Quadratic roots, narrow range

Percentage Accurate: 55.4% → 99.3%

Time: 13.6s

Alternatives: 7

Speedup: 29.0×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.4% 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: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.4%

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

   1. *-commutative 57.4%

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

3. Simplified 57.4%

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

   1. add-cbrt-cube 57.4%

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

   2. pow3 57.4%

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

6. Applied egg-rr 57.4%

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

   1. flip-+ 57.4%

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

   2. pow2 57.4%

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

   3. add-sqr-sqrt 59.1%

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

   4. pow2 59.1%

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

   5. *-commutative 59.1%

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

   6. unpow3 59.1%

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

   7. add-cbrt-cube 59.1%

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

   8. *-commutative 59.1%

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

8. Applied egg-rr 59.1%

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

   1. *-un-lft-identity 59.1%

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

   2. associate-/l/ 59.1%

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

   3. associate--r- 99.4%

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

   4. neg-mul-1 99.4%

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

   5. unpow-prod-down 99.4%

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

   6. metadata-eval 99.4%

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

   7. *-un-lft-identity 99.4%

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

   8. *-commutative 99.4%

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

10. Applied egg-rr 99.4%

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

    1. *-un-lft-identity 99.4%

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

    2. +-commutative 99.4%

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

    3. fma-define 99.4%

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

    4. +-inverses 99.4%

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

    5. *-commutative 99.4%

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

12. Applied egg-rr 99.4%

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

    1. fma-undefine 99.4%

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

    2. +-rgt-identity 99.4%

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

    3. *-commutative 99.4%

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

    4. *-commutative 99.4%

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

    5. *-commutative 99.4%

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

    6. *-commutative 99.4%

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

14. Simplified 99.4%

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

Alternative 2: 88.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{c} + a \cdot \left(0.5 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{a}}}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   1.0
   (/
    (+
     (* -0.5 (/ b c))
     (* a (+ (* 0.5 (/ (* c a) (pow b 3.0))) (* 0.5 (/ 1.0 b)))))
    a))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (1.0 / (((-0.5 * (b / c)) + (a * ((0.5 * ((c * a) / pow(b, 3.0))) + (0.5 * (1.0 / b))))) / a)) / (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 = (1.0d0 / ((((-0.5d0) * (b / c)) + (a * ((0.5d0 * ((c * a) / (b ** 3.0d0))) + (0.5d0 * (1.0d0 / b))))) / a)) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / (((-0.5 * (b / c)) + (a * ((0.5 * ((c * a) / Math.pow(b, 3.0))) + (0.5 * (1.0 / b))))) / a)) / (a * 2.0);
}

Python

def code(a, b, c):
	return (1.0 / (((-0.5 * (b / c)) + (a * ((0.5 * ((c * a) / math.pow(b, 3.0))) + (0.5 * (1.0 / b))))) / a)) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(Float64(Float64(-0.5 * Float64(b / c)) + Float64(a * Float64(Float64(0.5 * Float64(Float64(c * a) / (b ^ 3.0))) + Float64(0.5 * Float64(1.0 / b))))) / a)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / (((-0.5 * (b / c)) + (a * ((0.5 * ((c * a) / (b ^ 3.0))) + (0.5 * (1.0 / b))))) / a)) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / N[(N[(N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(0.5 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.4%

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

   1. *-commutative 57.4%

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

3. Simplified 57.4%

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

   1. add-cbrt-cube 57.4%

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

   2. pow3 57.4%

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

6. Applied egg-rr 57.4%

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

   1. flip-+ 57.4%

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

   2. pow2 57.4%

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

   3. add-sqr-sqrt 59.1%

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

   4. pow2 59.1%

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

   5. *-commutative 59.1%

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

   6. unpow3 59.1%

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

   7. add-cbrt-cube 59.1%

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

   8. *-commutative 59.1%

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

8. Applied egg-rr 59.1%

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

   1. clear-num 59.1%

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

   2. inv-pow 59.1%

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

   3. associate--r- 99.4%

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

   4. neg-mul-1 99.4%

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

   5. unpow-prod-down 99.4%

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

   6. metadata-eval 99.4%

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

   7. *-un-lft-identity 99.4%

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

10. Applied egg-rr 99.4%

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

    1. unpow-1 99.4%

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

    2. associate-*r* 99.4%

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

    3. *-commutative 99.4%

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

    4. *-commutative 99.4%

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

    5. cancel-sign-sub-inv 99.4%

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

    6. metadata-eval 99.4%

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

    7. +-commutative 99.4%

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

    8. fma-define 99.4%

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

    9. *-commutative 99.4%

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

    10. +-inverses 99.4%

        \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}}{\color{blue}{0} + c \cdot \left(a \cdot 4\right)}}}{a \cdot 2} \]

    11. +-lft-identity 99.4%

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

    12. associate-*r* 99.4%

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

    13. *-commutative 99.4%

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

    14. associate-*r* 99.4%

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

    15. *-commutative 99.4%

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

12. Simplified 99.4%

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

13. Taylor expanded in a around 0 88.4%

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

14. Final simplification 88.4%

    \[\leadsto \frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{c} + a \cdot \left(0.5 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{a}}}{a \cdot 2} \]
15. Add Preprocessing

Alternative 3: 87.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{a} + c \cdot \left(0.5 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{c}}}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   1.0
   (/
    (+
     (* -0.5 (/ b a))
     (* c (+ (* 0.5 (/ (* c a) (pow b 3.0))) (* 0.5 (/ 1.0 b)))))
    c))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (1.0 / (((-0.5 * (b / a)) + (c * ((0.5 * ((c * a) / pow(b, 3.0))) + (0.5 * (1.0 / b))))) / c)) / (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 = (1.0d0 / ((((-0.5d0) * (b / a)) + (c * ((0.5d0 * ((c * a) / (b ** 3.0d0))) + (0.5d0 * (1.0d0 / b))))) / c)) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / (((-0.5 * (b / a)) + (c * ((0.5 * ((c * a) / Math.pow(b, 3.0))) + (0.5 * (1.0 / b))))) / c)) / (a * 2.0);
}

Python

def code(a, b, c):
	return (1.0 / (((-0.5 * (b / a)) + (c * ((0.5 * ((c * a) / math.pow(b, 3.0))) + (0.5 * (1.0 / b))))) / c)) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(Float64(Float64(-0.5 * Float64(b / a)) + Float64(c * Float64(Float64(0.5 * Float64(Float64(c * a) / (b ^ 3.0))) + Float64(0.5 * Float64(1.0 / b))))) / c)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / (((-0.5 * (b / a)) + (c * ((0.5 * ((c * a) / (b ^ 3.0))) + (0.5 * (1.0 / b))))) / c)) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / N[(N[(N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(0.5 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.4%

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

   1. *-commutative 57.4%

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

3. Simplified 57.4%

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

   1. add-cbrt-cube 57.4%

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

   2. pow3 57.4%

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

6. Applied egg-rr 57.4%

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

   1. flip-+ 57.4%

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

   2. pow2 57.4%

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

   3. add-sqr-sqrt 59.1%

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

   4. pow2 59.1%

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

   5. *-commutative 59.1%

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

   6. unpow3 59.1%

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

   7. add-cbrt-cube 59.1%

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

   8. *-commutative 59.1%

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

8. Applied egg-rr 59.1%

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

   1. clear-num 59.1%

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

   2. inv-pow 59.1%

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

   3. associate--r- 99.4%

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

   4. neg-mul-1 99.4%

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

   5. unpow-prod-down 99.4%

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

   6. metadata-eval 99.4%

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

   7. *-un-lft-identity 99.4%

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

10. Applied egg-rr 99.4%

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

    1. unpow-1 99.4%

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

    2. associate-*r* 99.4%

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

    3. *-commutative 99.4%

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

    4. *-commutative 99.4%

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

    5. cancel-sign-sub-inv 99.4%

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

    6. metadata-eval 99.4%

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

    7. +-commutative 99.4%

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

    8. fma-define 99.4%

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

    9. *-commutative 99.4%

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

    10. +-inverses 99.4%

        \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}}{\color{blue}{0} + c \cdot \left(a \cdot 4\right)}}}{a \cdot 2} \]

    11. +-lft-identity 99.4%

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

    12. associate-*r* 99.4%

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

    13. *-commutative 99.4%

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

    14. associate-*r* 99.4%

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

    15. *-commutative 99.4%

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

12. Simplified 99.4%

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

13. Taylor expanded in c around 0 88.4%

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

14. Final simplification 88.4%

    \[\leadsto \frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{a} + c \cdot \left(0.5 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{c}}}{a \cdot 2} \]
15. Add Preprocessing

Alternative 4: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 17:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 17.0)
   (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))
   (/ (/ 1.0 (/ (+ (* -0.5 (/ b a)) (* 0.5 (/ c b))) c)) (* a 2.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 17.0) {
		tmp = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / (((-0.5 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 2.0);
	}
	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 <= 17.0d0) then
        tmp = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (a * 2.0d0)
    else
        tmp = (1.0d0 / ((((-0.5d0) * (b / a)) + (0.5d0 * (c / b))) / c)) / (a * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 17.0) {
		tmp = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	} else {
		tmp = (1.0 / (((-0.5 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 2.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 17.0:
		tmp = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)
	else:
		tmp = (1.0 / (((-0.5 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 2.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 17.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(-0.5 * Float64(b / a)) + Float64(0.5 * Float64(c / b))) / c)) / Float64(a * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 17.0)
		tmp = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	else
		tmp = (1.0 / (((-0.5 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 17.0], 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[(1.0 / N[(N[(N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 17

   1. Initial program 78.0%

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

   if 17 < b

   1. Initial program 50.0%

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

      1. *-commutative 50.0%

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

   3. Simplified 50.0%

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

      1. add-cbrt-cube 50.0%

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

      2. pow3 50.0%

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

   6. Applied egg-rr 50.0%

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

      1. flip-+ 49.9%

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

      2. pow2 49.9%

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

      3. add-sqr-sqrt 51.6%

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

      4. pow2 51.6%

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

      5. *-commutative 51.6%

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

      6. unpow3 51.6%

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

      7. add-cbrt-cube 51.6%

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

      8. *-commutative 51.6%

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

   8. Applied egg-rr 51.6%

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

      1. clear-num 51.6%

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

      2. inv-pow 51.6%

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

      3. associate--r- 99.4%

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

      4. neg-mul-1 99.4%

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

      5. unpow-prod-down 99.4%

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

      6. metadata-eval 99.4%

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

      7. *-un-lft-identity 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. unpow-1 99.4%

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

       2. associate-*r* 99.4%

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

       3. *-commutative 99.4%

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

       4. *-commutative 99.4%

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

       5. cancel-sign-sub-inv 99.4%

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

       6. metadata-eval 99.4%

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

       7. +-commutative 99.4%

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

       8. fma-define 99.4%

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

       9. *-commutative 99.4%

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

       10. +-inverses 99.4%

           \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}}{\color{blue}{0} + c \cdot \left(a \cdot 4\right)}}}{a \cdot 2} \]

       11. +-lft-identity 99.4%

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

       12. associate-*r* 99.4%

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

       13. *-commutative 99.4%

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

       14. associate-*r* 99.4%

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

       15. *-commutative 99.4%

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

   12. Simplified 99.4%

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

   13. Taylor expanded in c around 0 87.4%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 17:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}{a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}{a}}}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (/ 1.0 (/ (+ (* -0.5 (/ b c)) (* 0.5 (/ a b))) a)) (* a 2.0)))

C

double code(double a, double b, double c) {
	return (1.0 / (((-0.5 * (b / c)) + (0.5 * (a / b))) / a)) / (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 = (1.0d0 / ((((-0.5d0) * (b / c)) + (0.5d0 * (a / b))) / a)) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / (((-0.5 * (b / c)) + (0.5 * (a / b))) / a)) / (a * 2.0);
}

Python

def code(a, b, c):
	return (1.0 / (((-0.5 * (b / c)) + (0.5 * (a / b))) / a)) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(Float64(Float64(-0.5 * Float64(b / c)) + Float64(0.5 * Float64(a / b))) / a)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / (((-0.5 * (b / c)) + (0.5 * (a / b))) / a)) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / N[(N[(N[(-0.5 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.4%

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

   1. *-commutative 57.4%

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

3. Simplified 57.4%

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

   1. add-cbrt-cube 57.4%

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

   2. pow3 57.4%

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

6. Applied egg-rr 57.4%

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

   1. flip-+ 57.4%

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

   2. pow2 57.4%

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

   3. add-sqr-sqrt 59.1%

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

   4. pow2 59.1%

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

   5. *-commutative 59.1%

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

   6. unpow3 59.1%

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

   7. add-cbrt-cube 59.1%

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

   8. *-commutative 59.1%

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

8. Applied egg-rr 59.1%

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

   1. clear-num 59.1%

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

   2. inv-pow 59.1%

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

   3. associate--r- 99.4%

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

   4. neg-mul-1 99.4%

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

   5. unpow-prod-down 99.4%

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

   6. metadata-eval 99.4%

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

   7. *-un-lft-identity 99.4%

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

10. Applied egg-rr 99.4%

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

    1. unpow-1 99.4%

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

    2. associate-*r* 99.4%

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

    3. *-commutative 99.4%

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

    4. *-commutative 99.4%

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

    5. cancel-sign-sub-inv 99.4%

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

    6. metadata-eval 99.4%

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

    7. +-commutative 99.4%

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

    8. fma-define 99.4%

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

    9. *-commutative 99.4%

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

    10. +-inverses 99.4%

        \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}}{\color{blue}{0} + c \cdot \left(a \cdot 4\right)}}}{a \cdot 2} \]

    11. +-lft-identity 99.4%

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

    12. associate-*r* 99.4%

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

    13. *-commutative 99.4%

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

    14. associate-*r* 99.4%

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

    15. *-commutative 99.4%

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

12. Simplified 99.4%

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

13. Taylor expanded in a around 0 81.9%

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-0.5 \cdot \frac{b}{c} + 0.5 \cdot \frac{a}{b}}{a}}}}{a \cdot 2} \]
14. Add Preprocessing

Alternative 6: 81.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (/ 1.0 (/ (+ (* -0.5 (/ b a)) (* 0.5 (/ c b))) c)) (* a 2.0)))

C

double code(double a, double b, double c) {
	return (1.0 / (((-0.5 * (b / a)) + (0.5 * (c / b))) / c)) / (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 = (1.0d0 / ((((-0.5d0) * (b / a)) + (0.5d0 * (c / b))) / c)) / (a * 2.0d0)
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / (((-0.5 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 2.0);
}

Python

def code(a, b, c):
	return (1.0 / (((-0.5 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 2.0)

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / Float64(Float64(Float64(-0.5 * Float64(b / a)) + Float64(0.5 * Float64(c / b))) / c)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / (((-0.5 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 2.0);
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / N[(N[(N[(-0.5 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.4%

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

   1. *-commutative 57.4%

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

3. Simplified 57.4%

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

   1. add-cbrt-cube 57.4%

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

   2. pow3 57.4%

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

6. Applied egg-rr 57.4%

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

   1. flip-+ 57.4%

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

   2. pow2 57.4%

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

   3. add-sqr-sqrt 59.1%

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

   4. pow2 59.1%

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

   5. *-commutative 59.1%

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

   6. unpow3 59.1%

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

   7. add-cbrt-cube 59.1%

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

   8. *-commutative 59.1%

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

8. Applied egg-rr 59.1%

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

   1. clear-num 59.1%

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

   2. inv-pow 59.1%

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

   3. associate--r- 99.4%

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

   4. neg-mul-1 99.4%

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

   5. unpow-prod-down 99.4%

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

   6. metadata-eval 99.4%

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

   7. *-un-lft-identity 99.4%

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

10. Applied egg-rr 99.4%

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

    1. unpow-1 99.4%

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

    2. associate-*r* 99.4%

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

    3. *-commutative 99.4%

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

    4. *-commutative 99.4%

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

    5. cancel-sign-sub-inv 99.4%

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

    6. metadata-eval 99.4%

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

    7. +-commutative 99.4%

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

    8. fma-define 99.4%

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

    9. *-commutative 99.4%

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

    10. +-inverses 99.4%

        \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, {b}^{2}\right)}}{\color{blue}{0} + c \cdot \left(a \cdot 4\right)}}}{a \cdot 2} \]

    11. +-lft-identity 99.4%

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

    12. associate-*r* 99.4%

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

    13. *-commutative 99.4%

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

    14. associate-*r* 99.4%

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

    15. *-commutative 99.4%

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

12. Simplified 99.4%

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

13. Taylor expanded in c around 0 81.9%

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-0.5 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}}{a \cdot 2} \]
14. Add Preprocessing

Alternative 7: 64.3% 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 57.4%

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

   1. *-commutative 57.4%

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

3. Simplified 57.5%

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

5. Taylor expanded in b around inf 63.1%

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

   1. associate-*r/ 63.1%

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

   2. mul-1-neg 63.1%

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

7. Simplified 63.1%

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

8. Final simplification 63.1%

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

Reproduce

herbie shell --seed 2024179 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))