Quadratic roots, medium range

Percentage Accurate: 31.5% → 99.7%

Time: 15.9s

Alternatives: 11

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: 31.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.8%

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

   1. +-commutative 29.8%

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

   2. flip-+ 29.8%

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

   3. add-sqr-sqrt 30.8%

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

   4. associate-*l* 30.8%

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

   5. cancel-sign-sub-inv 30.8%

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

   6. metadata-eval 30.8%

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

   7. sqr-neg 30.8%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 0.8%

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

   10. sqr-neg 0.8%

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

   11. sqrt-prod 1.5%

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

   12. add-sqr-sqrt 0.8%

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

   13. unsub-neg 0.8%

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

   14. +-commutative 0.8%

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

4. Applied egg-rr 30.8%

   \[\leadsto \frac{\color{blue}{\frac{\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right) - b \cdot b}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. +-commutative 30.8%

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

   2. associate--l+ 99.4%

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

   3. *-commutative 99.4%

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

   4. +-inverses 99.4%

      \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -4 + \color{blue}{0}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. mul0-rgt 99.4%

      \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -4 + \color{blue}{\left(a \cdot c\right) \cdot 0}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   6. distribute-lft-out 99.4%

      \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot \left(-4 + 0\right)}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   7. metadata-eval 99.4%

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

   8. +-commutative 99.4%

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

   9. *-commutative 99.4%

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

6. Simplified 99.4%

   \[\leadsto \frac{\color{blue}{\frac{\left(a \cdot c\right) \cdot -4}{b + {\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}^{0.5}}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. div-inv 99.2%

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

   2. associate-/l* 99.2%

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

   3. *-commutative 99.2%

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

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\left(a \cdot c\right) \cdot \frac{-4}{b + {\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}^{0.5}}\right) \cdot \frac{1}{a \cdot 2}} \]
9. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. associate-*r/ 99.2%

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

   3. times-frac 99.4%

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

   4. *-lft-identity 99.4%

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

   5. associate-/r* 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

   8. times-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. associate-*l* 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in a around 0 99.8%

    \[\leadsto \frac{-2 \cdot \color{blue}{c}}{b + {\left(a \cdot \left(c \cdot -4\right) + b \cdot b\right)}^{0.5}} \]
12. Step-by-step derivation

    1. unpow1/2 99.8%

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

13. Applied egg-rr 99.8%

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

Alternative 2: 95.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot c\right)\\ t_1 := b \cdot \left(b \cdot b\right)\\ \frac{-2 \cdot c}{b \cdot \left(2 + \left(-2 \cdot \left(a \cdot \frac{c}{b \cdot b} + \frac{a \cdot t\_0}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right) + -4 \cdot \frac{a \cdot \left(t\_0 \cdot \left(c \cdot a\right)\right)}{t\_1 \cdot t\_1}\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c c))) (t_1 (* b (* b b))))
   (/
    (* -2.0 c)
    (*
     b
     (+
      2.0
      (+
       (* -2.0 (+ (* a (/ c (* b b))) (/ (* a t_0) (* (* b b) (* b b)))))
       (* -4.0 (/ (* a (* t_0 (* c a))) (* t_1 t_1)))))))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * c);
	double t_1 = b * (b * b);
	return (-2.0 * c) / (b * (2.0 + ((-2.0 * ((a * (c / (b * b))) + ((a * t_0) / ((b * b) * (b * b))))) + (-4.0 * ((a * (t_0 * (c * a))) / (t_1 * t_1))))));
}

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
    real(8) :: t_1
    t_0 = a * (c * c)
    t_1 = b * (b * b)
    code = ((-2.0d0) * c) / (b * (2.0d0 + (((-2.0d0) * ((a * (c / (b * b))) + ((a * t_0) / ((b * b) * (b * b))))) + ((-4.0d0) * ((a * (t_0 * (c * a))) / (t_1 * t_1))))))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (c * c);
	double t_1 = b * (b * b);
	return (-2.0 * c) / (b * (2.0 + ((-2.0 * ((a * (c / (b * b))) + ((a * t_0) / ((b * b) * (b * b))))) + (-4.0 * ((a * (t_0 * (c * a))) / (t_1 * t_1))))));
}

Python

def code(a, b, c):
	t_0 = a * (c * c)
	t_1 = b * (b * b)
	return (-2.0 * c) / (b * (2.0 + ((-2.0 * ((a * (c / (b * b))) + ((a * t_0) / ((b * b) * (b * b))))) + (-4.0 * ((a * (t_0 * (c * a))) / (t_1 * t_1))))))

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * c))
	t_1 = Float64(b * Float64(b * b))
	return Float64(Float64(-2.0 * c) / Float64(b * Float64(2.0 + Float64(Float64(-2.0 * Float64(Float64(a * Float64(c / Float64(b * b))) + Float64(Float64(a * t_0) / Float64(Float64(b * b) * Float64(b * b))))) + Float64(-4.0 * Float64(Float64(a * Float64(t_0 * Float64(c * a))) / Float64(t_1 * t_1)))))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = a * (c * c);
	t_1 = b * (b * b);
	tmp = (-2.0 * c) / (b * (2.0 + ((-2.0 * ((a * (c / (b * b))) + ((a * t_0) / ((b * b) * (b * b))))) + (-4.0 * ((a * (t_0 * (c * a))) / (t_1 * t_1))))));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(-2.0 * c), $MachinePrecision] / N[(b * N[(2.0 + N[(N[(-2.0 * N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * t$95$0), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(N[(a * N[(t$95$0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 29.8%

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

   1. +-commutative 29.8%

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

   2. flip-+ 29.8%

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

   3. add-sqr-sqrt 30.8%

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

   4. associate-*l* 30.8%

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

   5. cancel-sign-sub-inv 30.8%

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

   6. metadata-eval 30.8%

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

   7. sqr-neg 30.8%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 0.8%

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

   10. sqr-neg 0.8%

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

   11. sqrt-prod 1.5%

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

   12. add-sqr-sqrt 0.8%

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

   13. unsub-neg 0.8%

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

   14. +-commutative 0.8%

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

4. Applied egg-rr 30.8%

   \[\leadsto \frac{\color{blue}{\frac{\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right) - b \cdot b}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. +-commutative 30.8%

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

   2. associate--l+ 99.4%

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

   3. *-commutative 99.4%

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

   4. +-inverses 99.4%

      \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -4 + \color{blue}{0}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. mul0-rgt 99.4%

      \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -4 + \color{blue}{\left(a \cdot c\right) \cdot 0}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   6. distribute-lft-out 99.4%

      \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot \left(-4 + 0\right)}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   7. metadata-eval 99.4%

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

   8. +-commutative 99.4%

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

   9. *-commutative 99.4%

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

6. Simplified 99.4%

   \[\leadsto \frac{\color{blue}{\frac{\left(a \cdot c\right) \cdot -4}{b + {\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}^{0.5}}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. div-inv 99.2%

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

   2. associate-/l* 99.2%

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

   3. *-commutative 99.2%

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

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\left(a \cdot c\right) \cdot \frac{-4}{b + {\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}^{0.5}}\right) \cdot \frac{1}{a \cdot 2}} \]
9. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. associate-*r/ 99.2%

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

   3. times-frac 99.4%

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

   4. *-lft-identity 99.4%

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

   5. associate-/r* 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

   8. times-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. associate-*l* 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in a around 0 99.8%

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

12. Taylor expanded in b around inf 95.3%

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

14. Simplified 95.3%

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

15. Final simplification 95.3%

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

Alternative 3: 94.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.8%

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

   1. +-commutative 29.8%

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

   2. flip-+ 29.8%

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

   3. add-sqr-sqrt 30.8%

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

   4. associate-*l* 30.8%

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

   5. cancel-sign-sub-inv 30.8%

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

   6. metadata-eval 30.8%

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

   7. sqr-neg 30.8%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 0.8%

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

   10. sqr-neg 0.8%

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

   11. sqrt-prod 1.5%

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

   12. add-sqr-sqrt 0.8%

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

   13. unsub-neg 0.8%

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

   14. +-commutative 0.8%

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

4. Applied egg-rr 30.8%

   \[\leadsto \frac{\color{blue}{\frac{\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right) - b \cdot b}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. +-commutative 30.8%

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

   2. associate--l+ 99.4%

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

   3. *-commutative 99.4%

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

   4. +-inverses 99.4%

      \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -4 + \color{blue}{0}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. mul0-rgt 99.4%

      \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -4 + \color{blue}{\left(a \cdot c\right) \cdot 0}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   6. distribute-lft-out 99.4%

      \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot \left(-4 + 0\right)}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   7. metadata-eval 99.4%

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

   8. +-commutative 99.4%

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

   9. *-commutative 99.4%

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

6. Simplified 99.4%

   \[\leadsto \frac{\color{blue}{\frac{\left(a \cdot c\right) \cdot -4}{b + {\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}^{0.5}}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. div-inv 99.2%

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

   2. associate-/l* 99.2%

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

   3. *-commutative 99.2%

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

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\left(a \cdot c\right) \cdot \frac{-4}{b + {\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}^{0.5}}\right) \cdot \frac{1}{a \cdot 2}} \]
9. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. associate-*r/ 99.2%

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

   3. times-frac 99.4%

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

   4. *-lft-identity 99.4%

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

   5. associate-/r* 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

   8. times-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. associate-*l* 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in a around 0 99.8%

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

12. Taylor expanded in c around 0 94.1%

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

    1. *-commutative 94.1%

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

    2. distribute-lft-out 94.1%

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

    3. associate-/l* 94.1%

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

    4. unpow2 94.1%

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

    5. cube-unmult 94.1%

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

14. Simplified 94.1%

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

Alternative 4: 94.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.8%

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

   1. +-commutative 29.8%

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

   2. flip-+ 29.8%

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

   3. add-sqr-sqrt 30.8%

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

   4. associate-*l* 30.8%

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

   5. cancel-sign-sub-inv 30.8%

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

   6. metadata-eval 30.8%

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

   7. sqr-neg 30.8%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 0.8%

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

   10. sqr-neg 0.8%

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

   11. sqrt-prod 1.5%

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

   12. add-sqr-sqrt 0.8%

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

   13. unsub-neg 0.8%

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

   14. +-commutative 0.8%

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

4. Applied egg-rr 30.8%

   \[\leadsto \frac{\color{blue}{\frac{\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right) - b \cdot b}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. +-commutative 30.8%

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

   2. associate--l+ 99.4%

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

   3. *-commutative 99.4%

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

   4. +-inverses 99.4%

      \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -4 + \color{blue}{0}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. mul0-rgt 99.4%

      \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -4 + \color{blue}{\left(a \cdot c\right) \cdot 0}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   6. distribute-lft-out 99.4%

      \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot \left(-4 + 0\right)}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   7. metadata-eval 99.4%

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

   8. +-commutative 99.4%

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

   9. *-commutative 99.4%

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

6. Simplified 99.4%

   \[\leadsto \frac{\color{blue}{\frac{\left(a \cdot c\right) \cdot -4}{b + {\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}^{0.5}}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. div-inv 99.2%

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

   2. associate-/l* 99.2%

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

   3. *-commutative 99.2%

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

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\left(a \cdot c\right) \cdot \frac{-4}{b + {\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}^{0.5}}\right) \cdot \frac{1}{a \cdot 2}} \]
9. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. associate-*r/ 99.2%

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

   3. times-frac 99.4%

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

   4. *-lft-identity 99.4%

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

   5. associate-/r* 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

   8. times-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. associate-*l* 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in a around 0 99.8%

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

12. Taylor expanded in c around 0 94.0%

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

    1. distribute-lft-out 94.0%

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

    2. *-commutative 94.0%

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

    3. unpow3 94.0%

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

    4. unpow2 94.0%

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

    5. times-frac 94.0%

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

    6. unpow2 94.0%

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

    7. unpow2 94.0%

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

14. Simplified 94.0%

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

Alternative 5: 91.0% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.8%

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

   1. +-commutative 29.8%

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

   2. flip-+ 29.8%

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

   3. add-sqr-sqrt 30.8%

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

   4. associate-*l* 30.8%

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

   5. cancel-sign-sub-inv 30.8%

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

   6. metadata-eval 30.8%

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

   7. sqr-neg 30.8%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 0.8%

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

   10. sqr-neg 0.8%

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

   11. sqrt-prod 1.5%

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

   12. add-sqr-sqrt 0.8%

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

   13. unsub-neg 0.8%

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

   14. +-commutative 0.8%

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

4. Applied egg-rr 30.8%

   \[\leadsto \frac{\color{blue}{\frac{\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right) - b \cdot b}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. +-commutative 30.8%

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

   2. associate--l+ 99.4%

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

   3. *-commutative 99.4%

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

   4. +-inverses 99.4%

      \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -4 + \color{blue}{0}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. mul0-rgt 99.4%

      \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -4 + \color{blue}{\left(a \cdot c\right) \cdot 0}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   6. distribute-lft-out 99.4%

      \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot \left(-4 + 0\right)}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   7. metadata-eval 99.4%

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

   8. +-commutative 99.4%

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

   9. *-commutative 99.4%

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

6. Simplified 99.4%

   \[\leadsto \frac{\color{blue}{\frac{\left(a \cdot c\right) \cdot -4}{b + {\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}^{0.5}}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. div-inv 99.2%

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

   2. associate-/l* 99.2%

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

   3. *-commutative 99.2%

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

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\left(a \cdot c\right) \cdot \frac{-4}{b + {\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}^{0.5}}\right) \cdot \frac{1}{a \cdot 2}} \]
9. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. associate-*r/ 99.2%

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

   3. times-frac 99.4%

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

   4. *-lft-identity 99.4%

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

   5. associate-/r* 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

   8. times-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. associate-*l* 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in a around 0 99.8%

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

12. Taylor expanded in a around 0 91.3%

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

    1. +-commutative 91.3%

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

    2. *-commutative 91.3%

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

    3. associate-*r/ 91.3%

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

    4. *-commutative 91.3%

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

    5. *-commutative 91.3%

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

    6. associate-*r* 91.3%

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

    7. associate-/l* 91.3%

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

    8. *-commutative 91.3%

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

    9. associate-*r/ 91.3%

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

14. Simplified 91.3%

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

Alternative 6: 90.9% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.8%

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

   1. +-commutative 29.8%

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

   2. flip-+ 29.8%

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

   3. add-sqr-sqrt 30.8%

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

   4. associate-*l* 30.8%

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

   5. cancel-sign-sub-inv 30.8%

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

   6. metadata-eval 30.8%

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

   7. sqr-neg 30.8%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 0.8%

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

   10. sqr-neg 0.8%

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

   11. sqrt-prod 1.5%

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

   12. add-sqr-sqrt 0.8%

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

   13. unsub-neg 0.8%

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

   14. +-commutative 0.8%

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

4. Applied egg-rr 30.8%

   \[\leadsto \frac{\color{blue}{\frac{\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right) - b \cdot b}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. +-commutative 30.8%

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

   2. associate--l+ 99.4%

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

   3. *-commutative 99.4%

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

   4. +-inverses 99.4%

      \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -4 + \color{blue}{0}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. mul0-rgt 99.4%

      \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -4 + \color{blue}{\left(a \cdot c\right) \cdot 0}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   6. distribute-lft-out 99.4%

      \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot \left(-4 + 0\right)}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   7. metadata-eval 99.4%

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

   8. +-commutative 99.4%

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

   9. *-commutative 99.4%

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

6. Simplified 99.4%

   \[\leadsto \frac{\color{blue}{\frac{\left(a \cdot c\right) \cdot -4}{b + {\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}^{0.5}}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. div-inv 99.2%

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

   2. associate-/l* 99.2%

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

   3. *-commutative 99.2%

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

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\left(a \cdot c\right) \cdot \frac{-4}{b + {\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}^{0.5}}\right) \cdot \frac{1}{a \cdot 2}} \]
9. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. associate-*r/ 99.2%

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

   3. times-frac 99.4%

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

   4. *-lft-identity 99.4%

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

   5. associate-/r* 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

   8. times-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. associate-*l* 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in a around 0 99.8%

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

12. Taylor expanded in a around 0 91.3%

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

    1. associate-/l* 91.3%

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

    2. associate-*l* 91.3%

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

    3. *-commutative 91.3%

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

    4. associate-*r* 91.3%

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

    5. *-commutative 91.3%

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

14. Simplified 91.3%

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

15. Final simplification 91.3%

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

Alternative 7: 90.4% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in a around 0 91.0%

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

   1. +-commutative 91.0%

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

   2. mul-1-neg 91.0%

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

   3. unsub-neg 91.0%

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

   4. mul-1-neg 91.0%

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

   5. associate-/l* 91.0%

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

   6. distribute-rgt-neg-in 91.0%

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

   7. neg-sub0 91.0%

      \[\leadsto a \cdot \color{blue}{\left(0 - \frac{{c}^{2}}{{b}^{3}}\right)} - \frac{c}{b} \]

   8. unpow2 91.0%

      \[\leadsto a \cdot \left(0 - \frac{\color{blue}{c \cdot c}}{{b}^{3}}\right) - \frac{c}{b} \]

   9. cube-unmult 91.0%

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

   10. times-frac 91.0%

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

5. Simplified 91.0%

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

6. Taylor expanded in b around 0 90.5%

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

   1. distribute-lft-out 90.5%

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

   2. cube-unmult 90.7%

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

   3. times-frac 90.6%

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

   4. unpow2 90.6%

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

   5. unpow2 90.6%

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

   6. *-commutative 90.6%

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

8. Simplified 90.6%

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

9. Taylor expanded in a around 0 90.7%

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

    1. unpow2 90.7%

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

    2. associate-/l* 90.7%

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

    3. unpow2 90.7%

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

11. Simplified 90.7%

    \[\leadsto \frac{-1}{b} \cdot \color{blue}{\left(c + a \cdot \frac{c \cdot c}{b \cdot b}\right)} \]
12. Add Preprocessing

Alternative 8: 90.3% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

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) / b) * (c * ((a * (c / (b * b))) + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in a around 0 91.0%

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

   1. +-commutative 91.0%

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

   2. mul-1-neg 91.0%

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

   3. unsub-neg 91.0%

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

   4. mul-1-neg 91.0%

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

   5. associate-/l* 91.0%

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

   6. distribute-rgt-neg-in 91.0%

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

   7. neg-sub0 91.0%

      \[\leadsto a \cdot \color{blue}{\left(0 - \frac{{c}^{2}}{{b}^{3}}\right)} - \frac{c}{b} \]

   8. unpow2 91.0%

      \[\leadsto a \cdot \left(0 - \frac{\color{blue}{c \cdot c}}{{b}^{3}}\right) - \frac{c}{b} \]

   9. cube-unmult 91.0%

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

   10. times-frac 91.0%

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

5. Simplified 91.0%

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

6. Taylor expanded in b around 0 90.5%

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

   1. distribute-lft-out 90.5%

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

   2. cube-unmult 90.7%

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

   3. times-frac 90.6%

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

   4. unpow2 90.6%

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

   5. unpow2 90.6%

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

   6. *-commutative 90.6%

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

8. Simplified 90.6%

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

9. Taylor expanded in c around 0 90.7%

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

    1. unpow2 90.7%

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

    2. associate-/l* 90.7%

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

11. Simplified 90.7%

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

12. Final simplification 90.7%

    \[\leadsto \frac{-1}{b} \cdot \left(c \cdot \left(a \cdot \frac{c}{b \cdot b} + 1\right)\right) \]
13. Add Preprocessing

Alternative 9: 81.0% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-1}{b} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in a around 0 91.0%

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

   1. +-commutative 91.0%

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

   2. mul-1-neg 91.0%

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

   3. unsub-neg 91.0%

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

   4. mul-1-neg 91.0%

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

   5. associate-/l* 91.0%

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

   6. distribute-rgt-neg-in 91.0%

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

   7. neg-sub0 91.0%

      \[\leadsto a \cdot \color{blue}{\left(0 - \frac{{c}^{2}}{{b}^{3}}\right)} - \frac{c}{b} \]

   8. unpow2 91.0%

      \[\leadsto a \cdot \left(0 - \frac{\color{blue}{c \cdot c}}{{b}^{3}}\right) - \frac{c}{b} \]

   9. cube-unmult 91.0%

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

   10. times-frac 91.0%

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

5. Simplified 91.0%

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

6. Taylor expanded in b around 0 90.5%

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

   1. distribute-lft-out 90.5%

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

   2. cube-unmult 90.7%

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

   3. times-frac 90.6%

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

   4. unpow2 90.6%

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

   5. unpow2 90.6%

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

   6. *-commutative 90.6%

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

8. Simplified 90.6%

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

9. Taylor expanded in a around 0 82.1%

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

10. Final simplification 82.1%

    \[\leadsto c \cdot \frac{-1}{b} \]
11. Add Preprocessing

Alternative 10: 81.2% 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 29.8%

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

3. Taylor expanded in b around inf 82.3%

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

   1. mul-1-neg 82.3%

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

   2. neg-sub0 82.3%

      \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

5. Simplified 82.3%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]

6. Final simplification 82.3%

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

Alternative 11: 1.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = b / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.8%

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

   1. +-commutative 29.8%

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

   2. flip-+ 29.8%

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

   3. add-sqr-sqrt 30.8%

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

   4. associate-*l* 30.8%

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

   5. cancel-sign-sub-inv 30.8%

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

   6. metadata-eval 30.8%

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

   7. sqr-neg 30.8%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 0.8%

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

   10. sqr-neg 0.8%

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

   11. sqrt-prod 1.5%

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

   12. add-sqr-sqrt 0.8%

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

   13. unsub-neg 0.8%

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

   14. +-commutative 0.8%

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

4. Applied egg-rr 30.8%

   \[\leadsto \frac{\color{blue}{\frac{\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right) - b \cdot b}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}}{2 \cdot a} \]
5. Step-by-step derivation

   1. +-commutative 30.8%

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

   2. associate--l+ 99.4%

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

   3. *-commutative 99.4%

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

   4. +-inverses 99.4%

      \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -4 + \color{blue}{0}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. mul0-rgt 99.4%

      \[\leadsto \frac{\frac{\left(a \cdot c\right) \cdot -4 + \color{blue}{\left(a \cdot c\right) \cdot 0}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   6. distribute-lft-out 99.4%

      \[\leadsto \frac{\frac{\color{blue}{\left(a \cdot c\right) \cdot \left(-4 + 0\right)}}{b + {\left(b \cdot b + -4 \cdot \left(a \cdot c\right)\right)}^{0.5}}}{2 \cdot a} \]

   7. metadata-eval 99.4%

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

   8. +-commutative 99.4%

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

   9. *-commutative 99.4%

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

6. Simplified 99.4%

   \[\leadsto \frac{\color{blue}{\frac{\left(a \cdot c\right) \cdot -4}{b + {\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}^{0.5}}}}{2 \cdot a} \]
7. Step-by-step derivation

   1. div-inv 99.2%

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

   2. associate-/l* 99.2%

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

   3. *-commutative 99.2%

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

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\left(a \cdot c\right) \cdot \frac{-4}{b + {\left(\left(a \cdot c\right) \cdot -4 + b \cdot b\right)}^{0.5}}\right) \cdot \frac{1}{a \cdot 2}} \]
9. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. associate-*r/ 99.2%

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

   3. times-frac 99.4%

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

   4. *-lft-identity 99.4%

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

   5. associate-/r* 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

   8. times-frac 99.7%

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

   9. metadata-eval 99.7%

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

   10. associate-*l* 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in a around 0 91.2%

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

    1. +-commutative 91.3%

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

    2. *-commutative 91.3%

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

    3. associate-*r/ 91.3%

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

    4. *-commutative 91.3%

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

    5. *-commutative 91.3%

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

    6. associate-*r* 91.3%

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

    7. associate-/l* 91.3%

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

    8. *-commutative 91.3%

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

    9. associate-*r/ 91.3%

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

13. Simplified 91.2%

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

14. Taylor expanded in a around inf 1.6%

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

Reproduce

herbie shell --seed 2024098 
(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)))