Quadratic roots, narrow range

Percentage Accurate: 55.4% → 92.0%

Time: 5.6s

Alternatives: 9

Speedup: 4.6×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    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: 55.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    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: 92.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -250:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \mathsf{fma}\left(-5, \frac{\left(a \cdot a\right) \cdot c}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2 \cdot \frac{a}{b \cdot b}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -250.0)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 2.0 a))
     (fma
      (/
       (*
        (* c c)
        (-
         (*
          c
          (fma
           -5.0
           (/ (* (* a a) c) (* (* b b) (* b b)))
           (* -2.0 (/ a (* b b)))))
         1.0))
       (* (* b b) b))
      a
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -250.0) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = fma((((c * c) * ((c * fma(-5.0, (((a * a) * c) / ((b * b) * (b * b))), (-2.0 * (a / (b * b))))) - 1.0)) / ((b * b) * b)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -250.0)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = fma(Float64(Float64(Float64(c * c) * Float64(Float64(c * fma(-5.0, Float64(Float64(Float64(a * a) * c) / Float64(Float64(b * b) * Float64(b * b))), Float64(-2.0 * Float64(a / Float64(b * b))))) - 1.0)) / Float64(Float64(b * b) * b)), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -250.0], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * N[(N[(c * N[(-5.0 * N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -250

   1. Initial program 55.4%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 56.9%

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

   if -250 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.4%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 90.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot c}{{b}^{5}}, -2, \frac{\left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot a}{{b}^{7}} \cdot -5\right), -\frac{c \cdot c}{\left(b \cdot b\right) \cdot b}\right), \color{blue}{a}, \frac{-c}{b}\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 90.9%

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

   11. Taylor expanded in c around 0

       \[\leadsto \mathsf{fma}\left(\frac{{c}^{2} \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + -2 \cdot \frac{a}{{b}^{2}}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{c}^{2} \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + -2 \cdot \frac{a}{{b}^{2}}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right) \]

       2. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + -2 \cdot \frac{a}{{b}^{2}}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + -2 \cdot \frac{a}{{b}^{2}}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right) \]

       4. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + -2 \cdot \frac{a}{{b}^{2}}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right) \]

   13. Applied rewrites 90.9%

       \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \mathsf{fma}\left(-5, \frac{\left(a \cdot a\right) \cdot c}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2 \cdot \frac{a}{b \cdot b}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 92.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -250:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(-5 \cdot \left(a \cdot \frac{c}{b \cdot b}\right) - 2\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} \cdot a - \frac{c \cdot c}{b \cdot b}\right) \cdot a - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -250.0)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 2.0 a))
     (/
      (-
       (*
        (-
         (*
          (/
           (* (* (* c c) c) (- (* -5.0 (* a (/ c (* b b)))) 2.0))
           (* (* (* b b) b) b))
          a)
         (/ (* c c) (* b b)))
        a)
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -250.0) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = ((((((((c * c) * c) * ((-5.0 * (a * (c / (b * b)))) - 2.0)) / (((b * b) * b) * b)) * a) - ((c * c) / (b * b))) * a) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -250.0)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(c * c) * c) * Float64(Float64(-5.0 * Float64(a * Float64(c / Float64(b * b)))) - 2.0)) / Float64(Float64(Float64(b * b) * b) * b)) * a) - Float64(Float64(c * c) / Float64(b * b))) * a) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -250.0], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(-5.0 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -250

   1. Initial program 55.4%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 56.9%

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

   if -250 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.4%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b} \]

   7. Applied rewrites 90.9%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 90.9%

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

   11. Taylor expanded in c around 0

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

       1. lower-*.f64 N/A

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

       2. pow3 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. associate-/l* N/A

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

       8. pow2 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 90.9

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

   13. Applied rewrites 90.9%

       \[\leadsto \frac{\left(\frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(-5 \cdot \left(a \cdot \frac{c}{b \cdot b}\right) - 2\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} \cdot a - \frac{c \cdot c}{b \cdot b}\right) \cdot a - c}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -250:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \frac{c}{b \cdot b}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -250.0)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 2.0 a))
     (fma
      (/ (* (* c c) (- (* -2.0 (* a (/ c (* b b)))) 1.0)) (* (* b b) b))
      a
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -250.0) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = fma((((c * c) * ((-2.0 * (a * (c / (b * b)))) - 1.0)) / ((b * b) * b)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -250.0)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = fma(Float64(Float64(Float64(c * c) * Float64(Float64(-2.0 * Float64(a * Float64(c / Float64(b * b)))) - 1.0)) / Float64(Float64(b * b) * b)), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -250.0], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * N[(N[(-2.0 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -250

   1. Initial program 55.4%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 56.9%

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

   if -250 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.4%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 90.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot c}{{b}^{5}}, -2, \frac{\left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot a}{{b}^{7}} \cdot -5\right), -\frac{c \cdot c}{\left(b \cdot b\right) \cdot b}\right), \color{blue}{a}, \frac{-c}{b}\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 90.9%

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

   11. Taylor expanded in c around 0

       \[\leadsto \mathsf{fma}\left(\frac{{c}^{2} \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} - 1\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. pow2 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 87.7

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

   13. Applied rewrites 87.7%

       \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \frac{c}{b \cdot b}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -250:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \frac{c}{b \cdot b}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -250.0)
   (/ (+ (sqrt (fma (* -4.0 a) c (* b b))) (- b)) (+ a a))
   (fma
    (/ (* (* c c) (- (* -2.0 (* a (/ c (* b b)))) 1.0)) (* (* b b) b))
    a
    (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -250.0) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) + -b) / (a + a);
	} else {
		tmp = fma((((c * c) * ((-2.0 * (a * (c / (b * b)))) - 1.0)) / ((b * b) * b)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -250.0)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) + Float64(-b)) / Float64(a + a));
	else
		tmp = fma(Float64(Float64(Float64(c * c) * Float64(Float64(-2.0 * Float64(a * Float64(c / Float64(b * b)))) - 1.0)) / Float64(Float64(b * b) * b)), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -250.0], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * N[(N[(-2.0 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -250

   1. Initial program 55.4%

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

   3. Applied rewrites 55.4%

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

   if -250 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.4%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 90.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot c}{{b}^{5}}, -2, \frac{\left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot a}{{b}^{7}} \cdot -5\right), -\frac{c \cdot c}{\left(b \cdot b\right) \cdot b}\right), \color{blue}{a}, \frac{-c}{b}\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 90.9%

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

   11. Taylor expanded in c around 0

       \[\leadsto \mathsf{fma}\left(\frac{{c}^{2} \cdot \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} - 1\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. pow2 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 87.7

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

   13. Applied rewrites 87.7%

       \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(-2 \cdot \left(a \cdot \frac{c}{b \cdot b}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -250:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot a}{b \cdot b} \cdot -2 - c \cdot c}{b \cdot b} \cdot a - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -250.0)
   (/ (+ (sqrt (fma (* -4.0 a) c (* b b))) (- b)) (+ a a))
   (/
    (-
     (* (/ (- (* (/ (* (* (* c c) c) a) (* b b)) -2.0) (* c c)) (* b b)) a)
     c)
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -250.0) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) + -b) / (a + a);
	} else {
		tmp = (((((((((c * c) * c) * a) / (b * b)) * -2.0) - (c * c)) / (b * b)) * a) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -250.0)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) + Float64(-b)) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(c * c) * c) * a) / Float64(b * b)) * -2.0) - Float64(c * c)) / Float64(b * b)) * a) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -250.0], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] - N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -250

   1. Initial program 55.4%

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

   3. Applied rewrites 55.4%

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

   if -250 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.4%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b} \]

   7. Applied rewrites 90.9%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 87.8%

       \[\leadsto \frac{\frac{\frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot a}{b \cdot b} \cdot -2 - c \cdot c}{b \cdot b} \cdot a - c}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -44.85:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-c \cdot c}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -44.85)
   (/ (+ (sqrt (fma (* -4.0 a) c (* b b))) (- b)) (+ a a))
   (fma (/ (- (* c c)) (* (* b b) b)) a (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -44.85) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) + -b) / (a + a);
	} else {
		tmp = fma((-(c * c) / ((b * b) * b)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -44.85)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) + Float64(-b)) / Float64(a + a));
	else
		tmp = fma(Float64(Float64(-Float64(c * c)) / Float64(Float64(b * b) * b)), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -44.85], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[((-N[(c * c), $MachinePrecision]) / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -44.8500000000000014

   1. Initial program 55.4%

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

   3. Applied rewrites 55.4%

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

   if -44.8500000000000014 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.4%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 90.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot c}{{b}^{5}}, -2, \frac{\left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot a}{{b}^{7}} \cdot -5\right), -\frac{c \cdot c}{\left(b \cdot b\right) \cdot b}\right), \color{blue}{a}, \frac{-c}{b}\right) \]

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}}, a, \frac{-c}{b}\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\frac{{c}^{2}}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({c}^{2}\right)}{{b}^{3}}, a, \frac{-c}{b}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left({c}^{2}\right)}{{b}^{3}}, a, \frac{-c}{b}\right) \]

      4. lower-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow3 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-c \cdot c}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-c \cdot c}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right) \]

      9. lift-*.f64 81.5

         \[\leadsto \mathsf{fma}\left(\frac{-c \cdot c}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right) \]

   10. Applied rewrites 81.5%

       \[\leadsto \mathsf{fma}\left(\frac{-c \cdot c}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -44.85:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -44.85)
   (/ (+ (sqrt (fma (* -4.0 a) c (* b b))) (- b)) (+ a a))
   (- (/ (fma (* c c) (/ a (* b b)) c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -44.85) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) + -b) / (a + a);
	} else {
		tmp = -(fma((c * c), (a / (b * b)), c) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -44.85)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) + Float64(-b)) / Float64(a + a));
	else
		tmp = Float64(-Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -44.85], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], (-N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / b), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -44.8500000000000014

   1. Initial program 55.4%

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

   3. Applied rewrites 55.4%

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

   if -44.8500000000000014 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.4%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

   5. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

          \[\leadsto -\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b} \]

      14. lift-*.f64 81.5

          \[\leadsto -\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b} \]

   7. Applied rewrites 81.5%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

4. Applied rewrites 90.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

5. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r/ N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. pow2 N/A

       \[\leadsto -\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b} \]

   14. lift-*.f64 81.5

       \[\leadsto -\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b} \]

7. Applied rewrites 81.5%

   \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}} \]
8. Add Preprocessing

Alternative 9: 64.4% accurate, 4.6× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    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 55.4%

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

2. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(c\right)}{\color{blue}{b}} \]

   4. lower-neg.f64 64.4

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

4. Applied rewrites 64.4%

   \[\leadsto \color{blue}{\frac{-c}{b}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))