Quadratic roots, narrow range

Percentage Accurate: 55.8% → 92.0%

Time: 6.2s

Alternatives: 15

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: 55.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot b\\ t_2 := t\_1 \cdot b\\ \mathbf{if}\;b \leq 0.82:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{t\_2} \cdot -2 - 1\right) \cdot c}{b} + \frac{\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot \frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{t\_2}, -5, \left(c \cdot c\right) \cdot \left(-a\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))) (t_1 (* (* b b) b)) (t_2 (* t_1 b)))
   (if (<= b 0.82)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 2.0 a))
     (+
      (/ (* (- (* (/ (* (* a a) (* c c)) t_2) -2.0) 1.0) c) b)
      (/
       (fma
        (* (* (* a a) a) (/ (* (* c c) (* c c)) t_2))
        -5.0
        (* (* c c) (- a)))
       t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double t_1 = (b * b) * b;
	double t_2 = t_1 * b;
	double tmp;
	if (b <= 0.82) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = (((((((a * a) * (c * c)) / t_2) * -2.0) - 1.0) * c) / b) + (fma((((a * a) * a) * (((c * c) * (c * c)) / t_2)), -5.0, ((c * c) * -a)) / t_1);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	t_1 = Float64(Float64(b * b) * b)
	t_2 = Float64(t_1 * b)
	tmp = 0.0
	if (b <= 0.82)
		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(a * a) * Float64(c * c)) / t_2) * -2.0) - 1.0) * c) / b) + Float64(fma(Float64(Float64(Float64(a * a) * a) * Float64(Float64(Float64(c * c) * Float64(c * c)) / t_2)), -5.0, Float64(Float64(c * c) * Float64(-a))) / t_1));
	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]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * b), $MachinePrecision]}, If[LessEqual[b, 0.82], 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[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * -2.0), $MachinePrecision] - 1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * -5.0 + N[(N[(c * c), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.819999999999999951

   1. Initial program 55.8%

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

      1. 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 57.3%

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

   if 0.819999999999999951 < b

   1. Initial program 55.8%

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

   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 91.0%

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

   6. Applied rewrites 91.1%

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

   7. Taylor expanded in c around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) \cdot c}{b} + \frac{\mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\color{blue}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot a\right)}}, \frac{-1}{4}, \frac{-\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) \cdot c}{b} + \frac{\mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\color{blue}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot a\right)}}, \frac{-1}{4}, \frac{-\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

   9. Applied rewrites 91.0%

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

   10. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 91.0%

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

Alternative 2: 91.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ t_1 := \left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \mathbf{if}\;b \leq 0.82:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot c}{t\_1}, -2, \left(a \cdot \frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{t\_1 \cdot \left(b \cdot b\right)}\right) \cdot -5\right) \cdot a - \frac{c \cdot c}{b \cdot b}\right) \cdot a - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))) (t_1 (* (* (* b b) b) b)))
   (if (<= b 0.82)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 2.0 a))
     (/
      (-
       (*
        (-
         (*
          (fma
           (/ (* (* c c) c) t_1)
           -2.0
           (* (* a (/ (* (* c c) (* c c)) (* t_1 (* b b)))) -5.0))
          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 t_1 = ((b * b) * b) * b;
	double tmp;
	if (b <= 0.82) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = ((((fma((((c * c) * c) / t_1), -2.0, ((a * (((c * c) * (c * c)) / (t_1 * (b * b)))) * -5.0)) * 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))
	t_1 = Float64(Float64(Float64(b * b) * b) * b)
	tmp = 0.0
	if (b <= 0.82)
		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(fma(Float64(Float64(Float64(c * c) * c) / t_1), -2.0, Float64(Float64(a * Float64(Float64(Float64(c * c) * Float64(c * c)) / Float64(t_1 * Float64(b * b)))) * -5.0)) * 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]}, Block[{t$95$1 = N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, 0.82], 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] / t$95$1), $MachinePrecision] * -2.0 + N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -5.0), $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 b < 0.819999999999999951

   1. Initial program 55.8%

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

      1. 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 57.3%

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

   if 0.819999999999999951 < b

   1. Initial program 55.8%

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

   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 91.0%

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

   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 91.1%

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

Alternative 3: 91.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ t_1 := \left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \mathbf{if}\;b \leq 0.82:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\frac{a \cdot a}{t\_1}, -2, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{t\_1 \cdot \left(b \cdot b\right)} \cdot -5\right) \cdot c - \frac{a}{b \cdot b}\right) \cdot c - 1\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))) (t_1 (* (* (* b b) b) b)))
   (if (<= b 0.82)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 2.0 a))
     (/
      (*
       (-
        (*
         (-
          (*
           (fma
            (/ (* a a) t_1)
            -2.0
            (* (/ (* (* (* a a) a) c) (* t_1 (* b b))) -5.0))
           c)
          (/ a (* b b)))
         c)
        1.0)
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double t_1 = ((b * b) * b) * b;
	double tmp;
	if (b <= 0.82) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = (((((fma(((a * a) / t_1), -2.0, (((((a * a) * a) * c) / (t_1 * (b * b))) * -5.0)) * c) - (a / (b * b))) * c) - 1.0) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	t_1 = Float64(Float64(Float64(b * b) * b) * b)
	tmp = 0.0
	if (b <= 0.82)
		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(fma(Float64(Float64(a * a) / t_1), -2.0, Float64(Float64(Float64(Float64(Float64(a * a) * a) * c) / Float64(t_1 * Float64(b * b))) * -5.0)) * c) - Float64(a / Float64(b * b))) * c) - 1.0) * 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]}, Block[{t$95$1 = N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, 0.82], 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[(a * a), $MachinePrecision] / t$95$1), $MachinePrecision] * -2.0 + N[(N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * c), $MachinePrecision] / N[(t$95$1 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -5.0), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.819999999999999951

   1. Initial program 55.8%

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

      1. 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 57.3%

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

   if 0.819999999999999951 < b

   1. Initial program 55.8%

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

   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 91.0%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) - \frac{a}{{b}^{2}}\right) - 1\right)}{b} \]

   7. Applied rewrites 90.9%

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

Alternative 4: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot b\\ \mathbf{if}\;b \leq 0.82:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{\left(c \cdot c\right) \cdot c}{t\_1 \cdot b}, -2, -c\right)}{b} + \frac{\left(c \cdot c\right) \cdot \left(-a\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))) (t_1 (* (* b b) b)))
   (if (<= b 0.82)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 2.0 a))
     (+
      (/ (fma (* (* a a) (/ (* (* c c) c) (* t_1 b))) -2.0 (- c)) b)
      (/ (* (* c c) (- a)) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double t_1 = (b * b) * b;
	double tmp;
	if (b <= 0.82) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = (fma(((a * a) * (((c * c) * c) / (t_1 * b))), -2.0, -c) / b) + (((c * c) * -a) / t_1);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	t_1 = Float64(Float64(b * b) * b)
	tmp = 0.0
	if (b <= 0.82)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(Float64(a * a) * Float64(Float64(Float64(c * c) * c) / Float64(t_1 * b))), -2.0, Float64(-c)) / b) + Float64(Float64(Float64(c * c) * Float64(-a)) / t_1));
	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]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, 0.82], 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[(a * a), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] / N[(t$95$1 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + (-c)), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * (-a)), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.819999999999999951

   1. Initial program 55.8%

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

      1. 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 57.3%

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

   if 0.819999999999999951 < b

   1. Initial program 55.8%

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

   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 91.0%

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

   6. Applied rewrites 91.1%

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

   7. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 87.9

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

   9. Applied rewrites 87.9%

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

Alternative 5: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot b\\ \mathbf{if}\;b \leq 0.82:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{t\_1 \cdot b} \cdot -2 - 1\right) \cdot c}{b} + \frac{\left(c \cdot c\right) \cdot \left(-a\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))) (t_1 (* (* b b) b)))
   (if (<= b 0.82)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 2.0 a))
     (+
      (/ (* (- (* (/ (* (* a a) (* c c)) (* t_1 b)) -2.0) 1.0) c) b)
      (/ (* (* c c) (- a)) t_1)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double t_1 = (b * b) * b;
	double tmp;
	if (b <= 0.82) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = (((((((a * a) * (c * c)) / (t_1 * b)) * -2.0) - 1.0) * c) / b) + (((c * c) * -a) / t_1);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	t_1 = Float64(Float64(b * b) * b)
	tmp = 0.0
	if (b <= 0.82)
		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(a * a) * Float64(c * c)) / Float64(t_1 * b)) * -2.0) - 1.0) * c) / b) + Float64(Float64(Float64(c * c) * Float64(-a)) / t_1));
	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]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, 0.82], 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[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] - 1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * (-a)), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.819999999999999951

   1. Initial program 55.8%

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

      1. 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 57.3%

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

   if 0.819999999999999951 < b

   1. Initial program 55.8%

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

   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 91.0%

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

   6. Applied rewrites 91.1%

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

   7. Taylor expanded in c around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) \cdot c}{b} + \frac{\mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\color{blue}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot a\right)}}, \frac{-1}{4}, \frac{-\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} - 1\right) \cdot c}{b} + \frac{\mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{\color{blue}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot a\right)}}, \frac{-1}{4}, \frac{-\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b} \]

   9. Applied rewrites 91.0%

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

   10. Taylor expanded in a around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. mul-1-neg N/A

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

       4. *-commutative N/A

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

       5. distribute-rgt-neg-in N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. lower-neg.f64 N/A

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

       10. pow3 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 87.9

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

   12. Applied rewrites 87.9%

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

Alternative 6: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.82:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot a}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} \cdot -2 - \frac{c \cdot c}{b \cdot b}\right) \cdot a - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 0.82)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 2.0 a))
     (/
      (-
       (*
        (-
         (* (/ (* (* (* c c) c) a) (* (* (* b b) b) b)) -2.0)
         (/ (* 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 <= 0.82) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = ((((((((c * c) * c) * a) / (((b * b) * b) * b)) * -2.0) - ((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 (b <= 0.82)
		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) * a) / Float64(Float64(Float64(b * b) * b) * b)) * -2.0) - 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[b, 0.82], 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] * a), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * -2.0), $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 b < 0.819999999999999951

   1. Initial program 55.8%

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

      1. 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 57.3%

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

   if 0.819999999999999951 < b

   1. Initial program 55.8%

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

   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 91.0%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   7. Applied rewrites 88.0%

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

Alternative 7: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.82:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{-2 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} - \frac{a}{b \cdot b}\right) \cdot c - 1\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 0.82)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 2.0 a))
     (/
      (*
       (-
        (* (- (/ (* -2.0 (* (* a a) c)) (* (* (* b b) b) b)) (/ a (* b b))) c)
        1.0)
       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 <= 0.82) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (2.0 * a);
	} else {
		tmp = ((((((-2.0 * ((a * a) * c)) / (((b * b) * b) * b)) - (a / (b * b))) * c) - 1.0) * 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 (b <= 0.82)
		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(-2.0 * Float64(Float64(a * a) * c)) / Float64(Float64(Float64(b * b) * b) * b)) - Float64(a / Float64(b * b))) * c) - 1.0) * 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[b, 0.82], 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[(-2.0 * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.819999999999999951

   1. Initial program 55.8%

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

      1. 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 57.3%

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

   if 0.819999999999999951 < b

   1. Initial program 55.8%

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

   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 91.0%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}{b} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 87.8%

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

Alternative 8: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.82:\\ \;\;\;\;\frac{-b}{a + a} + \frac{\sqrt{\mathsf{fma}\left(\frac{b}{a}, b, -4 \cdot c\right) \cdot a}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{-2 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} - \frac{a}{b \cdot b}\right) \cdot c - 1\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.82)
   (+ (/ (- b) (+ a a)) (/ (sqrt (* (fma (/ b a) b (* -4.0 c)) a)) (+ a a)))
   (/
    (*
     (-
      (* (- (/ (* -2.0 (* (* a a) c)) (* (* (* b b) b) b)) (/ a (* b b))) c)
      1.0)
     c)
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.82) {
		tmp = (-b / (a + a)) + (sqrt((fma((b / a), b, (-4.0 * c)) * a)) / (a + a));
	} else {
		tmp = ((((((-2.0 * ((a * a) * c)) / (((b * b) * b) * b)) - (a / (b * b))) * c) - 1.0) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.82)
		tmp = Float64(Float64(Float64(-b) / Float64(a + a)) + Float64(sqrt(Float64(fma(Float64(b / a), b, Float64(-4.0 * c)) * a)) / Float64(a + a)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-2.0 * Float64(Float64(a * a) * c)) / Float64(Float64(Float64(b * b) * b) * b)) - Float64(a / Float64(b * b))) * c) - 1.0) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.82], N[(N[((-b) / N[(a + a), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(N[(N[(b / a), $MachinePrecision] * b + N[(-4.0 * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-2.0 * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.819999999999999951

   1. Initial program 55.8%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 55.6

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

   4. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 55.7

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

   6. Applied rewrites 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

   if 0.819999999999999951 < b

   1. Initial program 55.8%

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

   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 91.0%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}{b} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 87.8%

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

Alternative 9: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.55:\\ \;\;\;\;\frac{-b}{a + a} + \frac{\sqrt{\mathsf{fma}\left(\frac{b}{a}, b, -4 \cdot c\right) \cdot a}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot c\right) \cdot \left(-a\right)}{\left(b \cdot b\right) \cdot b} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.55)
   (+ (/ (- b) (+ a a)) (/ (sqrt (* (fma (/ b a) b (* -4.0 c)) a)) (+ a a)))
   (- (/ (* (* c c) (- a)) (* (* b b) b)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.55) {
		tmp = (-b / (a + a)) + (sqrt((fma((b / a), b, (-4.0 * c)) * a)) / (a + a));
	} else {
		tmp = (((c * c) * -a) / ((b * b) * b)) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.55)
		tmp = Float64(Float64(Float64(-b) / Float64(a + a)) + Float64(sqrt(Float64(fma(Float64(b / a), b, Float64(-4.0 * c)) * a)) / Float64(a + a)));
	else
		tmp = Float64(Float64(Float64(Float64(c * c) * Float64(-a)) / Float64(Float64(b * b) * b)) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.55], N[(N[((-b) / N[(a + a), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(N[(N[(b / a), $MachinePrecision] * b + N[(-4.0 * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * (-a)), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.55000000000000004

   1. Initial program 55.8%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 55.6

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

   4. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 55.7

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

   6. Applied rewrites 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

   if 1.55000000000000004 < b

   1. Initial program 55.8%

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

   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 91.0%

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

   6. Applied rewrites 91.1%

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

   7. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

         \[\leadsto -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} - \frac{c}{\color{blue}{b}} \]

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 81.5

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

   9. Applied rewrites 81.5%

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

Alternative 10: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(\frac{b}{a}, b, -4 \cdot c\right) \cdot a}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot c\right) \cdot \left(-a\right)}{\left(b \cdot b\right) \cdot b} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.6)
   (/ (+ (- b) (sqrt (* (fma (/ b a) b (* -4.0 c)) a))) (+ a a))
   (- (/ (* (* c c) (- a)) (* (* b b) b)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6) {
		tmp = (-b + sqrt((fma((b / a), b, (-4.0 * c)) * a))) / (a + a);
	} else {
		tmp = (((c * c) * -a) / ((b * b) * b)) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.6)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(fma(Float64(b / a), b, Float64(-4.0 * c)) * a))) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(Float64(c * c) * Float64(-a)) / Float64(Float64(b * b) * b)) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.6], N[(N[((-b) + N[Sqrt[N[(N[(N[(b / a), $MachinePrecision] * b + N[(-4.0 * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * (-a)), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001

   1. Initial program 55.8%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 55.6

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

   4. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 55.7

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

   6. Applied rewrites 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 55.7

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

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(\frac{b}{a}, b, -4 \cdot c\right) \cdot a}}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(2 \cdot a\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(\frac{b}{a}, b, -4 \cdot c\right) \cdot a}}{\mathsf{Rewrite=>}\left(count-2-rev, \left(a + a\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(\frac{b}{a}, b, -4 \cdot c\right) \cdot a}}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(a + a\right)\right)} \]

   8. Applied rewrites 55.7%

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

   if 1.6000000000000001 < b

   1. Initial program 55.8%

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

   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 91.0%

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

   6. Applied rewrites 91.1%

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

   7. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

         \[\leadsto -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} - \frac{c}{\color{blue}{b}} \]

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 81.5

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

   9. Applied rewrites 81.5%

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

Alternative 11: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot c\right) \cdot \left(-a\right)}{\left(b \cdot b\right) \cdot b} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.6)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (- (/ (* (* c c) (- a)) (* (* b b) b)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = (((c * c) * -a) / ((b * b) * b)) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.6)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(c * c) * Float64(-a)) / Float64(Float64(b * b) * b)) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.6], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * (-a)), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001

   1. Initial program 55.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 55.9

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

   3. Applied rewrites 55.9%

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

   if 1.6000000000000001 < b

   1. Initial program 55.8%

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

   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 91.0%

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

   6. Applied rewrites 91.1%

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

   7. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

         \[\leadsto -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} - \frac{c}{\color{blue}{b}} \]

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 81.5

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

   9. Applied rewrites 81.5%

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

Alternative 12: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot c\right) \cdot \left(-a\right)}{\left(b \cdot b\right) \cdot b} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.6)
   (/ (+ (sqrt (fma (* -4.0 a) c (* b b))) (- b)) (+ a a))
   (- (/ (* (* c c) (- a)) (* (* b b) b)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.6) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) + -b) / (a + a);
	} else {
		tmp = (((c * c) * -a) / ((b * b) * b)) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.6)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) + Float64(-b)) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(Float64(c * c) * Float64(-a)) / Float64(Float64(b * b) * b)) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.6], 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] * (-a)), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6000000000000001

   1. Initial program 55.8%

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

   3. Applied rewrites 55.8%

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

   if 1.6000000000000001 < b

   1. Initial program 55.8%

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

   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 91.0%

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

   6. Applied rewrites 91.1%

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

   7. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

         \[\leadsto -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} - \frac{c}{\color{blue}{b}} \]

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 81.5

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

   9. Applied rewrites 81.5%

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

Alternative 13: 81.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

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 91.0%

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

6. Applied rewrites 91.1%

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

7. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

      \[\leadsto -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} - \frac{c}{\color{blue}{b}} \]

   2. associate-*r/ N/A

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

   3. lower-/.f64 N/A

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

   4. mul-1-neg N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. pow3 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 81.5

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

9. Applied rewrites 81.5%

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

Alternative 14: 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 55.8%

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

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 91.0%

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

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 15: 64.1% 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 55.8%

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

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 64.1

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

4. Applied rewrites 64.1%

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

Reproduce

herbie shell --seed 2025142 
(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)))