Quadratic roots, medium range

Percentage Accurate: 31.5% → 95.4%

Time: 13.7s

Alternatives: 8

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: 95.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{20}{a \cdot {b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (-
   (-
    (* -0.25 (* (pow (* a c) 4.0) (/ 20.0 (* a (pow b 7.0)))))
    (/ (* a (pow c 2.0)) (pow b 3.0)))
   (/ c b))))

C

double code(double a, double b, double c) {
	return (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + (((-0.25 * (pow((a * c), 4.0) * (20.0 / (a * pow(b, 7.0))))) - ((a * pow(c, 2.0)) / pow(b, 3.0))) - (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) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + ((((-0.25d0) * (((a * c) ** 4.0d0) * (20.0d0 / (a * (b ** 7.0d0))))) - ((a * (c ** 2.0d0)) / (b ** 3.0d0))) - (c / b))
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + (((-0.25 * (Math.pow((a * c), 4.0) * (20.0 / (a * Math.pow(b, 7.0))))) - ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) - (c / b));
}

Python

def code(a, b, c):
	return (-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + (((-0.25 * (math.pow((a * c), 4.0) * (20.0 / (a * math.pow(b, 7.0))))) - ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) - (c / b))

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(Float64(-0.25 * Float64((Float64(a * c) ^ 4.0) * Float64(20.0 / Float64(a * (b ^ 7.0))))) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) - Float64(c / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + (((-0.25 * (((a * c) ^ 4.0) * (20.0 / (a * (b ^ 7.0))))) - ((a * (c ^ 2.0)) / (b ^ 3.0))) - (c / b));
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.25 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.4%

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

   1. *-commutative 31.4%

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

3. Simplified 31.4%

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

4. Taylor expanded in b around inf 95.4%

   \[\leadsto \color{blue}{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]
5. Step-by-step derivation

   1. unpow2 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{\left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

   2. *-commutative 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \left(-2 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot -2\right)}}{a \cdot {b}^{7}}\right)\right) \]

   3. *-commutative 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot -2\right)} \cdot \left(\left({a}^{2} \cdot {c}^{2}\right) \cdot -2\right)}{a \cdot {b}^{7}}\right)\right) \]

   4. swap-sqr 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(-2 \cdot -2\right)}}{a \cdot {b}^{7}}\right)\right) \]

   5. pow-prod-down 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \left(\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(-2 \cdot -2\right)}{a \cdot {b}^{7}}\right)\right) \]

   6. pow-prod-down 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \left({\left(a \cdot c\right)}^{2} \cdot \color{blue}{{\left(a \cdot c\right)}^{2}}\right) \cdot \left(-2 \cdot -2\right)}{a \cdot {b}^{7}}\right)\right) \]

   7. pow-sqr 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{{\left(a \cdot c\right)}^{\left(2 \cdot 2\right)}} \cdot \left(-2 \cdot -2\right)}{a \cdot {b}^{7}}\right)\right) \]

   8. metadata-eval 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(a \cdot c\right)}^{\color{blue}{4}} \cdot \left(-2 \cdot -2\right)}{a \cdot {b}^{7}}\right)\right) \]

   9. metadata-eval 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(a \cdot c\right)}^{4} \cdot \color{blue}{4}}{a \cdot {b}^{7}}\right)\right) \]

6. Applied egg-rr 95.4%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + \color{blue}{{\left(a \cdot c\right)}^{4} \cdot 4}}{a \cdot {b}^{7}}\right)\right) \]
7. Step-by-step derivation

   1. pow-prod-down 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{{\left(a \cdot c\right)}^{4}} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   2. metadata-eval 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot {\left(a \cdot c\right)}^{\color{blue}{\left(2 \cdot 2\right)}} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   3. pow-sqr 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{2} \cdot {\left(a \cdot c\right)}^{2}\right)} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   4. pow-prod-down 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left(\color{blue}{\left({a}^{2} \cdot {c}^{2}\right)} \cdot {\left(a \cdot c\right)}^{2}\right) + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   5. pow-prod-down 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left(\left({a}^{2} \cdot {c}^{2}\right) \cdot \color{blue}{\left({a}^{2} \cdot {c}^{2}\right)}\right) + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   6. pow1 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left(\color{blue}{{\left({a}^{2} \cdot {c}^{2}\right)}^{1}} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   7. metadata-eval 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({\left({a}^{2} \cdot {c}^{2}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   8. pow1 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({\left({a}^{2} \cdot {c}^{2}\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{\left({a}^{2} \cdot {c}^{2}\right)}^{1}}\right) + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   9. metadata-eval 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({\left({a}^{2} \cdot {c}^{2}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left({a}^{2} \cdot {c}^{2}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right) + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   10. sqr-pow 95.4%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{{\left({a}^{2} \cdot {c}^{2}\right)}^{2}} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   11. pow-prod-down 95.4%

       \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot {\color{blue}{\left({\left(a \cdot c\right)}^{2}\right)}}^{2} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

8. Applied egg-rr 95.4%

   \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{{\left({\left(a \cdot c\right)}^{2}\right)}^{2}} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]
9. Step-by-step derivation

   1. unpow2 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{2} \cdot {\left(a \cdot c\right)}^{2}\right)} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   2. metadata-eval 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({\left(a \cdot c\right)}^{\color{blue}{\left(\frac{4}{2}\right)}} \cdot {\left(a \cdot c\right)}^{2}\right) + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   3. metadata-eval 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \left({\left(a \cdot c\right)}^{\left(\frac{4}{2}\right)} \cdot {\left(a \cdot c\right)}^{\color{blue}{\left(\frac{4}{2}\right)}}\right) + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

   4. sqr-pow 95.4%

      \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{{\left(a \cdot c\right)}^{4}} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

10. Simplified 95.4%

    \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \frac{16 \cdot \color{blue}{{\left(a \cdot c\right)}^{4}} + {\left(a \cdot c\right)}^{4} \cdot 4}{a \cdot {b}^{7}}\right)\right) \]

11. Taylor expanded in c around 0 95.4%

    \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\frac{{c}^{4} \cdot \left(4 \cdot {a}^{4} + 16 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
12. Step-by-step derivation

13. Simplified 95.4%

    \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.25 \cdot \color{blue}{\left({\left(a \cdot c\right)}^{4} \cdot \frac{20}{a \cdot {b}^{7}}\right)}\right)\right) \]

14. Final simplification 95.4%

    \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \left({\left(a \cdot c\right)}^{4} \cdot \frac{20}{a \cdot {b}^{7}}\right) - \frac{a \cdot {c}^{2}}{{b}^{3}}\right) - \frac{c}{b}\right) \]

Alternative 2: 93.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+ (/ c b) (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) - ((c / b) + ((a * pow(c, 2.0)) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-2.0d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) - ((c / b) + ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-2.0 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) - ((c / b) + ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-2.0 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) - ((c / b) + ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) - Float64(Float64(c / b) + Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-2.0 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) - ((c / b) + ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] + N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.4%

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

   1. *-commutative 31.4%

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

3. Simplified 31.4%

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

4. Taylor expanded in b around inf 93.8%

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

5. Final simplification 93.8%

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

Alternative 3: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{-c}{b} - a \cdot \frac{{c}^{2}}{{b}^{3}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return (-c / b) - (a * (pow(c, 2.0) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-c / b) - (a * ((c ** 2.0d0) / (b ** 3.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.4%

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

   1. *-commutative 31.4%

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

3. Simplified 31.4%

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

4. Taylor expanded in b around inf 90.1%

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

   1. distribute-lft-out 90.1%

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

   2. associate-/l* 90.1%

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

   3. associate-/l* 90.1%

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

6. Simplified 90.1%

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

7. Taylor expanded in a around 0 90.4%

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

   1. associate-*r/ 90.4%

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

   2. mul-1-neg 90.4%

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

   3. mul-1-neg 90.4%

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

   4. distribute-neg-out 90.4%

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

9. Simplified 90.4%

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

10. Final simplification 90.4%

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

Alternative 4: 90.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.4%

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

   1. *-commutative 31.4%

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

3. Simplified 31.4%

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

4. Taylor expanded in b around inf 90.1%

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

   1. distribute-lft-out 90.1%

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

   2. associate-/l* 90.1%

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

   3. associate-/l* 90.1%

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

6. Simplified 90.1%

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

   1. associate-/l* 90.1%

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

   2. add-sqr-sqrt 90.1%

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

   3. unpow3 90.1%

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

   4. times-frac 90.1%

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

   5. sqrt-prod 90.1%

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

   6. unpow2 90.1%

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

   7. sqrt-prod 90.1%

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

   8. add-sqr-sqrt 90.1%

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

   9. unpow2 90.1%

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

   10. sqrt-prod 90.1%

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

   11. add-sqr-sqrt 90.1%

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

   12. pow1 90.1%

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

   13. metadata-eval 90.1%

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

   14. pow1 90.1%

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

   15. metadata-eval 90.1%

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

   16. pow-sqr 90.1%

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

   17. metadata-eval 90.1%

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

   18. metadata-eval 90.1%

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

8. Applied egg-rr 90.1%

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

   1. div-inv 90.0%

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

   2. associate-/r/ 90.0%

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

   3. *-commutative 90.0%

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

   4. associate-*l* 90.0%

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

10. Applied egg-rr 90.1%

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

    1. associate-*r* 90.1%

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

    2. *-commutative 90.1%

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

    3. associate-*r* 90.1%

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

    4. associate-*l* 90.1%

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

    5. associate-*r/ 90.1%

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

    6. *-commutative 90.1%

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

    7. associate-*r/ 90.1%

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

    8. associate-*r/ 90.1%

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

    9. metadata-eval 90.1%

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

12. Simplified 90.1%

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

13. Final simplification 90.1%

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

Alternative 5: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.4%

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

   1. *-commutative 31.4%

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

3. Simplified 31.4%

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

4. Taylor expanded in b around inf 90.1%

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

   1. distribute-lft-out 90.1%

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

   2. associate-/l* 90.1%

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

   3. associate-/l* 90.1%

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

6. Simplified 90.1%

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

   1. associate-/l* 90.1%

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

   2. add-sqr-sqrt 90.1%

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

   3. unpow3 90.1%

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

   4. times-frac 90.1%

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

   5. sqrt-prod 90.1%

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

   6. unpow2 90.1%

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

   7. sqrt-prod 90.1%

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

   8. add-sqr-sqrt 90.1%

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

   9. unpow2 90.1%

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

   10. sqrt-prod 90.1%

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

   11. add-sqr-sqrt 90.1%

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

   12. pow1 90.1%

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

   13. metadata-eval 90.1%

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

   14. pow1 90.1%

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

   15. metadata-eval 90.1%

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

   16. pow-sqr 90.1%

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

   17. metadata-eval 90.1%

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

   18. metadata-eval 90.1%

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

8. Applied egg-rr 90.1%

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

   1. *-commutative 90.1%

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

   2. associate-/r/ 90.1%

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

   3. *-commutative 90.1%

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

   4. times-frac 90.1%

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

10. Applied egg-rr 90.3%

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

11. Final simplification 90.3%

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

Alternative 6: 90.4% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.4%

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

   1. *-commutative 31.4%

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

3. Simplified 31.4%

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

4. Taylor expanded in b around inf 90.1%

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

   1. distribute-lft-out 90.1%

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

   2. associate-/l* 90.1%

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

   3. associate-/l* 90.1%

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

6. Simplified 90.1%

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

   1. associate-/l* 90.1%

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

   2. add-sqr-sqrt 90.1%

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

   3. unpow3 90.1%

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

   4. times-frac 90.1%

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

   5. sqrt-prod 90.1%

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

   6. unpow2 90.1%

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

   7. sqrt-prod 90.1%

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

   8. add-sqr-sqrt 90.1%

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

   9. unpow2 90.1%

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

   10. sqrt-prod 90.1%

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

   11. add-sqr-sqrt 90.1%

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

   12. pow1 90.1%

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

   13. metadata-eval 90.1%

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

   14. pow1 90.1%

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

   15. metadata-eval 90.1%

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

   16. pow-sqr 90.1%

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

   17. metadata-eval 90.1%

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

   18. metadata-eval 90.1%

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

8. Applied egg-rr 90.1%

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

   1. *-commutative 90.1%

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

   2. unpow2 90.1%

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

   3. times-frac 90.1%

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

10. Applied egg-rr 90.1%

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

11. Final simplification 90.1%

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

Alternative 7: 81.3% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.4%

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

   1. *-commutative 31.4%

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

3. Simplified 31.4%

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

4. Taylor expanded in b around inf 81.2%

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

   1. mul-1-neg 81.2%

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

6. Simplified 81.2%

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

7. Final simplification 81.2%

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

Alternative 8: 1.6% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.4%

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

   1. *-commutative 31.4%

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

3. Simplified 31.4%

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

4. Taylor expanded in b around inf 21.0%

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

   1. associate-/l* 21.0%

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

   2. associate-*r/ 21.0%

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

   3. *-commutative 21.0%

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

6. Simplified 21.0%

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

7. Taylor expanded in b around 0 80.9%

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

   1. associate-*r/ 81.1%

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

   2. associate-*l* 81.1%

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

   3. *-commutative 81.1%

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

   4. associate-*l/ 80.9%

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

   5. associate-/l* 80.9%

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

   6. *-commutative 80.9%

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

9. Simplified 80.9%

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

    1. div-inv 80.8%

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

    2. *-un-lft-identity 80.8%

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

    3. times-frac 80.8%

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

    4. /-rgt-identity 80.8%

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

    5. clear-num 80.9%

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

    6. *-un-lft-identity 80.9%

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

    7. times-frac 80.9%

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

    8. /-rgt-identity 80.9%

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

    9. add-sqr-sqrt 80.6%

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

    10. sqrt-unprod 80.9%

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

    11. swap-sqr 80.9%

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

    12. metadata-eval 80.9%

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

    13. metadata-eval 80.9%

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

    14. swap-sqr 80.9%

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

    15. sqrt-unprod 0.0%

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

    16. add-sqr-sqrt 1.6%

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

11. Applied egg-rr 1.6%

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

    1. associate-*r/ 1.6%

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

    2. associate-*r/ 1.6%

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

    3. associate-*r/ 1.6%

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

    4. associate-*l/ 1.6%

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

    5. /-rgt-identity 1.6%

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

    6. associate-*r/ 1.6%

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

    7. /-rgt-identity 1.6%

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

    8. *-inverses 1.6%

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

    9. *-rgt-identity 1.6%

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

13. Simplified 1.6%

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

14. Final simplification 1.6%

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

Reproduce

herbie shell --seed 2023336 
(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)))