Quadratic roots, narrow range

Percentage Accurate: 55.9% → 91.9%

Time: 11.4s

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

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

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: 55.9% accurate, 1.0× speedup

Math

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

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: 91.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \left(a \cdot a\right) \cdot a\\ t_1 := \left(c \cdot c\right) \cdot c\\ t_2 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_3 := {\left(a \cdot c\right)}^{4}\\ t_4 := t\_3 \cdot {b}^{-6}\\ t_5 := t\_3 \cdot 20\\ t_6 := \left(a \cdot a\right) \cdot \left(c \cdot c\right)\\ t_7 := \frac{t\_6}{b \cdot b}\\ t_8 := t\_1 \cdot t\_0\\ t_9 := t\_6 \cdot 0\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1:\\ \;\;\;\;\frac{\frac{t\_2 - b \cdot b}{\sqrt{t\_2} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-8 \cdot t\_0\right) \cdot t\_1, {b}^{-4}, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(t\_7, -4, \mathsf{fma}\left(-4 \cdot t\_8, {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(t\_8 \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2 \cdot \left(\left(t\_9 \cdot c\right) \cdot a\right), {b}^{-4}, \mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot \left(t\_9 \cdot \left(c \cdot c\right)\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, t\_7, \frac{\mathsf{fma}\left(-1, t\_5, -0.5 \cdot t\_5\right)}{{b}^{6}} + \mathsf{fma}\left(t\_7, 4, \mathsf{fma}\left(t\_4, 4, \mathsf{fma}\left(8, t\_7, \mathsf{fma}\left(16, t\_8 \cdot {b}^{-4}, t\_4 \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b, \left(b \cdot \left(1 + \mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{a \cdot c}{{b}^{2}}, -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right) \cdot b\right)\right)\right)}}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a a) a))
        (t_1 (* (* c c) c))
        (t_2 (fma (* c -4.0) a (* b b)))
        (t_3 (pow (* a c) 4.0))
        (t_4 (* t_3 (pow b -6.0)))
        (t_5 (* t_3 20.0))
        (t_6 (* (* a a) (* c c)))
        (t_7 (/ t_6 (* b b)))
        (t_8 (* t_1 t_0))
        (t_9 (* t_6 0.0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.0)
     (/ (/ (- t_2 (* b b)) (+ (sqrt t_2) b)) (* 2.0 a))
     (/
      (/
       (*
        (fma
         (* (* -8.0 t_0) t_1)
         (pow b -4.0)
         (fma
          (* -4.0 a)
          c
          (fma
           t_7
           -4.0
           (fma
            (* -4.0 t_8)
            (pow b -4.0)
            (fma
             (* -2.0 a)
             c
             (fma
              (* (* (* (* t_8 0.0) c) a) (pow b -6.0))
              -2.0
              (fma
               (* -2.0 (* (* t_9 c) a))
               (pow b -4.0)
               (fma
                (* (* (* a a) (* t_9 (* c c))) (pow b -6.0))
                -2.0
                (fma
                 -2.0
                 t_7
                 (+
                  (/ (fma -1.0 t_5 (* -0.5 t_5)) (pow b 6.0))
                  (fma
                   t_7
                   4.0
                   (fma
                    t_4
                    4.0
                    (fma
                     8.0
                     t_7
                     (fma 16.0 (* t_8 (pow b -4.0)) (* t_4 32.0)))))))))))))))
        b)
       (fma
        (* -4.0 c)
        a
        (fma
         b
         b
         (fma
          b
          b
          (*
           (*
            b
            (+
             1.0
             (fma
              -4.0
              (/ (* (pow a 3.0) (pow c 3.0)) (pow b 6.0))
              (fma
               -2.0
               (/ (* a c) (pow b 2.0))
               (* -2.0 (/ (* (pow a 2.0) (pow c 2.0)) (pow b 4.0)))))))
           b)))))
      (+ a a)))))

C

double code(double a, double b, double c) {
	double t_0 = (a * a) * a;
	double t_1 = (c * c) * c;
	double t_2 = fma((c * -4.0), a, (b * b));
	double t_3 = pow((a * c), 4.0);
	double t_4 = t_3 * pow(b, -6.0);
	double t_5 = t_3 * 20.0;
	double t_6 = (a * a) * (c * c);
	double t_7 = t_6 / (b * b);
	double t_8 = t_1 * t_0;
	double t_9 = t_6 * 0.0;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.0) {
		tmp = ((t_2 - (b * b)) / (sqrt(t_2) + b)) / (2.0 * a);
	} else {
		tmp = ((fma(((-8.0 * t_0) * t_1), pow(b, -4.0), fma((-4.0 * a), c, fma(t_7, -4.0, fma((-4.0 * t_8), pow(b, -4.0), fma((-2.0 * a), c, fma(((((t_8 * 0.0) * c) * a) * pow(b, -6.0)), -2.0, fma((-2.0 * ((t_9 * c) * a)), pow(b, -4.0), fma((((a * a) * (t_9 * (c * c))) * pow(b, -6.0)), -2.0, fma(-2.0, t_7, ((fma(-1.0, t_5, (-0.5 * t_5)) / pow(b, 6.0)) + fma(t_7, 4.0, fma(t_4, 4.0, fma(8.0, t_7, fma(16.0, (t_8 * pow(b, -4.0)), (t_4 * 32.0))))))))))))))) * b) / fma((-4.0 * c), a, fma(b, b, fma(b, b, ((b * (1.0 + fma(-4.0, ((pow(a, 3.0) * pow(c, 3.0)) / pow(b, 6.0)), fma(-2.0, ((a * c) / pow(b, 2.0)), (-2.0 * ((pow(a, 2.0) * pow(c, 2.0)) / pow(b, 4.0))))))) * b))))) / (a + a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * a) * a)
	t_1 = Float64(Float64(c * c) * c)
	t_2 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_3 = Float64(a * c) ^ 4.0
	t_4 = Float64(t_3 * (b ^ -6.0))
	t_5 = Float64(t_3 * 20.0)
	t_6 = Float64(Float64(a * a) * Float64(c * c))
	t_7 = Float64(t_6 / Float64(b * b))
	t_8 = Float64(t_1 * t_0)
	t_9 = Float64(t_6 * 0.0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.0)
		tmp = Float64(Float64(Float64(t_2 - Float64(b * b)) / Float64(sqrt(t_2) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(-8.0 * t_0) * t_1), (b ^ -4.0), fma(Float64(-4.0 * a), c, fma(t_7, -4.0, fma(Float64(-4.0 * t_8), (b ^ -4.0), fma(Float64(-2.0 * a), c, fma(Float64(Float64(Float64(Float64(t_8 * 0.0) * c) * a) * (b ^ -6.0)), -2.0, fma(Float64(-2.0 * Float64(Float64(t_9 * c) * a)), (b ^ -4.0), fma(Float64(Float64(Float64(a * a) * Float64(t_9 * Float64(c * c))) * (b ^ -6.0)), -2.0, fma(-2.0, t_7, Float64(Float64(fma(-1.0, t_5, Float64(-0.5 * t_5)) / (b ^ 6.0)) + fma(t_7, 4.0, fma(t_4, 4.0, fma(8.0, t_7, fma(16.0, Float64(t_8 * (b ^ -4.0)), Float64(t_4 * 32.0))))))))))))))) * b) / fma(Float64(-4.0 * c), a, fma(b, b, fma(b, b, Float64(Float64(b * Float64(1.0 + fma(-4.0, Float64(Float64((a ^ 3.0) * (c ^ 3.0)) / (b ^ 6.0)), fma(-2.0, Float64(Float64(a * c) / (b ^ 2.0)), Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 2.0)) / (b ^ 4.0))))))) * b))))) / Float64(a + a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * 20.0), $MachinePrecision]}, Block[{t$95$6 = N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$6 / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$1 * t$95$0), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$6 * 0.0), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(t$95$2 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$2], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-8.0 * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision] + N[(N[(-4.0 * a), $MachinePrecision] * c + N[(t$95$7 * -4.0 + N[(N[(-4.0 * t$95$8), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision] + N[(N[(-2.0 * a), $MachinePrecision] * c + N[(N[(N[(N[(N[(t$95$8 * 0.0), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(-2.0 * N[(N[(t$95$9 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision] + N[(N[(N[(N[(a * a), $MachinePrecision] * N[(t$95$9 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(-2.0 * t$95$7 + N[(N[(N[(-1.0 * t$95$5 + N[(-0.5 * t$95$5), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$7 * 4.0 + N[(t$95$4 * 4.0 + N[(8.0 * t$95$7 + N[(16.0 * N[(t$95$8 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * 32.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b + N[(b * b + N[(N[(b * N[(1.0 + N[(-4.0 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1

   1. Initial program 55.9%

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

      1. 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. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 57.5%

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

   if -1 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.9%

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

      1. 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. +-commutative N/A

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

      3. flip3-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 55.8%

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

   4. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(-8 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(-4 \cdot \left(a \cdot c\right) + \left(-4 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(-2 \cdot \left(a \cdot c\right) + \left(-2 \cdot \frac{a \cdot \left(c \cdot \left(-8 \cdot \left({a}^{3} \cdot {c}^{3}\right) + 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot \left(c \cdot \left(-4 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}} + \left(-2 \cdot \frac{{a}^{2} \cdot \left({c}^{2} \cdot \left(-4 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(-1 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(\frac{-1}{2} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(4 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(4 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}} + \left(8 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(16 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \left(-b\right)\right)}}{2 \cdot a} \]

   6. Applied rewrites 91.2%

      \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-8, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-4, a \cdot c, \mathsf{fma}\left(-4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-2, a \cdot c, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-8, {a}^{3} \cdot {c}^{3}, 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot \left({c}^{2} \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(4, \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(8, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(16, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \left(-b\right)\right)}}{2 \cdot a} \]

   8. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(-8 \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot c\right), {b}^{-4}, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, -4, \mathsf{fma}\left(-4 \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right), {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2 \cdot \left(\left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right), {b}^{-4}, \mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot \left(c \cdot c\right)\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{\mathsf{fma}\left(-1, {\left(a \cdot c\right)}^{4} \cdot 20, -0.5 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 20\right)\right)}{{b}^{6}} + \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 4, \mathsf{fma}\left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}, 4, \mathsf{fma}\left(8, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(16, \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-4}, \left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}\right) \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} \cdot b\right)\right)\right)}}{a + a}} \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(-8 \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot c\right), {b}^{-4}, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, -4, \mathsf{fma}\left(-4 \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right), {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2 \cdot \left(\left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right), {b}^{-4}, \mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot \left(c \cdot c\right)\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{\mathsf{fma}\left(-1, {\left(a \cdot c\right)}^{4} \cdot 20, \frac{-1}{2} \cdot \left({\left(a \cdot c\right)}^{4} \cdot 20\right)\right)}{{b}^{6}} + \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 4, \mathsf{fma}\left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}, 4, \mathsf{fma}\left(8, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(16, \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-4}, \left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}\right) \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b, \color{blue}{\left(b \cdot \left(1 + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)} \cdot b\right)\right)\right)}}{a + a} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(-8 \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot c\right), {b}^{-4}, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, -4, \mathsf{fma}\left(-4 \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right), {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2 \cdot \left(\left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right), {b}^{-4}, \mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot \left(c \cdot c\right)\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{\mathsf{fma}\left(-1, {\left(a \cdot c\right)}^{4} \cdot 20, \frac{-1}{2} \cdot \left({\left(a \cdot c\right)}^{4} \cdot 20\right)\right)}{{b}^{6}} + \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 4, \mathsf{fma}\left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}, 4, \mathsf{fma}\left(8, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(16, \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-4}, \left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}\right) \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b, \left(b \cdot \color{blue}{\left(1 + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)}\right) \cdot b\right)\right)\right)}}{a + a} \]

       2. lower-+.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(-8 \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot c\right), {b}^{-4}, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, -4, \mathsf{fma}\left(-4 \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right), {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2 \cdot \left(\left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right), {b}^{-4}, \mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot \left(c \cdot c\right)\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{\mathsf{fma}\left(-1, {\left(a \cdot c\right)}^{4} \cdot 20, \frac{-1}{2} \cdot \left({\left(a \cdot c\right)}^{4} \cdot 20\right)\right)}{{b}^{6}} + \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 4, \mathsf{fma}\left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}, 4, \mathsf{fma}\left(8, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(16, \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-4}, \left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}\right) \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b, \left(b \cdot \left(1 + \color{blue}{\left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)}\right)\right) \cdot b\right)\right)\right)}}{a + a} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(-8 \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot c\right), {b}^{-4}, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, -4, \mathsf{fma}\left(-4 \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right), {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2 \cdot \left(\left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right), {b}^{-4}, \mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot \left(c \cdot c\right)\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{\mathsf{fma}\left(-1, {\left(a \cdot c\right)}^{4} \cdot 20, \frac{-1}{2} \cdot \left({\left(a \cdot c\right)}^{4} \cdot 20\right)\right)}{{b}^{6}} + \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 4, \mathsf{fma}\left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}, 4, \mathsf{fma}\left(8, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(16, \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-4}, \left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}\right) \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b, \left(b \cdot \left(1 + \mathsf{fma}\left(-4, \color{blue}{\frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}}, -2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right) \cdot b\right)\right)\right)}}{a + a} \]

   11. Applied rewrites 91.4%

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\left(-8 \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot c\right), {b}^{-4}, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, -4, \mathsf{fma}\left(-4 \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right), {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2 \cdot \left(\left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right), {b}^{-4}, \mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot \left(c \cdot c\right)\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{\mathsf{fma}\left(-1, {\left(a \cdot c\right)}^{4} \cdot 20, -0.5 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 20\right)\right)}{{b}^{6}} + \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 4, \mathsf{fma}\left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}, 4, \mathsf{fma}\left(8, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(16, \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-4}, \left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}\right) \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b, \color{blue}{\left(b \cdot \left(1 + \mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{a \cdot c}{{b}^{2}}, -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)} \cdot b\right)\right)\right)}}{a + a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 91.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \left(c \cdot c\right) \cdot c\\ t_1 := {\left(a \cdot c\right)}^{4}\\ t_2 := \left(a \cdot a\right) \cdot a\\ t_3 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_4 := \frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{b \cdot b}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1:\\ \;\;\;\;\frac{\frac{t\_3 - b \cdot b}{\sqrt{t\_3} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left({b}^{-4}, t\_0 \cdot \left(t\_2 \cdot -8\right), \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(t\_4, -4, \mathsf{fma}\left(\left(-4 \cdot t\_0\right) \cdot t\_2, {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(0 \cdot c\right) \cdot a, {b}^{-6} \cdot -2, \mathsf{fma}\left(\left(-2 \cdot \left(0 \cdot c\right)\right) \cdot a, {b}^{-4}, \mathsf{fma}\left({b}^{-6} \cdot 0, -2, \mathsf{fma}\left(-2, t\_4, \mathsf{fma}\left(\left(t\_1 \cdot 20\right) \cdot -1.5, {b}^{-6}, \mathsf{fma}\left(4, t\_4, \mathsf{fma}\left(t\_1 \cdot {b}^{-6}, 4, \mathsf{fma}\left(8, t\_4, \mathsf{fma}\left(\left(t\_0 \cdot t\_2\right) \cdot {b}^{-4}, 16, t\_1 \cdot \left({b}^{-6} \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} \cdot b\right)\right)\right)}}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c c) c))
        (t_1 (pow (* a c) 4.0))
        (t_2 (* (* a a) a))
        (t_3 (fma (* c -4.0) a (* b b)))
        (t_4 (/ (* (* (* a a) c) c) (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.0)
     (/ (/ (- t_3 (* b b)) (+ (sqrt t_3) b)) (* 2.0 a))
     (/
      (/
       (*
        (fma
         (pow b -4.0)
         (* t_0 (* t_2 -8.0))
         (fma
          (* -4.0 c)
          a
          (fma
           t_4
           -4.0
           (fma
            (* (* -4.0 t_0) t_2)
            (pow b -4.0)
            (fma
             (* -2.0 a)
             c
             (fma
              (* (* 0.0 c) a)
              (* (pow b -6.0) -2.0)
              (fma
               (* (* -2.0 (* 0.0 c)) a)
               (pow b -4.0)
               (fma
                (* (pow b -6.0) 0.0)
                -2.0
                (fma
                 -2.0
                 t_4
                 (fma
                  (* (* t_1 20.0) -1.5)
                  (pow b -6.0)
                  (fma
                   4.0
                   t_4
                   (fma
                    (* t_1 (pow b -6.0))
                    4.0
                    (fma
                     8.0
                     t_4
                     (fma
                      (* (* t_0 t_2) (pow b -4.0))
                      16.0
                      (* t_1 (* (pow b -6.0) 32.0))))))))))))))))
        b)
       (fma
        (* -4.0 c)
        a
        (fma b b (fma b b (* (sqrt (fma (* -4.0 c) a (* b b))) b)))))
      (+ a a)))))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) * c;
	double t_1 = pow((a * c), 4.0);
	double t_2 = (a * a) * a;
	double t_3 = fma((c * -4.0), a, (b * b));
	double t_4 = (((a * a) * c) * c) / (b * b);
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.0) {
		tmp = ((t_3 - (b * b)) / (sqrt(t_3) + b)) / (2.0 * a);
	} else {
		tmp = ((fma(pow(b, -4.0), (t_0 * (t_2 * -8.0)), fma((-4.0 * c), a, fma(t_4, -4.0, fma(((-4.0 * t_0) * t_2), pow(b, -4.0), fma((-2.0 * a), c, fma(((0.0 * c) * a), (pow(b, -6.0) * -2.0), fma(((-2.0 * (0.0 * c)) * a), pow(b, -4.0), fma((pow(b, -6.0) * 0.0), -2.0, fma(-2.0, t_4, fma(((t_1 * 20.0) * -1.5), pow(b, -6.0), fma(4.0, t_4, fma((t_1 * pow(b, -6.0)), 4.0, fma(8.0, t_4, fma(((t_0 * t_2) * pow(b, -4.0)), 16.0, (t_1 * (pow(b, -6.0) * 32.0)))))))))))))))) * b) / fma((-4.0 * c), a, fma(b, b, fma(b, b, (sqrt(fma((-4.0 * c), a, (b * b))) * b))))) / (a + a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) * c)
	t_1 = Float64(a * c) ^ 4.0
	t_2 = Float64(Float64(a * a) * a)
	t_3 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_4 = Float64(Float64(Float64(Float64(a * a) * c) * c) / Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.0)
		tmp = Float64(Float64(Float64(t_3 - Float64(b * b)) / Float64(sqrt(t_3) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(fma((b ^ -4.0), Float64(t_0 * Float64(t_2 * -8.0)), fma(Float64(-4.0 * c), a, fma(t_4, -4.0, fma(Float64(Float64(-4.0 * t_0) * t_2), (b ^ -4.0), fma(Float64(-2.0 * a), c, fma(Float64(Float64(0.0 * c) * a), Float64((b ^ -6.0) * -2.0), fma(Float64(Float64(-2.0 * Float64(0.0 * c)) * a), (b ^ -4.0), fma(Float64((b ^ -6.0) * 0.0), -2.0, fma(-2.0, t_4, fma(Float64(Float64(t_1 * 20.0) * -1.5), (b ^ -6.0), fma(4.0, t_4, fma(Float64(t_1 * (b ^ -6.0)), 4.0, fma(8.0, t_4, fma(Float64(Float64(t_0 * t_2) * (b ^ -4.0)), 16.0, Float64(t_1 * Float64((b ^ -6.0) * 32.0)))))))))))))))) * b) / fma(Float64(-4.0 * c), a, fma(b, b, fma(b, b, Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) * b))))) / Float64(a + a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(t$95$3 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$3], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Power[b, -4.0], $MachinePrecision] * N[(t$95$0 * N[(t$95$2 * -8.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a + N[(t$95$4 * -4.0 + N[(N[(N[(-4.0 * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision] + N[(N[(-2.0 * a), $MachinePrecision] * c + N[(N[(N[(0.0 * c), $MachinePrecision] * a), $MachinePrecision] * N[(N[Power[b, -6.0], $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(N[(-2.0 * N[(0.0 * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision] + N[(N[(N[Power[b, -6.0], $MachinePrecision] * 0.0), $MachinePrecision] * -2.0 + N[(-2.0 * t$95$4 + N[(N[(N[(t$95$1 * 20.0), $MachinePrecision] * -1.5), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision] + N[(4.0 * t$95$4 + N[(N[(t$95$1 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * 4.0 + N[(8.0 * t$95$4 + N[(N[(N[(t$95$0 * t$95$2), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision] * 16.0 + N[(t$95$1 * N[(N[Power[b, -6.0], $MachinePrecision] * 32.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b + N[(b * b + N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1

   1. Initial program 55.9%

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

      1. 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. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 57.5%

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

   if -1 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.9%

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

      1. 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. +-commutative N/A

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

      3. flip3-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 55.8%

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

   4. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(-8 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(-4 \cdot \left(a \cdot c\right) + \left(-4 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(-2 \cdot \left(a \cdot c\right) + \left(-2 \cdot \frac{a \cdot \left(c \cdot \left(-8 \cdot \left({a}^{3} \cdot {c}^{3}\right) + 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot \left(c \cdot \left(-4 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}} + \left(-2 \cdot \frac{{a}^{2} \cdot \left({c}^{2} \cdot \left(-4 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(-1 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(\frac{-1}{2} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(4 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(4 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}} + \left(8 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(16 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \left(-b\right)\right)}}{2 \cdot a} \]

   6. Applied rewrites 91.2%

      \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-8, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-4, a \cdot c, \mathsf{fma}\left(-4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-2, a \cdot c, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-8, {a}^{3} \cdot {c}^{3}, 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot \left({c}^{2} \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(4, \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(8, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(16, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \left(-b\right)\right)}}{2 \cdot a} \]

   8. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(-8 \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot c\right), {b}^{-4}, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, -4, \mathsf{fma}\left(-4 \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right), {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2 \cdot \left(\left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right), {b}^{-4}, \mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot \left(c \cdot c\right)\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{\mathsf{fma}\left(-1, {\left(a \cdot c\right)}^{4} \cdot 20, -0.5 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 20\right)\right)}{{b}^{6}} + \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 4, \mathsf{fma}\left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}, 4, \mathsf{fma}\left(8, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(16, \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-4}, \left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}\right) \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} \cdot b\right)\right)\right)}}{a + a}} \]

   10. Applied rewrites 91.3%

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({b}^{-4}, \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot -8\right), \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{b \cdot b}, -4, \mathsf{fma}\left(\left(-4 \cdot \left(\left(c \cdot c\right) \cdot c\right)\right) \cdot \left(\left(a \cdot a\right) \cdot a\right), {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(0 \cdot c\right) \cdot a, {b}^{-6} \cdot -2, \mathsf{fma}\left(\left(-2 \cdot \left(0 \cdot c\right)\right) \cdot a, {b}^{-4}, \mathsf{fma}\left({b}^{-6} \cdot 0, -2, \mathsf{fma}\left(-2, \frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{b \cdot b}, \mathsf{fma}\left(\left({\left(a \cdot c\right)}^{4} \cdot 20\right) \cdot -1.5, {b}^{-6}, \mathsf{fma}\left(4, \frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{b \cdot b}, \mathsf{fma}\left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}, 4, \mathsf{fma}\left(8, \frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{b \cdot b}, \mathsf{fma}\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-4}, 16, {\left(a \cdot c\right)}^{4} \cdot \left({b}^{-6} \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} \cdot b\right)\right)\right)}}{a + a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \left(c \cdot c\right) \cdot c\\ t_1 := {\left(a \cdot c\right)}^{4}\\ t_2 := \left(a \cdot a\right) \cdot a\\ t_3 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_4 := \frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{b \cdot b}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1:\\ \;\;\;\;\frac{\frac{t\_3 - b \cdot b}{\sqrt{t\_3} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({b}^{-4}, t\_0 \cdot \left(t\_2 \cdot -8\right), \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(t\_4, -4, \mathsf{fma}\left(\left(-4 \cdot t\_0\right) \cdot t\_2, {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(0 \cdot c\right) \cdot a, {b}^{-6} \cdot -2, \mathsf{fma}\left(\left(-2 \cdot \left(0 \cdot c\right)\right) \cdot a, {b}^{-4}, \mathsf{fma}\left({b}^{-6} \cdot 0, -2, \mathsf{fma}\left(-2, t\_4, \mathsf{fma}\left(\left(t\_1 \cdot 20\right) \cdot -1.5, {b}^{-6}, \mathsf{fma}\left(4, t\_4, \mathsf{fma}\left(t\_1 \cdot {b}^{-6}, 4, \mathsf{fma}\left(8, t\_4, \mathsf{fma}\left(\left(t\_0 \cdot t\_2\right) \cdot {b}^{-4}, 16, t\_1 \cdot \left({b}^{-6} \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}, b \cdot b\right)\right) \cdot \left(a + a\right)}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c c) c))
        (t_1 (pow (* a c) 4.0))
        (t_2 (* (* a a) a))
        (t_3 (fma (* c -4.0) a (* b b)))
        (t_4 (/ (* (* (* a a) c) c) (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.0)
     (/ (/ (- t_3 (* b b)) (+ (sqrt t_3) b)) (* 2.0 a))
     (/
      (*
       (fma
        (pow b -4.0)
        (* t_0 (* t_2 -8.0))
        (fma
         (* -4.0 c)
         a
         (fma
          t_4
          -4.0
          (fma
           (* (* -4.0 t_0) t_2)
           (pow b -4.0)
           (fma
            (* -2.0 a)
            c
            (fma
             (* (* 0.0 c) a)
             (* (pow b -6.0) -2.0)
             (fma
              (* (* -2.0 (* 0.0 c)) a)
              (pow b -4.0)
              (fma
               (* (pow b -6.0) 0.0)
               -2.0
               (fma
                -2.0
                t_4
                (fma
                 (* (* t_1 20.0) -1.5)
                 (pow b -6.0)
                 (fma
                  4.0
                  t_4
                  (fma
                   (* t_1 (pow b -6.0))
                   4.0
                   (fma
                    8.0
                    t_4
                    (fma
                     (* (* t_0 t_2) (pow b -4.0))
                     16.0
                     (* t_1 (* (pow b -6.0) 32.0))))))))))))))))
       b)
      (*
       (fma
        (* -4.0 c)
        a
        (fma b (+ b (sqrt (fma (* -4.0 c) a (* b b)))) (* b b)))
       (+ a a))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) * c;
	double t_1 = pow((a * c), 4.0);
	double t_2 = (a * a) * a;
	double t_3 = fma((c * -4.0), a, (b * b));
	double t_4 = (((a * a) * c) * c) / (b * b);
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.0) {
		tmp = ((t_3 - (b * b)) / (sqrt(t_3) + b)) / (2.0 * a);
	} else {
		tmp = (fma(pow(b, -4.0), (t_0 * (t_2 * -8.0)), fma((-4.0 * c), a, fma(t_4, -4.0, fma(((-4.0 * t_0) * t_2), pow(b, -4.0), fma((-2.0 * a), c, fma(((0.0 * c) * a), (pow(b, -6.0) * -2.0), fma(((-2.0 * (0.0 * c)) * a), pow(b, -4.0), fma((pow(b, -6.0) * 0.0), -2.0, fma(-2.0, t_4, fma(((t_1 * 20.0) * -1.5), pow(b, -6.0), fma(4.0, t_4, fma((t_1 * pow(b, -6.0)), 4.0, fma(8.0, t_4, fma(((t_0 * t_2) * pow(b, -4.0)), 16.0, (t_1 * (pow(b, -6.0) * 32.0)))))))))))))))) * b) / (fma((-4.0 * c), a, fma(b, (b + sqrt(fma((-4.0 * c), a, (b * b)))), (b * b))) * (a + a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) * c)
	t_1 = Float64(a * c) ^ 4.0
	t_2 = Float64(Float64(a * a) * a)
	t_3 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_4 = Float64(Float64(Float64(Float64(a * a) * c) * c) / Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.0)
		tmp = Float64(Float64(Float64(t_3 - Float64(b * b)) / Float64(sqrt(t_3) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma((b ^ -4.0), Float64(t_0 * Float64(t_2 * -8.0)), fma(Float64(-4.0 * c), a, fma(t_4, -4.0, fma(Float64(Float64(-4.0 * t_0) * t_2), (b ^ -4.0), fma(Float64(-2.0 * a), c, fma(Float64(Float64(0.0 * c) * a), Float64((b ^ -6.0) * -2.0), fma(Float64(Float64(-2.0 * Float64(0.0 * c)) * a), (b ^ -4.0), fma(Float64((b ^ -6.0) * 0.0), -2.0, fma(-2.0, t_4, fma(Float64(Float64(t_1 * 20.0) * -1.5), (b ^ -6.0), fma(4.0, t_4, fma(Float64(t_1 * (b ^ -6.0)), 4.0, fma(8.0, t_4, fma(Float64(Float64(t_0 * t_2) * (b ^ -4.0)), 16.0, Float64(t_1 * Float64((b ^ -6.0) * 32.0)))))))))))))))) * b) / Float64(fma(Float64(-4.0 * c), a, fma(b, Float64(b + sqrt(fma(Float64(-4.0 * c), a, Float64(b * b)))), Float64(b * b))) * Float64(a + a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(t$95$3 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$3], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[b, -4.0], $MachinePrecision] * N[(t$95$0 * N[(t$95$2 * -8.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a + N[(t$95$4 * -4.0 + N[(N[(N[(-4.0 * t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision] + N[(N[(-2.0 * a), $MachinePrecision] * c + N[(N[(N[(0.0 * c), $MachinePrecision] * a), $MachinePrecision] * N[(N[Power[b, -6.0], $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(N[(-2.0 * N[(0.0 * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision] + N[(N[(N[Power[b, -6.0], $MachinePrecision] * 0.0), $MachinePrecision] * -2.0 + N[(-2.0 * t$95$4 + N[(N[(N[(t$95$1 * 20.0), $MachinePrecision] * -1.5), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision] + N[(4.0 * t$95$4 + N[(N[(t$95$1 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * 4.0 + N[(8.0 * t$95$4 + N[(N[(N[(t$95$0 * t$95$2), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision] * 16.0 + N[(t$95$1 * N[(N[Power[b, -6.0], $MachinePrecision] * 32.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * N[(b + N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1

   1. Initial program 55.9%

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

      1. 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. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 57.5%

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

   if -1 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.9%

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

      1. 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. +-commutative N/A

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

      3. flip3-+ N/A

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

      4. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 55.8%

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

   4. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(-8 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(-4 \cdot \left(a \cdot c\right) + \left(-4 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(-2 \cdot \left(a \cdot c\right) + \left(-2 \cdot \frac{a \cdot \left(c \cdot \left(-8 \cdot \left({a}^{3} \cdot {c}^{3}\right) + 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot \left(c \cdot \left(-4 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}} + \left(-2 \cdot \frac{{a}^{2} \cdot \left({c}^{2} \cdot \left(-4 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(-1 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(\frac{-1}{2} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(4 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(4 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}} + \left(8 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(16 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \left(-b\right)\right)}}{2 \cdot a} \]

   6. Applied rewrites 91.2%

      \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-8, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-4, a \cdot c, \mathsf{fma}\left(-4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-2, a \cdot c, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-8, {a}^{3} \cdot {c}^{3}, 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot \left({c}^{2} \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(4, \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(8, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(16, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \left(-b\right)\right)}}{2 \cdot a} \]

   8. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\left(-8 \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot c\right), {b}^{-4}, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, -4, \mathsf{fma}\left(-4 \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right), {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2 \cdot \left(\left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right), {b}^{-4}, \mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 0\right) \cdot \left(c \cdot c\right)\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{\mathsf{fma}\left(-1, {\left(a \cdot c\right)}^{4} \cdot 20, -0.5 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 20\right)\right)}{{b}^{6}} + \mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 4, \mathsf{fma}\left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}, 4, \mathsf{fma}\left(8, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(16, \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-4}, \left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}\right) \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b, \mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} \cdot b\right)\right)\right)}}{a + a}} \]

   10. Applied rewrites 91.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({b}^{-4}, \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot -8\right), \mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{b \cdot b}, -4, \mathsf{fma}\left(\left(-4 \cdot \left(\left(c \cdot c\right) \cdot c\right)\right) \cdot \left(\left(a \cdot a\right) \cdot a\right), {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(0 \cdot c\right) \cdot a, {b}^{-6} \cdot -2, \mathsf{fma}\left(\left(-2 \cdot \left(0 \cdot c\right)\right) \cdot a, {b}^{-4}, \mathsf{fma}\left({b}^{-6} \cdot 0, -2, \mathsf{fma}\left(-2, \frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{b \cdot b}, \mathsf{fma}\left(\left({\left(a \cdot c\right)}^{4} \cdot 20\right) \cdot -1.5, {b}^{-6}, \mathsf{fma}\left(4, \frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{b \cdot b}, \mathsf{fma}\left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}, 4, \mathsf{fma}\left(8, \frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot c}{b \cdot b}, \mathsf{fma}\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-4}, 16, {\left(a \cdot c\right)}^{4} \cdot \left({b}^{-6} \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot c, a, \mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}, b \cdot b\right)\right) \cdot \left(a + a\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.8:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{\frac{a}{b \cdot b} \cdot \left(c \cdot c\right) - \mathsf{fma}\left(\frac{{b}^{-6}}{a}, {\left(a \cdot c\right)}^{4} \cdot -5, \left({b}^{-4} \cdot -2\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.8)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (-
      (/ (- c) b)
      (/
       (-
        (* (/ a (* b b)) (* c c))
        (fma
         (/ (pow b -6.0) a)
         (* (pow (* a c) 4.0) -5.0)
         (* (* (pow b -4.0) -2.0) (* (* (* (* a a) c) c) c))))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.8) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = (-c / b) - ((((a / (b * b)) * (c * c)) - fma((pow(b, -6.0) / a), (pow((a * c), 4.0) * -5.0), ((pow(b, -4.0) * -2.0) * ((((a * a) * c) * c) * c)))) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(Float64(Float64(a / Float64(b * b)) * Float64(c * c)) - fma(Float64((b ^ -6.0) / a), Float64((Float64(a * c) ^ 4.0) * -5.0), Float64(Float64((b ^ -4.0) * -2.0) * Float64(Float64(Float64(Float64(a * a) * c) * c) * c)))) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[b, -6.0], $MachinePrecision] / a), $MachinePrecision] * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * -5.0), $MachinePrecision] + N[(N[(N[Power[b, -4.0], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.80000000000000004

   1. Initial program 55.9%

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

      1. 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. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 57.5%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   6. Applied rewrites 90.7%

      \[\leadsto \frac{\left(\left(-c\right) - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) + \mathsf{fma}\left(\left({\left(c \cdot a\right)}^{4} \cdot 20\right) \cdot -0.25, \frac{{b}^{-6}}{a}, \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   8. Applied rewrites 90.7%

      \[\leadsto \frac{-c}{b} - \color{blue}{\frac{\frac{a}{b \cdot b} \cdot \left(c \cdot c\right) - \mathsf{fma}\left(\frac{{b}^{-6}}{a}, {\left(a \cdot c\right)}^{4} \cdot -5, \left({b}^{-4} \cdot -2\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.8:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \left(\frac{a}{b \cdot b} \cdot \left(c \cdot c\right) - \mathsf{fma}\left(\frac{{b}^{-6}}{a}, {\left(a \cdot c\right)}^{4} \cdot -5, \left({b}^{-4} \cdot -2\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right)\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.8)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (/
      (-
       (- c)
       (-
        (* (/ a (* b b)) (* c c))
        (fma
         (/ (pow b -6.0) a)
         (* (pow (* a c) 4.0) -5.0)
         (* (* (pow b -4.0) -2.0) (* (* (* (* a a) c) c) c)))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.8) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = (-c - (((a / (b * b)) * (c * c)) - fma((pow(b, -6.0) / a), (pow((a * c), 4.0) * -5.0), ((pow(b, -4.0) * -2.0) * ((((a * a) * c) * c) * c))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(Float64(Float64(a / Float64(b * b)) * Float64(c * c)) - fma(Float64((b ^ -6.0) / a), Float64((Float64(a * c) ^ 4.0) * -5.0), Float64(Float64((b ^ -4.0) * -2.0) * Float64(Float64(Float64(Float64(a * a) * c) * c) * c))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[b, -6.0], $MachinePrecision] / a), $MachinePrecision] * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * -5.0), $MachinePrecision] + N[(N[(N[Power[b, -4.0], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.80000000000000004

   1. Initial program 55.9%

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

      1. 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. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 57.5%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   6. Applied rewrites 90.7%

      \[\leadsto \frac{\left(\left(-c\right) - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) + \mathsf{fma}\left(\left({\left(c \cdot a\right)}^{4} \cdot 20\right) \cdot -0.25, \frac{{b}^{-6}}{a}, \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(-c\right) - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) + \mathsf{fma}\left(\left({\left(c \cdot a\right)}^{4} \cdot 20\right) \cdot \frac{-1}{4}, \frac{{b}^{-6}}{a}, \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\left(\left(-c\right) - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) + \mathsf{fma}\left(\left({\left(c \cdot a\right)}^{4} \cdot 20\right) \cdot \frac{-1}{4}, \frac{{b}^{-6}}{a}, \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

      3. associate-+l- N/A

         \[\leadsto \frac{\left(-c\right) - \left(\left(c \cdot c\right) \cdot \frac{a}{b \cdot b} - \mathsf{fma}\left(\left({\left(c \cdot a\right)}^{4} \cdot 20\right) \cdot \frac{-1}{4}, \frac{{b}^{-6}}{a}, \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)\right)}{b} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\left(-c\right) - \left(\left(c \cdot c\right) \cdot \frac{a}{b \cdot b} - \mathsf{fma}\left(\left({\left(c \cdot a\right)}^{4} \cdot 20\right) \cdot \frac{-1}{4}, \frac{{b}^{-6}}{a}, \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)\right)}{b} \]

      5. lower--.f64 90.7

         \[\leadsto \frac{\left(-c\right) - \left(\left(c \cdot c\right) \cdot \frac{a}{b \cdot b} - \mathsf{fma}\left(\left({\left(c \cdot a\right)}^{4} \cdot 20\right) \cdot -0.25, \frac{{b}^{-6}}{a}, \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)\right)}{b} \]

   8. Applied rewrites 90.7%

      \[\leadsto \frac{\left(-c\right) - \left(\frac{a}{b \cdot b} \cdot \left(c \cdot c\right) - \mathsf{fma}\left(\frac{{b}^{-6}}{a}, {\left(a \cdot c\right)}^{4} \cdot -5, \left({b}^{-4} \cdot -2\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right)\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 91.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.8:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{b}^{-6}}{a}, {\left(a \cdot c\right)}^{4} \cdot -5, \left({b}^{-4} \cdot -2\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) + \left(-\mathsf{fma}\left(\frac{a}{b \cdot b}, c \cdot c, c\right)\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.8)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (/
      (+
       (fma
        (/ (pow b -6.0) a)
        (* (pow (* a c) 4.0) -5.0)
        (* (* (pow b -4.0) -2.0) (* (* (* (* a a) c) c) c)))
       (- (fma (/ a (* b b)) (* c c) c)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.8) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = (fma((pow(b, -6.0) / a), (pow((a * c), 4.0) * -5.0), ((pow(b, -4.0) * -2.0) * ((((a * a) * c) * c) * c))) + -fma((a / (b * b)), (c * c), c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(Float64((b ^ -6.0) / a), Float64((Float64(a * c) ^ 4.0) * -5.0), Float64(Float64((b ^ -4.0) * -2.0) * Float64(Float64(Float64(Float64(a * a) * c) * c) * c))) + Float64(-fma(Float64(a / Float64(b * b)), Float64(c * c), c))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Power[b, -6.0], $MachinePrecision] / a), $MachinePrecision] * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * -5.0), $MachinePrecision] + N[(N[(N[Power[b, -4.0], $MachinePrecision] * -2.0), $MachinePrecision] * N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision] + c), $MachinePrecision])), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.80000000000000004

   1. Initial program 55.9%

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

      1. 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. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 57.5%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   6. Applied rewrites 90.7%

      \[\leadsto \frac{\left(\left(-c\right) - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) + \mathsf{fma}\left(\left({\left(c \cdot a\right)}^{4} \cdot 20\right) \cdot -0.25, \frac{{b}^{-6}}{a}, \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   8. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{{b}^{-6}}{a}, {\left(a \cdot c\right)}^{4} \cdot -5, \left({b}^{-4} \cdot -2\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) + \left(-\mathsf{fma}\left(\frac{a}{b \cdot b}, c \cdot c, c\right)\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.8:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} - \frac{\frac{a}{b \cdot b} \cdot \left(c \cdot c\right) - -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.8)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (-
      (/ (- c) b)
      (/
       (-
        (* (/ a (* b b)) (* c c))
        (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0))))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.8) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = (-c / b) - ((((a / (b * b)) * (c * c)) - (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0)))) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) / b) - Float64(Float64(Float64(Float64(a / Float64(b * b)) * Float64(c * c)) - Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0)))) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] - N[(N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] - N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.80000000000000004

   1. Initial program 55.9%

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

      1. 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. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 57.5%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   6. Applied rewrites 90.7%

      \[\leadsto \frac{\left(\left(-c\right) - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) + \mathsf{fma}\left(\left({\left(c \cdot a\right)}^{4} \cdot 20\right) \cdot -0.25, \frac{{b}^{-6}}{a}, \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   8. Applied rewrites 90.7%

      \[\leadsto \frac{-c}{b} - \color{blue}{\frac{\frac{a}{b \cdot b} \cdot \left(c \cdot c\right) - \mathsf{fma}\left(\frac{{b}^{-6}}{a}, {\left(a \cdot c\right)}^{4} \cdot -5, \left({b}^{-4} \cdot -2\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right)}{b}} \]

   9. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-pow.f64 87.6

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

   11. Applied rewrites 87.6%

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

Alternative 8: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.8:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-c\right) - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) + -2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.8)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (/
      (+
       (- (- c) (* (* c c) (/ a (* b b))))
       (* -2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.8) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = ((-c - ((c * c) * (a / (b * b)))) + (-2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(-c) - Float64(Float64(c * c) * Float64(a / Float64(b * b)))) + Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-c) - N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.80000000000000004

   1. Initial program 55.9%

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

      1. 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. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 57.5%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   6. Applied rewrites 90.7%

      \[\leadsto \frac{\left(\left(-c\right) - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) + \mathsf{fma}\left(\left({\left(c \cdot a\right)}^{4} \cdot 20\right) \cdot -0.25, \frac{{b}^{-6}}{a}, \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 87.6

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

   9. Applied rewrites 87.6%

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

Alternative 9: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.8:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} - \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.8)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (*
      c
      (-
       (* c (- (* -2.0 (/ (* (pow a 2.0) c) (pow b 5.0))) (/ a (pow b 3.0))))
       (/ 1.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.8) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = c * ((c * ((-2.0 * ((pow(a, 2.0) * c) / pow(b, 5.0))) - (a / pow(b, 3.0)))) - (1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * c) / (b ^ 5.0))) - Float64(a / (b ^ 3.0)))) - Float64(1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.80000000000000004

   1. Initial program 55.9%

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

      1. 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. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 57.5%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   6. Applied rewrites 90.7%

      \[\leadsto \frac{\left(\left(-c\right) - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) + \mathsf{fma}\left(\left({\left(c \cdot a\right)}^{4} \cdot 20\right) \cdot -0.25, \frac{{b}^{-6}}{a}, \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   8. Applied rewrites 90.7%

      \[\leadsto \frac{-c}{b} - \color{blue}{\frac{\frac{a}{b \cdot b} \cdot \left(c \cdot c\right) - \mathsf{fma}\left(\frac{{b}^{-6}}{a}, {\left(a \cdot c\right)}^{4} \cdot -5, \left({b}^{-4} \cdot -2\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right)}{b}} \]

   9. Taylor expanded in c around 0

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

   11. Applied rewrites 87.4%

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

Alternative 10: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.8:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{a}{b \cdot b} \cdot \left(c \cdot c\right)}{b} - \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.8)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (- (/ (- (* (/ a (* b b)) (* c c))) b) (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.8) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = (-((a / (b * b)) * (c * c)) / b) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.8)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(a / Float64(b * b)) * Float64(c * c))) / b) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]) / b), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.80000000000000004

   1. Initial program 55.9%

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

      1. 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. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 57.5%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. add-flip N/A

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

      5. lower--.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

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

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

      16. distribute-frac-neg2 N/A

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

   3. Applied rewrites 55.0%

      \[\leadsto \color{blue}{\frac{b}{-2 \cdot a} - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.2

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

   6. Applied rewrites 81.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

   8. Applied rewrites 81.2%

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

Alternative 11: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.8:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{a}{b \cdot b} \cdot \left(c \cdot c\right)}{b} - \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.8)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (- (/ (- (* (/ a (* b b)) (* c c))) b) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.8) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = (-((a / (b * b)) * (c * c)) / b) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.8)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(a / Float64(b * b)) * Float64(c * c))) / b) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]) / b), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.80000000000000004

   1. Initial program 55.9%

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

      1. 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

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

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

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sqr-neg-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-neg-in N/A

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 56.0

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

   3. Applied rewrites 56.0%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. add-flip N/A

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

      5. lower--.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

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

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

      16. distribute-frac-neg2 N/A

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

   3. Applied rewrites 55.0%

      \[\leadsto \color{blue}{\frac{b}{-2 \cdot a} - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.2

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

   6. Applied rewrites 81.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

   8. Applied rewrites 81.2%

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

Alternative 12: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{a}{b \cdot b} \cdot \left(c \cdot c\right)}{b} - \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.8)
   (/ (- (sqrt (fma (* c -4.0) a (* b b))) b) (+ a a))
   (- (/ (- (* (/ a (* b b)) (* c c))) b) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.8) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) / (a + a);
	} else {
		tmp = (-((a / (b * b)) * (c * c)) / b) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.8)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(a / Float64(b * b)) * Float64(c * c))) / b) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[((-N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]) / b), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.80000000000000004

   1. Initial program 55.9%

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

   3. Applied rewrites 55.9%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. add-flip N/A

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

      5. lower--.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

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

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

      16. distribute-frac-neg2 N/A

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

   3. Applied rewrites 55.0%

      \[\leadsto \color{blue}{\frac{b}{-2 \cdot a} - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.2

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

   6. Applied rewrites 81.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

   8. Applied rewrites 81.2%

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

Alternative 13: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.8:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\frac{a}{b \cdot b} \cdot \left(c \cdot c\right)\right) - c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.8)
   (/ (- (sqrt (fma (* c -4.0) a (* b b))) b) (+ a a))
   (/ (- (- (* (/ a (* b b)) (* c c))) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.8) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) / (a + a);
	} else {
		tmp = (-((a / (b * b)) * (c * c)) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.8)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(-Float64(Float64(a / Float64(b * b)) * Float64(c * c))) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.8], N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[((-N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]) - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.80000000000000004

   1. Initial program 55.9%

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

   3. Applied rewrites 55.9%

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

   if -0.80000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. add-flip N/A

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

      5. lower--.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

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

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

      16. distribute-frac-neg2 N/A

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

   3. Applied rewrites 55.0%

      \[\leadsto \color{blue}{\frac{b}{-2 \cdot a} - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.2

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

   6. Applied rewrites 81.2%

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

   8. Applied rewrites 81.2%

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

Alternative 14: 81.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. add-flip N/A

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

   5. lower--.f64 N/A

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

   6. frac-2neg N/A

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

   7. lower-/.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. remove-double-neg N/A

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

   10. lift-*.f64 N/A

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

   11. distribute-lft-neg-in N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. mult-flip N/A

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

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

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

   16. distribute-frac-neg2 N/A

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

3. Applied rewrites 55.0%

   \[\leadsto \color{blue}{\frac{b}{-2 \cdot a} - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 81.2

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

6. Applied rewrites 81.2%

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

8. Applied rewrites 81.2%

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

Alternative 15: 64.0% accurate, 4.6× speedup

Math

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

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 55.9%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 64.0

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

4. Applied rewrites 64.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 64.0

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

6. Applied rewrites 64.0%

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

Reproduce

herbie shell --seed 2025176 
(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)))