Quadratic roots, medium range

Percentage Accurate: 31.5% → 95.7%

Time: 14.3s

Alternatives: 5

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: 95.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 4 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} - \frac{\frac{c}{b}}{\frac{b}{c}} \cdot \frac{a}{b}\right) - \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (-
   (-
    (*
     -0.25
     (/
      (+ (* 16.0 (* (pow a 4.0) (pow c 4.0))) (* 4.0 (pow (* a c) 4.0)))
      (* a (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 * (((16.0 * (pow(a, 4.0) * pow(c, 4.0))) + (4.0 * pow((a * c), 4.0))) / (a * 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) * (((16.0d0 * ((a ** 4.0d0) * (c ** 4.0d0))) + (4.0d0 * ((a * c) ** 4.0d0))) / (a * (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 * (((16.0 * (Math.pow(a, 4.0) * Math.pow(c, 4.0))) + (4.0 * Math.pow((a * c), 4.0))) / (a * 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 * (((16.0 * (math.pow(a, 4.0) * math.pow(c, 4.0))) + (4.0 * math.pow((a * c), 4.0))) / (a * 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(16.0 * Float64((a ^ 4.0) * (c ^ 4.0))) + Float64(4.0 * (Float64(a * c) ^ 4.0))) / Float64(a * (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 * (((16.0 * ((a ^ 4.0) * (c ^ 4.0))) + (4.0 * ((a * c) ^ 4.0))) / (a * (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[(16.0 * N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $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 30.0%

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

   1. *-commutative 30.0%

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

3. Simplified 30.0%

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

5. Taylor expanded in b around inf 95.2%

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

   1. *-commutative 95.2%

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

   2. unpow-prod-down 95.2%

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

   3. pow-prod-down 95.2%

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

   4. pow-pow 95.2%

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

   5. metadata-eval 95.2%

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

   6. metadata-eval 95.2%

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

7. Applied egg-rr 95.2%

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

   1. *-commutative 95.2%

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

   2. unpow3 95.2%

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

   3. times-frac 95.2%

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

   4. unpow2 95.2%

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

   5. frac-times 95.2%

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

   6. pow2 95.2%

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

9. Applied egg-rr 95.2%

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

    1. unpow2 95.2%

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

    2. clear-num 95.2%

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

    3. un-div-inv 95.2%

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

11. Applied egg-rr 95.2%

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

12. Final simplification 95.2%

    \[\leadsto -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(\left(-0.25 \cdot \frac{16 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 4 \cdot {\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} - \frac{\frac{c}{b}}{\frac{b}{c}} \cdot \frac{a}{b}\right) - \frac{c}{b}\right) \]
13. Add Preprocessing

Alternative 2: 94.1% 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 30.0%

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

   1. *-commutative 30.0%

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

3. Simplified 30.0%

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

5. Taylor expanded in b around inf 93.6%

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

6. Final simplification 93.6%

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

Alternative 3: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \frac{\frac{a}{\frac{b}{c}}}{b} \cdot \frac{a \cdot c}{b}\right)\right)}{a \cdot 2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -4.0
   (/ (pow (* a c) 3.0) (pow b 5.0))
   (* -2.0 (fma a (/ c b) (* (/ (/ a (/ b c)) b) (/ (* a c) b)))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return fma(-4.0, (pow((a * c), 3.0) / pow(b, 5.0)), (-2.0 * fma(a, (c / b), (((a / (b / c)) / b) * ((a * c) / b))))) / (a * 2.0);
}

Julia

function code(a, b, c)
	return Float64(fma(-4.0, Float64((Float64(a * c) ^ 3.0) / (b ^ 5.0)), Float64(-2.0 * fma(a, Float64(c / b), Float64(Float64(Float64(a / Float64(b / c)) / b) * Float64(Float64(a * c) / b))))) / Float64(a * 2.0))
end

Wolfram

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

Derivation

1. Initial program 30.0%

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

   1. *-commutative 30.0%

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

3. Simplified 30.0%

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

5. Taylor expanded in b around inf 93.3%

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

   1. fma-define 93.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{5}}, -2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{a \cdot 2} \]

   2. cube-prod 93.3%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{\color{blue}{{\left(a \cdot c\right)}^{3}}}{{b}^{5}}, -2 \cdot \frac{a \cdot c}{b} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{a \cdot 2} \]

   3. distribute-lft-out 93.3%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, \color{blue}{-2 \cdot \left(\frac{a \cdot c}{b} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}\right)}{a \cdot 2} \]

   4. associate-/l* 93.5%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \left(\color{blue}{a \cdot \frac{c}{b}} + \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)\right)}{a \cdot 2} \]

   5. fma-define 93.5%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \color{blue}{\mathsf{fma}\left(a, \frac{c}{b}, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}\right)}{a \cdot 2} \]

7. Simplified 93.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)\right)}}{a \cdot 2} \]
8. Step-by-step derivation

   1. pow-prod-down 93.5%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \frac{\color{blue}{{\left(a \cdot c\right)}^{2}}}{{b}^{3}}\right)\right)}{a \cdot 2} \]

   2. pow2 93.5%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \frac{\color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}}{{b}^{3}}\right)\right)}{a \cdot 2} \]

   3. unpow3 93.5%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \frac{\left(a \cdot c\right) \cdot \left(a \cdot c\right)}{\color{blue}{\left(b \cdot b\right) \cdot b}}\right)\right)}{a \cdot 2} \]

   4. times-frac 93.5%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\frac{a \cdot c}{b \cdot b} \cdot \frac{a \cdot c}{b}}\right)\right)}{a \cdot 2} \]

   5. pow2 93.5%

      \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \frac{a \cdot c}{\color{blue}{{b}^{2}}} \cdot \frac{a \cdot c}{b}\right)\right)}{a \cdot 2} \]

9. Applied egg-rr 93.5%

   \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\frac{a \cdot c}{{b}^{2}} \cdot \frac{a \cdot c}{b}}\right)\right)}{a \cdot 2} \]
10. Step-by-step derivation

    1. *-commutative 93.5%

       \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \frac{\color{blue}{c \cdot a}}{{b}^{2}} \cdot \frac{a \cdot c}{b}\right)\right)}{a \cdot 2} \]

    2. unpow2 93.5%

       \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \frac{c \cdot a}{\color{blue}{b \cdot b}} \cdot \frac{a \cdot c}{b}\right)\right)}{a \cdot 2} \]

    3. times-frac 93.5%

       \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\left(\frac{c}{b} \cdot \frac{a}{b}\right)} \cdot \frac{a \cdot c}{b}\right)\right)}{a \cdot 2} \]

11. Applied egg-rr 93.5%

    \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\left(\frac{c}{b} \cdot \frac{a}{b}\right)} \cdot \frac{a \cdot c}{b}\right)\right)}{a \cdot 2} \]
12. Step-by-step derivation

    1. *-commutative 93.5%

       \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\left(\frac{a}{b} \cdot \frac{c}{b}\right)} \cdot \frac{a \cdot c}{b}\right)\right)}{a \cdot 2} \]

    2. associate-*r/ 93.5%

       \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\frac{\frac{a}{b} \cdot c}{b}} \cdot \frac{a \cdot c}{b}\right)\right)}{a \cdot 2} \]

    3. associate-/r/ 93.5%

       \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \frac{\color{blue}{\frac{a}{\frac{b}{c}}}}{b} \cdot \frac{a \cdot c}{b}\right)\right)}{a \cdot 2} \]

13. Simplified 93.5%

    \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \color{blue}{\frac{\frac{a}{\frac{b}{c}}}{b}} \cdot \frac{a \cdot c}{b}\right)\right)}{a \cdot 2} \]

14. Final simplification 93.5%

    \[\leadsto \frac{\mathsf{fma}\left(-4, \frac{{\left(a \cdot c\right)}^{3}}{{b}^{5}}, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b}, \frac{\frac{a}{\frac{b}{c}}}{b} \cdot \frac{a \cdot c}{b}\right)\right)}{a \cdot 2} \]
15. Add Preprocessing

Alternative 4: 91.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.0%

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

   1. *-commutative 30.0%

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

3. Simplified 30.0%

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

5. Taylor expanded in b around inf 90.8%

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

   1. distribute-lft-out 90.8%

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

   2. associate-/l* 90.8%

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

7. Simplified 90.8%

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

   1. unpow2 90.8%

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

   2. unpow3 90.8%

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

   3. times-frac 90.8%

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

   4. pow2 90.8%

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

9. Applied egg-rr 90.8%

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

10. Final simplification 90.8%

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

Alternative 5: 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(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 30.0%

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

   1. *-commutative 30.0%

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

3. Simplified 30.0%

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

5. Taylor expanded in b around inf 82.2%

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

   1. mul-1-neg 82.2%

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

   2. distribute-neg-frac 82.2%

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

7. Simplified 82.2%

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

8. Final simplification 82.2%

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

Reproduce

herbie shell --seed 2024040 
(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)))