Quadratic roots, medium range

Percentage Accurate: 31.8% → 99.7%

Time: 14.2s

Alternatives: 6

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: 31.8% 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: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

   1. *-commutative 33.5%

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

3. Simplified 33.5%

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

   1. add-exp-log 33.5%

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

6. Applied egg-rr 33.5%

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

   1. flip-+ 33.5%

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

   2. pow2 33.5%

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

   3. add-sqr-sqrt 34.4%

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

   4. pow2 34.4%

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

   5. *-commutative 34.4%

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

   6. rem-exp-log 34.5%

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

   7. *-commutative 34.5%

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

   8. pow2 34.5%

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

   9. *-commutative 34.5%

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

   10. rem-exp-log 34.5%

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

   11. *-commutative 34.5%

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

8. Applied egg-rr 34.5%

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

   1. *-un-lft-identity 34.5%

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

   2. associate-/l/ 34.5%

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

   3. associate--r- 99.3%

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

   4. neg-mul-1 99.3%

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

   5. unpow-prod-down 99.3%

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

   6. metadata-eval 99.3%

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

   7. *-un-lft-identity 99.3%

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

   8. associate-*r* 99.3%

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

   9. *-commutative 99.3%

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

   10. associate-*r* 99.3%

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

10. Applied egg-rr 99.3%

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

    1. *-lft-identity 99.3%

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

    2. associate-/r* 99.6%

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

    3. +-inverses 99.6%

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

    4. associate-*l* 99.6%

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

    5. *-commutative 99.6%

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

    6. *-commutative 99.6%

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

    7. *-commutative 99.6%

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

    8. cancel-sign-sub-inv 99.6%

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

    9. metadata-eval 99.6%

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

    10. +-commutative 99.6%

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

    11. fma-define 99.6%

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

    12. *-commutative 99.6%

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

12. Simplified 99.6%

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

13. Taylor expanded in c around 0 99.7%

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

    1. *-commutative 99.7%

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

15. Simplified 99.7%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 4.0);
	return (t_0 / (-b - sqrt((pow(b, 2.0) - t_0)))) / (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
    real(8) :: t_0
    t_0 = c * (a * 4.0d0)
    code = (t_0 / (-b - sqrt(((b ** 2.0d0) - t_0)))) / (2.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

   1. *-commutative 33.5%

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

3. Simplified 33.5%

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

   1. add-exp-log 33.5%

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

6. Applied egg-rr 33.5%

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

   1. flip-+ 33.5%

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

   2. pow2 33.5%

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

   3. add-sqr-sqrt 34.4%

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

   4. pow2 34.4%

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

   5. *-commutative 34.4%

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

   6. rem-exp-log 34.5%

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

   7. *-commutative 34.5%

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

   8. pow2 34.5%

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

   9. *-commutative 34.5%

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

   10. rem-exp-log 34.5%

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

   11. *-commutative 34.5%

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

8. Applied egg-rr 34.5%

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

9. Taylor expanded in b around 0 99.4%

   \[\leadsto \frac{\frac{\color{blue}{4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 4\right)}}}{a \cdot 2} \]
10. 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-*l* 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 3: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

   1. *-commutative 33.5%

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

3. Simplified 33.5%

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

   1. add-exp-log 33.5%

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

6. Applied egg-rr 33.5%

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

   1. flip-+ 33.5%

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

   2. pow2 33.5%

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

   3. add-sqr-sqrt 34.4%

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

   4. pow2 34.4%

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

   5. *-commutative 34.4%

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

   6. rem-exp-log 34.5%

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

   7. *-commutative 34.5%

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

   8. pow2 34.5%

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

   9. *-commutative 34.5%

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

   10. rem-exp-log 34.5%

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

   11. *-commutative 34.5%

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

8. Applied egg-rr 34.5%

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

   1. *-un-lft-identity 34.5%

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

   2. associate-/l/ 34.5%

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

   3. associate--r- 99.3%

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

   4. neg-mul-1 99.3%

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

   5. unpow-prod-down 99.3%

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

   6. metadata-eval 99.3%

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

   7. *-un-lft-identity 99.3%

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

   8. associate-*r* 99.3%

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

   9. *-commutative 99.3%

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

   10. associate-*r* 99.3%

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

10. Applied egg-rr 99.3%

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

    1. *-lft-identity 99.3%

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

    2. associate-/r* 99.6%

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

    3. +-inverses 99.6%

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

    4. associate-*l* 99.6%

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

    5. *-commutative 99.6%

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

    6. *-commutative 99.6%

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

    7. *-commutative 99.6%

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

    8. cancel-sign-sub-inv 99.6%

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

    9. metadata-eval 99.6%

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

    10. +-commutative 99.6%

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

    11. fma-define 99.6%

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

    12. *-commutative 99.6%

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

12. Simplified 99.6%

    \[\leadsto \color{blue}{\frac{\frac{0 + c \cdot \left(a \cdot 4\right)}{a \cdot 2}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}}} \]
13. Step-by-step derivation

    1. unpow2 99.6%

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

14. Applied egg-rr 99.6%

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

15. Final simplification 99.6%

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

Alternative 4: 90.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((c * (a * 4.0)) / (2.0 * a)) / (b * ((2.0 * ((c * a) / pow(b, 2.0))) - 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 * (a * 4.0d0)) / (2.0d0 * a)) / (b * ((2.0d0 * ((c * a) / (b ** 2.0d0))) - 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

   1. *-commutative 33.5%

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

3. Simplified 33.5%

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

   1. add-exp-log 33.5%

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

6. Applied egg-rr 33.5%

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

   1. flip-+ 33.5%

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

   2. pow2 33.5%

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

   3. add-sqr-sqrt 34.4%

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

   4. pow2 34.4%

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

   5. *-commutative 34.4%

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

   6. rem-exp-log 34.5%

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

   7. *-commutative 34.5%

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

   8. pow2 34.5%

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

   9. *-commutative 34.5%

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

   10. rem-exp-log 34.5%

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

   11. *-commutative 34.5%

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

8. Applied egg-rr 34.5%

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

   1. *-un-lft-identity 34.5%

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

   2. associate-/l/ 34.5%

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

   3. associate--r- 99.3%

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

   4. neg-mul-1 99.3%

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

   5. unpow-prod-down 99.3%

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

   6. metadata-eval 99.3%

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

   7. *-un-lft-identity 99.3%

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

   8. associate-*r* 99.3%

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

   9. *-commutative 99.3%

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

   10. associate-*r* 99.3%

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

10. Applied egg-rr 99.3%

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

    1. *-lft-identity 99.3%

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

    2. associate-/r* 99.6%

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

    3. +-inverses 99.6%

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

    4. associate-*l* 99.6%

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

    5. *-commutative 99.6%

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

    6. *-commutative 99.6%

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

    7. *-commutative 99.6%

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

    8. cancel-sign-sub-inv 99.6%

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

    9. metadata-eval 99.6%

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

    10. +-commutative 99.6%

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

    11. fma-define 99.6%

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

    12. *-commutative 99.6%

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

12. Simplified 99.6%

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

13. Taylor expanded in b around inf 88.9%

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

14. Final simplification 88.9%

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

Alternative 5: 90.9% accurate, 6.1× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((c * (a * 4.0)) / (2.0 * a)) / (2.0 * ((a * (c / b)) - 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 * (a * 4.0d0)) / (2.0d0 * a)) / (2.0d0 * ((a * (c / b)) - b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.5%

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

   1. *-commutative 33.5%

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

3. Simplified 33.5%

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

   1. add-exp-log 33.5%

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

6. Applied egg-rr 33.5%

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

   1. flip-+ 33.5%

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

   2. pow2 33.5%

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

   3. add-sqr-sqrt 34.4%

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

   4. pow2 34.4%

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

   5. *-commutative 34.4%

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

   6. rem-exp-log 34.5%

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

   7. *-commutative 34.5%

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

   8. pow2 34.5%

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

   9. *-commutative 34.5%

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

   10. rem-exp-log 34.5%

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

   11. *-commutative 34.5%

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

8. Applied egg-rr 34.5%

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

   1. *-un-lft-identity 34.5%

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

   2. associate-/l/ 34.5%

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

   3. associate--r- 99.3%

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

   4. neg-mul-1 99.3%

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

   5. unpow-prod-down 99.3%

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

   6. metadata-eval 99.3%

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

   7. *-un-lft-identity 99.3%

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

   8. associate-*r* 99.3%

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

   9. *-commutative 99.3%

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

   10. associate-*r* 99.3%

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

10. Applied egg-rr 99.3%

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

    1. *-lft-identity 99.3%

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

    2. associate-/r* 99.6%

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

    3. +-inverses 99.6%

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

    4. associate-*l* 99.6%

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

    5. *-commutative 99.6%

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

    6. *-commutative 99.6%

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

    7. *-commutative 99.6%

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

    8. cancel-sign-sub-inv 99.6%

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

    9. metadata-eval 99.6%

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

    10. +-commutative 99.6%

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

    11. fma-define 99.6%

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

    12. *-commutative 99.6%

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

12. Simplified 99.6%

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

13. Taylor expanded in c around 0 88.9%

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

    1. distribute-lft-out-- 88.9%

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

    2. associate-*r/ 88.9%

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

    3. *-commutative 88.9%

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

15. Simplified 88.9%

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

16. Final simplification 88.9%

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

Alternative 6: 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(c / Float64(-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 33.5%

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

   1. *-commutative 33.5%

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

3. Simplified 33.5%

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

5. Taylor expanded in b around inf 79.5%

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

   1. associate-*r/ 79.5%

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

   2. mul-1-neg 79.5%

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

7. Simplified 79.5%

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

8. Final simplification 79.5%

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

Reproduce

herbie shell --seed 2024172 
(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)))