Quadratic roots, wide range

Percentage Accurate: 17.8% → 97.6%

Time: 11.0s

Alternatives: 4

Speedup: 29.0×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.2× speedup

Math

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

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) (/ a 20.0)) (pow b 7.0)))
    (* (/ (/ c b) (/ b c)) (/ a b)))
   (/ 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) / (a / 20.0)) / pow(b, 7.0))) - (((c / b) / (b / c)) * (a / b))) - (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) / (a / 20.0d0)) / (b ** 7.0d0))) - (((c / b) / (b / c)) * (a / b))) - (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) / (a / 20.0)) / Math.pow(b, 7.0))) - (((c / b) / (b / c)) * (a / b))) - (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) / (a / 20.0)) / math.pow(b, 7.0))) - (((c / b) / (b / c)) * (a / b))) - (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((Float64(a * c) ^ 4.0) / Float64(a / 20.0)) / (b ^ 7.0))) - Float64(Float64(Float64(c / b) / Float64(b / c)) * Float64(a / b))) - 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) / (a / 20.0)) / (b ^ 7.0))) - (((c / b) / (b / c)) * (a / b))) - (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[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[(a / 20.0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c / b), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.9%

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

   1. *-commutative 16.9%

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

3. Simplified 16.9%

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

4. Taylor expanded in b around inf 98.0%

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

5. Taylor expanded in c around 0 98.0%

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

   1. associate-/r* 98.0%

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

   2. distribute-rgt-in 98.0%

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

   3. associate-*r* 98.0%

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

   4. associate-*r* 98.0%

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

7. Simplified 98.0%

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

   1. *-commutative 98.0%

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

   2. unpow3 98.0%

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

   3. times-frac 98.0%

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

   4. unpow2 98.0%

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

   5. frac-times 98.0%

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

   6. pow1 98.0%

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

   7. metadata-eval 98.0%

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

   8. pow1 98.0%

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

   9. metadata-eval 98.0%

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

   10. pow-sqr 98.0%

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

   11. metadata-eval 98.0%

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

   12. metadata-eval 98.0%

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

9. Applied egg-rr 98.0%

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

    1. unpow2 98.0%

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

    2. clear-num 98.0%

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

    3. un-div-inv 98.0%

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

11. Applied egg-rr 98.0%

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

12. Final simplification 98.0%

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

Alternative 2: 96.8% 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 16.9%

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

   1. *-commutative 16.9%

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

3. Simplified 16.9%

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

4. Taylor expanded in b around inf 97.3%

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

5. Final simplification 97.3%

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

Alternative 3: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.9%

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

   1. *-commutative 16.9%

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

3. Simplified 16.9%

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

4. Taylor expanded in b around inf 96.0%

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

   1. distribute-lft-out 96.0%

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

6. Simplified 96.0%

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

   1. *-commutative 98.0%

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

   2. unpow3 98.0%

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

   3. times-frac 98.0%

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

   4. unpow2 98.0%

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

   5. frac-times 98.0%

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

   6. pow1 98.0%

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

   7. metadata-eval 98.0%

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

   8. pow1 98.0%

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

   9. metadata-eval 98.0%

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

   10. pow-sqr 98.0%

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

   11. metadata-eval 98.0%

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

   12. metadata-eval 98.0%

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

8. Applied egg-rr 96.0%

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

9. Final simplification 96.0%

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

Alternative 4: 90.4% 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 16.9%

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

   1. *-commutative 16.9%

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

3. Simplified 16.9%

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

4. Taylor expanded in b around inf 91.1%

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

   1. mul-1-neg 91.1%

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

6. Simplified 91.1%

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

7. Final simplification 91.1%

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

Reproduce

herbie shell --seed 2023301 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))