Quadratic roots, medium range

Percentage Accurate: 31.5% → 95.6%

Time: 12.9s

Alternatives: 7

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: 95.6% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (-
   (-
    (*
     -0.25
     (/
      (+ (* 16.0 (* (pow a 4.0) (pow c 4.0))) (* 4.0 (pow (* a c) 4.0)))
      (* 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 * (((16.0 * (pow(a, 4.0) * pow(c, 4.0))) + (4.0 * pow((a * c), 4.0))) / (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) * (((16.0d0 * ((a ** 4.0d0) * (c ** 4.0d0))) + (4.0d0 * ((a * c) ** 4.0d0))) / (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 * (((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)))) - ((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 * (((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)))) - ((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(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(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 * (((16.0 * ((a ^ 4.0) * (c ^ 4.0))) + (4.0 * ((a * c) ^ 4.0))) / (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[(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[(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 32.8%

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

   1. *-commutative 32.8%

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

3. Simplified 32.8%

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

4. Taylor expanded in b around inf 95.0%

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

   1. unpow2 95.0%

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

   2. *-commutative 95.0%

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

   3. *-commutative 95.0%

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

   4. swap-sqr 95.0%

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

   5. pow-prod-down 95.0%

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

   6. pow-prod-down 95.0%

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

   7. pow-sqr 95.0%

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

   8. metadata-eval 95.0%

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

   9. metadata-eval 95.0%

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

6. Applied egg-rr 95.0%

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

7. Final simplification 95.0%

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

Alternative 2: 95.6% accurate, 0.2× speedup

Math

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

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 32.8%

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

   1. *-commutative 32.8%

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

3. Simplified 32.8%

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

4. Taylor expanded in b around inf 95.0%

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

   1. unpow2 95.0%

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

   2. *-commutative 95.0%

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

   3. *-commutative 95.0%

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

   4. swap-sqr 95.0%

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

   5. pow-prod-down 95.0%

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

   6. pow-prod-down 95.0%

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

   7. pow-sqr 95.0%

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

   8. metadata-eval 95.0%

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

   9. metadata-eval 95.0%

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

6. Applied egg-rr 95.0%

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

   1. *-commutative 89.8%

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

   2. unpow3 89.8%

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

   3. times-frac 89.8%

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

   4. unpow2 89.8%

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

   5. frac-times 89.8%

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

   6. pow1 89.8%

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

   7. metadata-eval 89.8%

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

   8. pow1 89.8%

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

   9. metadata-eval 89.8%

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

   10. pow-sqr 89.8%

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

   11. metadata-eval 89.8%

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

   12. metadata-eval 89.8%

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

8. Applied egg-rr 95.0%

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

   1. unpow2 95.0%

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

   2. clear-num 95.0%

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

   3. un-div-inv 95.0%

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

10. Applied egg-rr 95.0%

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

11. Final simplification 95.0%

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

Alternative 3: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -16:\\ \;\;\;\;\frac{\frac{t_0 - {b}^{2}}{b + \sqrt{t_0}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* a (* c -4.0)))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0)) -16.0)
     (/ (/ (- t_0 (pow b 2.0)) (+ b (sqrt t_0))) (* a 2.0))
     (- (/ (- c) b) (* (/ a b) (pow (/ c b) 2.0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, (a * (c * -4.0)));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)) <= -16.0) {
		tmp = ((t_0 - pow(b, 2.0)) / (b + sqrt(t_0))) / (a * 2.0);
	} else {
		tmp = (-c / b) - ((a / b) * pow((c / b), 2.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(a * Float64(c * -4.0)))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0)) <= -16.0)
		tmp = Float64(Float64(Float64(t_0 - (b ^ 2.0)) / Float64(b + sqrt(t_0))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / b) * (Float64(c / b) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -16.0], N[(N[(N[(t$95$0 - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -16

   1. Initial program 75.4%

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

      1. *-commutative 75.4%

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

   3. Simplified 75.4%

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

      1. associate-*l* 75.4%

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

      2. cancel-sign-sub-inv 75.4%

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

      3. metadata-eval 75.4%

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

      4. associate-*l* 75.4%

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

      5. *-commutative 75.4%

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

      6. *-commutative 75.4%

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

      7. fma-udef 75.2%

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

      8. add-cube-cbrt 73.0%

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

      9. pow3 73.0%

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

      10. *-commutative 73.0%

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

      11. associate-*r* 73.0%

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

   5. Applied egg-rr 73.0%

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

      1. +-commutative 73.0%

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

      2. flip-+ 73.2%

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

      3. add-sqr-sqrt 73.0%

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

      4. rem-cube-cbrt 75.8%

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

      5. *-commutative 75.8%

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

      6. sqr-neg 75.8%

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

      7. pow1 75.8%

         \[\leadsto \frac{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - \color{blue}{{b}^{1}} \cdot b}{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)}\right)}^{3}} - \left(-b\right)}}{a \cdot 2} \]

      8. metadata-eval 75.8%

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

      9. pow1 75.8%

         \[\leadsto \frac{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - {b}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{{b}^{1}}}{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)}\right)}^{3}} - \left(-b\right)}}{a \cdot 2} \]

      10. metadata-eval 75.8%

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

      11. pow-sqr 75.8%

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

      12. metadata-eval 75.8%

          \[\leadsto \frac{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - {b}^{\left(2 \cdot \color{blue}{1}\right)}}{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)}\right)}^{3}} - \left(-b\right)}}{a \cdot 2} \]

      13. metadata-eval 75.8%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 1.6%

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

      16. sqr-neg 1.6%

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

   7. Applied egg-rr 75.7%

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

   if -16 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 25.3%

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

      1. *-commutative 25.3%

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

   3. Simplified 25.3%

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

   4. Taylor expanded in b around inf 94.1%

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

      1. distribute-lft-out 94.1%

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

   6. Simplified 94.1%

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

      1. *-commutative 94.1%

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

      2. unpow3 94.1%

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

      3. times-frac 94.1%

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

      4. unpow2 94.1%

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

      5. frac-times 94.1%

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

      6. pow1 94.1%

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

      7. metadata-eval 94.1%

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

      8. pow1 94.1%

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

      9. metadata-eval 94.1%

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

      10. pow-sqr 94.1%

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

      11. metadata-eval 94.1%

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

      12. metadata-eval 94.1%

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

   8. Applied egg-rr 94.1%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -16:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \]

Alternative 4: 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 32.8%

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

   1. *-commutative 32.8%

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

3. Simplified 32.8%

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

4. Taylor expanded in b around inf 93.2%

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

5. Final simplification 93.2%

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

Alternative 5: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t_0 \leq -16:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))))
   (if (<= t_0 -16.0) t_0 (- (/ (- c) b) (* (/ a b) (pow (/ c b) 2.0))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -16.0) {
		tmp = t_0;
	} else {
		tmp = (-c / b) - ((a / b) * pow((c / b), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    if (t_0 <= (-16.0d0)) then
        tmp = t_0
    else
        tmp = (-c / b) - ((a / b) * ((c / b) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -16.0) {
		tmp = t_0;
	} else {
		tmp = (-c / b) - ((a / b) * Math.pow((c / b), 2.0));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -16.0:
		tmp = t_0
	else:
		tmp = (-c / b) - ((a / b) * math.pow((c / b), 2.0))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -16.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(a / b) * (Float64(c / b) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -16.0)
		tmp = t_0;
	else
		tmp = (-c / b) - ((a / b) * ((c / b) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -16.0], t$95$0, N[(N[((-c) / b), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) < -16

   1. Initial program 75.4%

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

   if -16 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 25.3%

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

      1. *-commutative 25.3%

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

   3. Simplified 25.3%

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

   4. Taylor expanded in b around inf 94.1%

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

      1. distribute-lft-out 94.1%

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

   6. Simplified 94.1%

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

      1. *-commutative 94.1%

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

      2. unpow3 94.1%

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

      3. times-frac 94.1%

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

      4. unpow2 94.1%

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

      5. frac-times 94.1%

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

      6. pow1 94.1%

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

      7. metadata-eval 94.1%

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

      8. pow1 94.1%

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

      9. metadata-eval 94.1%

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

      10. pow-sqr 94.1%

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

      11. metadata-eval 94.1%

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

      12. metadata-eval 94.1%

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

   8. Applied egg-rr 94.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2} \leq -16:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{a}{b} \cdot {\left(\frac{c}{b}\right)}^{2}\\ \end{array} \]

Alternative 6: 90.9% 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 32.8%

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

   1. *-commutative 32.8%

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

3. Simplified 32.8%

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

4. Taylor expanded in b around inf 89.8%

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

   1. distribute-lft-out 89.8%

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

6. Simplified 89.8%

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

   1. *-commutative 89.8%

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

   2. unpow3 89.8%

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

   3. times-frac 89.8%

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

   4. unpow2 89.8%

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

   5. frac-times 89.8%

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

   6. pow1 89.8%

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

   7. metadata-eval 89.8%

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

   8. pow1 89.8%

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

   9. metadata-eval 89.8%

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

   10. pow-sqr 89.8%

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

   11. metadata-eval 89.8%

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

   12. metadata-eval 89.8%

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

8. Applied egg-rr 89.8%

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

9. Final simplification 89.8%

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

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 32.8%

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

   1. *-commutative 32.8%

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

3. Simplified 32.8%

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

4. Taylor expanded in b around inf 79.9%

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

   1. mul-1-neg 79.9%

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

6. Simplified 79.9%

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

7. Final simplification 79.9%

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

Reproduce

herbie shell --seed 2023305 
(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)))