 99.4%

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

   4. difference-of-squares 99.4%

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

   5. +-commutative 99.4%

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

   6. neg-mul-1 99.4%

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

   7. distribute-rgt1-in 99.4%

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

   8. metadata-eval 99.4%

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

   9. mul0-lft 99.4%

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

   10. *-commutative 99.4%

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

   11. *-commutative 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in b around 0 99.4%

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

    1. *-commutative 99.4%

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

    2. *-commutative 99.4%

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

    3. associate-*r* 99.4%

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

    4. *-commutative 99.4%

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

13. Simplified 99.4%

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

14. Final simplification 99.4%

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

Alternative 2: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(-\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9e-6)
   (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))
   (/ (- (- c) (* a (pow (- (/ c b)) 2.0))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9e-6) {
		tmp = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	} else {
		tmp = (-c - (a * pow(-(c / b), 2.0))) / 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 <= 9d-6) then
        tmp = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (a * 2.0d0)
    else
        tmp = (-c - (a * (-(c / b) ** 2.0d0))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 9e-6) {
		tmp = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	} else {
		tmp = (-c - (a * Math.pow(-(c / b), 2.0))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9e-6:
		tmp = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)
	else:
		tmp = (-c - (a * math.pow(-(c / b), 2.0))) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9e-6)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(a * (Float64(-Float64(c / b)) ^ 2.0))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 9e-6)
		tmp = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	else
		tmp = (-c - (a * (-(c / b) ^ 2.0))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9e-6], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(a * N[Power[(-N[(c / b), $MachinePrecision]), 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.00000000000000023e-6

   1. Initial program 85.3%

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

   if 9.00000000000000023e-6 < b

   1. Initial program 27.5%

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

      1. *-commutative 27.5%

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

   3. Simplified 27.5%

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

   5. Taylor expanded in a around 0 97.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   6. Taylor expanded in c around inf 97.1%

      \[\leadsto -1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + \color{blue}{{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{2}}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5} \cdot c}\right)}\right) \]

   7. Taylor expanded in b around inf 92.8%

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

      1. mul-1-neg 92.8%

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

      2. unsub-neg 92.8%

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

      3. mul-1-neg 92.8%

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

      4. associate-/l* 92.8%

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

      5. unpow2 92.8%

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

      6. unpow2 92.8%

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

      7. times-frac 92.8%

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

      8. sqr-neg 92.8%

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

      9. unpow2 92.8%

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

      10. distribute-neg-frac2 92.8%

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

   9. Simplified 92.8%

      \[\leadsto \color{blue}{\frac{\left(-c\right) - a \cdot {\left(\frac{c}{-b}\right)}^{2}}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - a \cdot {\left(-\frac{c}{b}\right)}^{2}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{c \cdot \left(4 \cdot a\right) + 0 \cdot \left(b + b\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (+ (* c (* 4.0 a)) (* 0.0 (+ b b))) (* 2.0 (- (* a (/ c b)) b)))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (((c * (4.0 * a)) + (0.0 * (b + b))) / (2.0 * ((a * (c / b)) - b))) / (a * 2.0);
}

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 * (4.0d0 * a)) + (0.0d0 * (b + b))) / (2.0d0 * ((a * (c / b)) - b))) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (((c * (4.0 * a)) + (0.0 * (b + b))) / (2.0 * ((a * (c / b)) - b))) / (a * 2.0);
}

Python

def code(a, b, c):
	return (((c * (4.0 * a)) + (0.0 * (b + b))) / (2.0 * ((a * (c / b)) - b))) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(c * Float64(4.0 * a)) + Float64(0.0 * Float64(b + b))) / Float64(2.0 * Float64(Float64(a * Float64(c / b)) - b))) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((c * (4.0 * a)) + (0.0 * (b + b))) / (2.0 * ((a * (c / b)) - b))) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision] + N[(0.0 * N[(b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.7%

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

   1. *-commutative 30.7%

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

3. Simplified 30.7%

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

   1. add-sqr-sqrt 30.8%

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

   2. distribute-rgt-neg-in 30.8%

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

6. Applied egg-rr 30.8%

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

   1. flip-+ 31.0%

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

   2. pow2 31.0%

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

   3. distribute-rgt-neg-out 31.0%

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

   4. add-sqr-sqrt 30.8%

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

   5. add-sqr-sqrt 31.6%

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

   6. pow2 31.6%

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

   7. *-commutative 31.6%

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

   8. distribute-rgt-neg-out 31.6%

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

   9. add-sqr-sqrt 31.6%

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

   10. pow2 31.6%

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

   11. *-commutative 31.6%

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

8. Applied egg-rr 31.6%

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

   1. associate--r- 99.4%

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

   2. unpow2 99.4%

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

   3. unpow2 99.4%

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

   4. difference-of-squares 99.4%

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

   5. +-commutative 99.4%

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

   6. neg-mul-1 99.4%

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

   7. distribute-rgt1-in 99.4%

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

   8. metadata-eval 99.4%

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

   9. mul0-lft 99.4%

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

   10. *-commutative 99.4%

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

   11. *-commutative 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in c around 0 90.8%

    \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 4\right)}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}}{a \cdot 2} \]
12. Step-by-step derivation

    1. distribute-lft-out-- 90.8%

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

    2. associate-*r/ 90.8%

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

13. Simplified 90.8%

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

14. Final simplification 90.8%

    \[\leadsto \frac{\frac{c \cdot \left(4 \cdot a\right) + 0 \cdot \left(b + b\right)}{2 \cdot \left(a \cdot \frac{c}{b} - b\right)}}{a \cdot 2} \]
15. Add Preprocessing

Alternative 4: 81.0% accurate, 29.0× 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(-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 30.7%

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

   1. *-commutative 30.7%

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

3. Simplified 30.6%

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

5. Taylor expanded in b around inf 81.7%

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

   1. associate-*r/ 81.7%

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

   2. mul-1-neg 81.7%

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

7. Simplified 81.7%

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

8. Final simplification 81.7%

   \[\leadsto -\frac{c}{b} \]
9. Add Preprocessing

Alternative 5: 3.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 30.7%

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

   1. *-commutative 30.7%

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

3. Simplified 30.7%

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

   1. add-sqr-sqrt 30.8%

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

   2. distribute-rgt-neg-in 30.8%

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

6. Applied egg-rr 30.8%

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

7. Taylor expanded in a around 0 3.2%

   \[\leadsto \color{blue}{0.5 \cdot \frac{b + -1 \cdot b}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.5 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.5 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

9. Simplified 3.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024177 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))