Quadratic roots, narrow range

Percentage Accurate: 55.5% → 92.1%

Time: 10.5s

Alternatives: 23

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: 55.5% 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: 92.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(c \cdot c\right)\\ t_1 := \frac{t\_0}{b \cdot b}\\ t_2 := t\_0 \cdot 0\\ t_3 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_4 := {\left(a \cdot c\right)}^{4}\\ t_5 := \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ t_6 := t\_5 \cdot {b}^{-4}\\ t_7 := t\_4 \cdot {b}^{-6}\\ t_8 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_9 := t\_4 \cdot 20\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, \sqrt{t\_3} \cdot t\_3\right)}{t\_8 + \mathsf{fma}\left(b, b, \sqrt{t\_8} \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_6, -8, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(t\_1, -4, \mathsf{fma}\left(t\_6, -4, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(t\_5 \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(\left(\left(t\_2 \cdot c\right) \cdot a\right) \cdot {b}^{-4}, -2, \mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot \left(t\_2 \cdot \left(c \cdot c\right)\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, t\_1, \frac{\mathsf{fma}\left(-1, t\_9, -0.5 \cdot t\_9\right)}{{b}^{6}} + \mathsf{fma}\left(t\_1, 4, \mathsf{fma}\left(t\_7, 4, \mathsf{fma}\left(8, t\_1, \mathsf{fma}\left(16, t\_6, t\_7 \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(b, b + \left(\mathsf{fma}\left(t\_5 \cdot {b}^{-6}, -4, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, t\_0 \cdot {b}^{-4}\right)\right) + 1\right) \cdot b, \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a a) (* c c)))
        (t_1 (/ t_0 (* b b)))
        (t_2 (* t_0 0.0))
        (t_3 (fma (* -4.0 c) a (* b b)))
        (t_4 (pow (* a c) 4.0))
        (t_5 (* (* (* c c) c) (* (* a a) a)))
        (t_6 (* t_5 (pow b -4.0)))
        (t_7 (* t_4 (pow b -6.0)))
        (t_8 (fma (* c -4.0) a (* b b)))
        (t_9 (* t_4 20.0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (/
       (fma (* (- b) b) b (* (sqrt t_3) t_3))
       (+ t_8 (fma b b (* (sqrt t_8) b))))
      (* 2.0 a))
     (/
      (/
       (*
        (fma
         t_6
         -8.0
         (fma
          (* -4.0 a)
          c
          (fma
           t_1
           -4.0
           (fma
            t_6
            -4.0
            (fma
             (* -2.0 a)
             c
             (fma
              (* (* (* (* t_5 0.0) c) a) (pow b -6.0))
              -2.0
              (fma
               (* (* (* t_2 c) a) (pow b -4.0))
               -2.0
               (fma
                (* (* (* a a) (* t_2 (* c c))) (pow b -6.0))
                -2.0
                (fma
                 -2.0
                 t_1
                 (+
                  (/ (fma -1.0 t_9 (* -0.5 t_9)) (pow b 6.0))
                  (fma
                   t_1
                   4.0
                   (fma
                    t_7
                    4.0
                    (fma 8.0 t_1 (fma 16.0 t_6 (* t_7 32.0)))))))))))))))
        b)
       (fma
        b
        (+
         b
         (*
          (+
           (fma
            (* t_5 (pow b -6.0))
            -4.0
            (* -2.0 (fma a (/ c (* b b)) (* t_0 (pow b -4.0)))))
           1.0)
          b))
        (fma (* a c) -4.0 (* b b))))
      (+ a a)))))

C

double code(double a, double b, double c) {
	double t_0 = (a * a) * (c * c);
	double t_1 = t_0 / (b * b);
	double t_2 = t_0 * 0.0;
	double t_3 = fma((-4.0 * c), a, (b * b));
	double t_4 = pow((a * c), 4.0);
	double t_5 = ((c * c) * c) * ((a * a) * a);
	double t_6 = t_5 * pow(b, -4.0);
	double t_7 = t_4 * pow(b, -6.0);
	double t_8 = fma((c * -4.0), a, (b * b));
	double t_9 = t_4 * 20.0;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (fma((-b * b), b, (sqrt(t_3) * t_3)) / (t_8 + fma(b, b, (sqrt(t_8) * b)))) / (2.0 * a);
	} else {
		tmp = ((fma(t_6, -8.0, fma((-4.0 * a), c, fma(t_1, -4.0, fma(t_6, -4.0, fma((-2.0 * a), c, fma(((((t_5 * 0.0) * c) * a) * pow(b, -6.0)), -2.0, fma((((t_2 * c) * a) * pow(b, -4.0)), -2.0, fma((((a * a) * (t_2 * (c * c))) * pow(b, -6.0)), -2.0, fma(-2.0, t_1, ((fma(-1.0, t_9, (-0.5 * t_9)) / pow(b, 6.0)) + fma(t_1, 4.0, fma(t_7, 4.0, fma(8.0, t_1, fma(16.0, t_6, (t_7 * 32.0))))))))))))))) * b) / fma(b, (b + ((fma((t_5 * pow(b, -6.0)), -4.0, (-2.0 * fma(a, (c / (b * b)), (t_0 * pow(b, -4.0))))) + 1.0) * b)), fma((a * c), -4.0, (b * b)))) / (a + a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * a) * Float64(c * c))
	t_1 = Float64(t_0 / Float64(b * b))
	t_2 = Float64(t_0 * 0.0)
	t_3 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_4 = Float64(a * c) ^ 4.0
	t_5 = Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) * a))
	t_6 = Float64(t_5 * (b ^ -4.0))
	t_7 = Float64(t_4 * (b ^ -6.0))
	t_8 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_9 = Float64(t_4 * 20.0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, Float64(sqrt(t_3) * t_3)) / Float64(t_8 + fma(b, b, Float64(sqrt(t_8) * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(fma(t_6, -8.0, fma(Float64(-4.0 * a), c, fma(t_1, -4.0, fma(t_6, -4.0, fma(Float64(-2.0 * a), c, fma(Float64(Float64(Float64(Float64(t_5 * 0.0) * c) * a) * (b ^ -6.0)), -2.0, fma(Float64(Float64(Float64(t_2 * c) * a) * (b ^ -4.0)), -2.0, fma(Float64(Float64(Float64(a * a) * Float64(t_2 * Float64(c * c))) * (b ^ -6.0)), -2.0, fma(-2.0, t_1, Float64(Float64(fma(-1.0, t_9, Float64(-0.5 * t_9)) / (b ^ 6.0)) + fma(t_1, 4.0, fma(t_7, 4.0, fma(8.0, t_1, fma(16.0, t_6, Float64(t_7 * 32.0))))))))))))))) * b) / fma(b, Float64(b + Float64(Float64(fma(Float64(t_5 * (b ^ -6.0)), -4.0, Float64(-2.0 * fma(a, Float64(c / Float64(b * b)), Float64(t_0 * (b ^ -4.0))))) + 1.0) * b)), fma(Float64(a * c), -4.0, Float64(b * b)))) / Float64(a + a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * 0.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$4 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$4 * 20.0), $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], -1.5], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[(N[Sqrt[t$95$3], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$8 + N[(b * b + N[(N[Sqrt[t$95$8], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$6 * -8.0 + N[(N[(-4.0 * a), $MachinePrecision] * c + N[(t$95$1 * -4.0 + N[(t$95$6 * -4.0 + N[(N[(-2.0 * a), $MachinePrecision] * c + N[(N[(N[(N[(N[(t$95$5 * 0.0), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(t$95$2 * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(a * a), $MachinePrecision] * N[(t$95$2 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(-2.0 * t$95$1 + N[(N[(N[(-1.0 * t$95$9 + N[(-0.5 * t$95$9), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * 4.0 + N[(t$95$7 * 4.0 + N[(8.0 * t$95$1 + N[(16.0 * t$95$6 + N[(t$95$7 * 32.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(b * N[(b + N[(N[(N[(N[(t$95$5 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -4.0 + N[(-2.0 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 56.3%

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

      13. lift-pow.f64 N/A

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

      14. cube-mult N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 57.3%

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

   5. Applied rewrites 57.3%

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

   if -1.5 < (/.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 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

   4. Taylor expanded in b around inf

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

   6. Applied rewrites 91.2%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

   9. Applied rewrites 91.3%

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

   11. Applied rewrites 91.4%

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

Alternative 2: 92.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(c \cdot c\right)\\ t_1 := \frac{t\_0}{b \cdot b}\\ t_2 := t\_0 \cdot 0\\ t_3 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_4 := \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ t_5 := t\_4 \cdot {b}^{-4}\\ t_6 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_7 := {\left(a \cdot c\right)}^{4}\\ t_8 := t\_7 \cdot 20\\ t_9 := t\_7 \cdot {b}^{-6}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, \sqrt{t\_3} \cdot t\_3\right)}{t\_6 + \mathsf{fma}\left(b, b, \sqrt{t\_6} \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_5, -8, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(t\_1, -4, \mathsf{fma}\left(t\_5, -4, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(t\_4 \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(\left(\left(t\_2 \cdot c\right) \cdot a\right) \cdot {b}^{-4}, -2, \mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot \left(t\_2 \cdot \left(c \cdot c\right)\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, t\_1, \frac{\mathsf{fma}\left(-1, t\_8, -0.5 \cdot t\_8\right)}{{b}^{6}} + \mathsf{fma}\left(t\_1, 4, \mathsf{fma}\left(t\_9, 4, \mathsf{fma}\left(8, t\_1, \mathsf{fma}\left(16, t\_5, t\_9 \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(b, b + \left(\mathsf{fma}\left(t\_4 \cdot {b}^{-6}, -4, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, t\_0 \cdot {b}^{-4}\right)\right) + 1\right) \cdot b, \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right) \cdot \left(a + a\right)}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a a) (* c c)))
        (t_1 (/ t_0 (* b b)))
        (t_2 (* t_0 0.0))
        (t_3 (fma (* -4.0 c) a (* b b)))
        (t_4 (* (* (* c c) c) (* (* a a) a)))
        (t_5 (* t_4 (pow b -4.0)))
        (t_6 (fma (* c -4.0) a (* b b)))
        (t_7 (pow (* a c) 4.0))
        (t_8 (* t_7 20.0))
        (t_9 (* t_7 (pow b -6.0))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (/
       (fma (* (- b) b) b (* (sqrt t_3) t_3))
       (+ t_6 (fma b b (* (sqrt t_6) b))))
      (* 2.0 a))
     (/
      (*
       (fma
        t_5
        -8.0
        (fma
         (* -4.0 a)
         c
         (fma
          t_1
          -4.0
          (fma
           t_5
           -4.0
           (fma
            (* -2.0 a)
            c
            (fma
             (* (* (* (* t_4 0.0) c) a) (pow b -6.0))
             -2.0
             (fma
              (* (* (* t_2 c) a) (pow b -4.0))
              -2.0
              (fma
               (* (* (* a a) (* t_2 (* c c))) (pow b -6.0))
               -2.0
               (fma
                -2.0
                t_1
                (+
                 (/ (fma -1.0 t_8 (* -0.5 t_8)) (pow b 6.0))
                 (fma
                  t_1
                  4.0
                  (fma
                   t_9
                   4.0
                   (fma 8.0 t_1 (fma 16.0 t_5 (* t_9 32.0)))))))))))))))
       b)
      (*
       (fma
        b
        (+
         b
         (*
          (+
           (fma
            (* t_4 (pow b -6.0))
            -4.0
            (* -2.0 (fma a (/ c (* b b)) (* t_0 (pow b -4.0)))))
           1.0)
          b))
        (fma (* a c) -4.0 (* b b)))
       (+ a a))))))

C

double code(double a, double b, double c) {
	double t_0 = (a * a) * (c * c);
	double t_1 = t_0 / (b * b);
	double t_2 = t_0 * 0.0;
	double t_3 = fma((-4.0 * c), a, (b * b));
	double t_4 = ((c * c) * c) * ((a * a) * a);
	double t_5 = t_4 * pow(b, -4.0);
	double t_6 = fma((c * -4.0), a, (b * b));
	double t_7 = pow((a * c), 4.0);
	double t_8 = t_7 * 20.0;
	double t_9 = t_7 * pow(b, -6.0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (fma((-b * b), b, (sqrt(t_3) * t_3)) / (t_6 + fma(b, b, (sqrt(t_6) * b)))) / (2.0 * a);
	} else {
		tmp = (fma(t_5, -8.0, fma((-4.0 * a), c, fma(t_1, -4.0, fma(t_5, -4.0, fma((-2.0 * a), c, fma(((((t_4 * 0.0) * c) * a) * pow(b, -6.0)), -2.0, fma((((t_2 * c) * a) * pow(b, -4.0)), -2.0, fma((((a * a) * (t_2 * (c * c))) * pow(b, -6.0)), -2.0, fma(-2.0, t_1, ((fma(-1.0, t_8, (-0.5 * t_8)) / pow(b, 6.0)) + fma(t_1, 4.0, fma(t_9, 4.0, fma(8.0, t_1, fma(16.0, t_5, (t_9 * 32.0))))))))))))))) * b) / (fma(b, (b + ((fma((t_4 * pow(b, -6.0)), -4.0, (-2.0 * fma(a, (c / (b * b)), (t_0 * pow(b, -4.0))))) + 1.0) * b)), fma((a * c), -4.0, (b * b))) * (a + a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * a) * Float64(c * c))
	t_1 = Float64(t_0 / Float64(b * b))
	t_2 = Float64(t_0 * 0.0)
	t_3 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_4 = Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) * a))
	t_5 = Float64(t_4 * (b ^ -4.0))
	t_6 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_7 = Float64(a * c) ^ 4.0
	t_8 = Float64(t_7 * 20.0)
	t_9 = Float64(t_7 * (b ^ -6.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, Float64(sqrt(t_3) * t_3)) / Float64(t_6 + fma(b, b, Float64(sqrt(t_6) * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(t_5, -8.0, fma(Float64(-4.0 * a), c, fma(t_1, -4.0, fma(t_5, -4.0, fma(Float64(-2.0 * a), c, fma(Float64(Float64(Float64(Float64(t_4 * 0.0) * c) * a) * (b ^ -6.0)), -2.0, fma(Float64(Float64(Float64(t_2 * c) * a) * (b ^ -4.0)), -2.0, fma(Float64(Float64(Float64(a * a) * Float64(t_2 * Float64(c * c))) * (b ^ -6.0)), -2.0, fma(-2.0, t_1, Float64(Float64(fma(-1.0, t_8, Float64(-0.5 * t_8)) / (b ^ 6.0)) + fma(t_1, 4.0, fma(t_9, 4.0, fma(8.0, t_1, fma(16.0, t_5, Float64(t_9 * 32.0))))))))))))))) * b) / Float64(fma(b, Float64(b + Float64(Float64(fma(Float64(t_4 * (b ^ -6.0)), -4.0, Float64(-2.0 * fma(a, Float64(c / Float64(b * b)), Float64(t_0 * (b ^ -4.0))))) + 1.0) * b)), fma(Float64(a * c), -4.0, Float64(b * b))) * Float64(a + a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * 0.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$8 = N[(t$95$7 * 20.0), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$7 * N[Power[b, -6.0], $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], -1.5], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[(N[Sqrt[t$95$3], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$6 + N[(b * b + N[(N[Sqrt[t$95$6], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$5 * -8.0 + N[(N[(-4.0 * a), $MachinePrecision] * c + N[(t$95$1 * -4.0 + N[(t$95$5 * -4.0 + N[(N[(-2.0 * a), $MachinePrecision] * c + N[(N[(N[(N[(N[(t$95$4 * 0.0), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(t$95$2 * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(a * a), $MachinePrecision] * N[(t$95$2 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(-2.0 * t$95$1 + N[(N[(N[(-1.0 * t$95$8 + N[(-0.5 * t$95$8), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * 4.0 + N[(t$95$9 * 4.0 + N[(8.0 * t$95$1 + N[(16.0 * t$95$5 + N[(t$95$9 * 32.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(N[(b * N[(b + N[(N[(N[(N[(t$95$4 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -4.0 + N[(-2.0 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0 + 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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 56.3%

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

      13. lift-pow.f64 N/A

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

      14. cube-mult N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 57.3%

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

   5. Applied rewrites 57.3%

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

   if -1.5 < (/.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 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

   4. Taylor expanded in b around inf

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

   6. Applied rewrites 91.2%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

   9. Applied rewrites 91.3%

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

   11. Applied rewrites 91.4%

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

Alternative 3: 92.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \left(c \cdot c\right) \cdot \left(a \cdot a\right)\\ t_1 := \frac{t\_0}{b \cdot b}\\ t_2 := t\_0 \cdot 0\\ t_3 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_4 := \sqrt{t\_3}\\ t_5 := {\left(a \cdot c\right)}^{4}\\ t_6 := \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ t_7 := t\_6 \cdot {b}^{-4}\\ t_8 := t\_5 \cdot {b}^{-6}\\ t_9 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_10 := t\_5 \cdot 20\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, t\_4 \cdot t\_3\right)}{t\_9 + \mathsf{fma}\left(b, b, \sqrt{t\_9} \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_7, -8, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(t\_1, -4, \mathsf{fma}\left(t\_7, -4, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(t\_6 \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(\left(\left(t\_2 \cdot c\right) \cdot a\right) \cdot {b}^{-4}, -2, \mathsf{fma}\left(\left(\left(t\_2 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, t\_1, \frac{\mathsf{fma}\left(-1, t\_10, -0.5 \cdot t\_10\right)}{{b}^{6}} + \mathsf{fma}\left(t\_1, 4, \mathsf{fma}\left(t\_8, 4, \mathsf{fma}\left(8, t\_1, \mathsf{fma}\left(16, t\_7, t\_8 \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(b, b, b \cdot \left(b + t\_4\right)\right)\right)}}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c c) (* a a)))
        (t_1 (/ t_0 (* b b)))
        (t_2 (* t_0 0.0))
        (t_3 (fma (* -4.0 c) a (* b b)))
        (t_4 (sqrt t_3))
        (t_5 (pow (* a c) 4.0))
        (t_6 (* (* (* c c) c) (* (* a a) a)))
        (t_7 (* t_6 (pow b -4.0)))
        (t_8 (* t_5 (pow b -6.0)))
        (t_9 (fma (* c -4.0) a (* b b)))
        (t_10 (* t_5 20.0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (/ (fma (* (- b) b) b (* t_4 t_3)) (+ t_9 (fma b b (* (sqrt t_9) b))))
      (* 2.0 a))
     (/
      (/
       (*
        (fma
         t_7
         -8.0
         (fma
          (* -4.0 a)
          c
          (fma
           t_1
           -4.0
           (fma
            t_7
            -4.0
            (fma
             (* -2.0 a)
             c
             (fma
              (* (* (* (* t_6 0.0) c) a) (pow b -6.0))
              -2.0
              (fma
               (* (* (* t_2 c) a) (pow b -4.0))
               -2.0
               (fma
                (* (* (* t_2 (* c c)) (* a a)) (pow b -6.0))
                -2.0
                (fma
                 -2.0
                 t_1
                 (+
                  (/ (fma -1.0 t_10 (* -0.5 t_10)) (pow b 6.0))
                  (fma
                   t_1
                   4.0
                   (fma
                    t_8
                    4.0
                    (fma 8.0 t_1 (fma 16.0 t_7 (* t_8 32.0)))))))))))))))
        b)
       (fma (* -4.0 a) c (fma b b (* b (+ b t_4)))))
      (+ a a)))))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) * (a * a);
	double t_1 = t_0 / (b * b);
	double t_2 = t_0 * 0.0;
	double t_3 = fma((-4.0 * c), a, (b * b));
	double t_4 = sqrt(t_3);
	double t_5 = pow((a * c), 4.0);
	double t_6 = ((c * c) * c) * ((a * a) * a);
	double t_7 = t_6 * pow(b, -4.0);
	double t_8 = t_5 * pow(b, -6.0);
	double t_9 = fma((c * -4.0), a, (b * b));
	double t_10 = t_5 * 20.0;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (fma((-b * b), b, (t_4 * t_3)) / (t_9 + fma(b, b, (sqrt(t_9) * b)))) / (2.0 * a);
	} else {
		tmp = ((fma(t_7, -8.0, fma((-4.0 * a), c, fma(t_1, -4.0, fma(t_7, -4.0, fma((-2.0 * a), c, fma(((((t_6 * 0.0) * c) * a) * pow(b, -6.0)), -2.0, fma((((t_2 * c) * a) * pow(b, -4.0)), -2.0, fma((((t_2 * (c * c)) * (a * a)) * pow(b, -6.0)), -2.0, fma(-2.0, t_1, ((fma(-1.0, t_10, (-0.5 * t_10)) / pow(b, 6.0)) + fma(t_1, 4.0, fma(t_8, 4.0, fma(8.0, t_1, fma(16.0, t_7, (t_8 * 32.0))))))))))))))) * b) / fma((-4.0 * a), c, fma(b, b, (b * (b + t_4))))) / (a + a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) * Float64(a * a))
	t_1 = Float64(t_0 / Float64(b * b))
	t_2 = Float64(t_0 * 0.0)
	t_3 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_4 = sqrt(t_3)
	t_5 = Float64(a * c) ^ 4.0
	t_6 = Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) * a))
	t_7 = Float64(t_6 * (b ^ -4.0))
	t_8 = Float64(t_5 * (b ^ -6.0))
	t_9 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_10 = Float64(t_5 * 20.0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, Float64(t_4 * t_3)) / Float64(t_9 + fma(b, b, Float64(sqrt(t_9) * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(fma(t_7, -8.0, fma(Float64(-4.0 * a), c, fma(t_1, -4.0, fma(t_7, -4.0, fma(Float64(-2.0 * a), c, fma(Float64(Float64(Float64(Float64(t_6 * 0.0) * c) * a) * (b ^ -6.0)), -2.0, fma(Float64(Float64(Float64(t_2 * c) * a) * (b ^ -4.0)), -2.0, fma(Float64(Float64(Float64(t_2 * Float64(c * c)) * Float64(a * a)) * (b ^ -6.0)), -2.0, fma(-2.0, t_1, Float64(Float64(fma(-1.0, t_10, Float64(-0.5 * t_10)) / (b ^ 6.0)) + fma(t_1, 4.0, fma(t_8, 4.0, fma(8.0, t_1, fma(16.0, t_7, Float64(t_8 * 32.0))))))))))))))) * b) / fma(Float64(-4.0 * a), c, fma(b, b, Float64(b * Float64(b + t_4))))) / Float64(a + a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * 0.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$6 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$5 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(t$95$5 * 20.0), $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], -1.5], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[(t$95$4 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$9 + N[(b * b + N[(N[Sqrt[t$95$9], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$7 * -8.0 + N[(N[(-4.0 * a), $MachinePrecision] * c + N[(t$95$1 * -4.0 + N[(t$95$7 * -4.0 + N[(N[(-2.0 * a), $MachinePrecision] * c + N[(N[(N[(N[(N[(t$95$6 * 0.0), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(t$95$2 * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(t$95$2 * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(-2.0 * t$95$1 + N[(N[(N[(-1.0 * t$95$10 + N[(-0.5 * t$95$10), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * 4.0 + N[(t$95$8 * 4.0 + N[(8.0 * t$95$1 + N[(16.0 * t$95$7 + N[(t$95$8 * 32.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b + N[(b * N[(b + t$95$4), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 56.3%

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

      13. lift-pow.f64 N/A

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

      14. cube-mult N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 57.3%

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

   5. Applied rewrites 57.3%

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

   if -1.5 < (/.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 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

   4. Taylor expanded in b around inf

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

   6. Applied rewrites 91.2%

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

   8. Applied rewrites 91.2%

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

Alternative 4: 92.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \left(c \cdot c\right) \cdot \left(a \cdot a\right)\\ t_1 := \frac{t\_0}{b \cdot b}\\ t_2 := t\_0 \cdot 0\\ t_3 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_4 := \sqrt{t\_3}\\ t_5 := \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ t_6 := t\_5 \cdot {b}^{-4}\\ t_7 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_8 := {\left(a \cdot c\right)}^{4}\\ t_9 := t\_8 \cdot 20\\ t_10 := t\_8 \cdot {b}^{-6}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, t\_4 \cdot t\_3\right)}{t\_7 + \mathsf{fma}\left(b, b, \sqrt{t\_7} \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_6, -8, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(t\_1, -4, \mathsf{fma}\left(t\_6, -4, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(t\_5 \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(\left(\left(t\_2 \cdot c\right) \cdot a\right) \cdot {b}^{-4}, -2, \mathsf{fma}\left(\left(\left(t\_2 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, t\_1, \frac{\mathsf{fma}\left(-1, t\_9, -0.5 \cdot t\_9\right)}{{b}^{6}} + \mathsf{fma}\left(t\_1, 4, \mathsf{fma}\left(t\_10, 4, \mathsf{fma}\left(8, t\_1, \mathsf{fma}\left(16, t\_6, t\_10 \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(b, b, b \cdot \left(b + t\_4\right)\right)\right) \cdot \left(a + a\right)}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c c) (* a a)))
        (t_1 (/ t_0 (* b b)))
        (t_2 (* t_0 0.0))
        (t_3 (fma (* -4.0 c) a (* b b)))
        (t_4 (sqrt t_3))
        (t_5 (* (* (* c c) c) (* (* a a) a)))
        (t_6 (* t_5 (pow b -4.0)))
        (t_7 (fma (* c -4.0) a (* b b)))
        (t_8 (pow (* a c) 4.0))
        (t_9 (* t_8 20.0))
        (t_10 (* t_8 (pow b -6.0))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (/ (fma (* (- b) b) b (* t_4 t_3)) (+ t_7 (fma b b (* (sqrt t_7) b))))
      (* 2.0 a))
     (/
      (*
       (fma
        t_6
        -8.0
        (fma
         (* -4.0 a)
         c
         (fma
          t_1
          -4.0
          (fma
           t_6
           -4.0
           (fma
            (* -2.0 a)
            c
            (fma
             (* (* (* (* t_5 0.0) c) a) (pow b -6.0))
             -2.0
             (fma
              (* (* (* t_2 c) a) (pow b -4.0))
              -2.0
              (fma
               (* (* (* t_2 (* c c)) (* a a)) (pow b -6.0))
               -2.0
               (fma
                -2.0
                t_1
                (+
                 (/ (fma -1.0 t_9 (* -0.5 t_9)) (pow b 6.0))
                 (fma
                  t_1
                  4.0
                  (fma
                   t_10
                   4.0
                   (fma 8.0 t_1 (fma 16.0 t_6 (* t_10 32.0)))))))))))))))
       b)
      (* (fma (* -4.0 a) c (fma b b (* b (+ b t_4)))) (+ a a))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) * (a * a);
	double t_1 = t_0 / (b * b);
	double t_2 = t_0 * 0.0;
	double t_3 = fma((-4.0 * c), a, (b * b));
	double t_4 = sqrt(t_3);
	double t_5 = ((c * c) * c) * ((a * a) * a);
	double t_6 = t_5 * pow(b, -4.0);
	double t_7 = fma((c * -4.0), a, (b * b));
	double t_8 = pow((a * c), 4.0);
	double t_9 = t_8 * 20.0;
	double t_10 = t_8 * pow(b, -6.0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (fma((-b * b), b, (t_4 * t_3)) / (t_7 + fma(b, b, (sqrt(t_7) * b)))) / (2.0 * a);
	} else {
		tmp = (fma(t_6, -8.0, fma((-4.0 * a), c, fma(t_1, -4.0, fma(t_6, -4.0, fma((-2.0 * a), c, fma(((((t_5 * 0.0) * c) * a) * pow(b, -6.0)), -2.0, fma((((t_2 * c) * a) * pow(b, -4.0)), -2.0, fma((((t_2 * (c * c)) * (a * a)) * pow(b, -6.0)), -2.0, fma(-2.0, t_1, ((fma(-1.0, t_9, (-0.5 * t_9)) / pow(b, 6.0)) + fma(t_1, 4.0, fma(t_10, 4.0, fma(8.0, t_1, fma(16.0, t_6, (t_10 * 32.0))))))))))))))) * b) / (fma((-4.0 * a), c, fma(b, b, (b * (b + t_4)))) * (a + a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) * Float64(a * a))
	t_1 = Float64(t_0 / Float64(b * b))
	t_2 = Float64(t_0 * 0.0)
	t_3 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_4 = sqrt(t_3)
	t_5 = Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) * a))
	t_6 = Float64(t_5 * (b ^ -4.0))
	t_7 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_8 = Float64(a * c) ^ 4.0
	t_9 = Float64(t_8 * 20.0)
	t_10 = Float64(t_8 * (b ^ -6.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, Float64(t_4 * t_3)) / Float64(t_7 + fma(b, b, Float64(sqrt(t_7) * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(t_6, -8.0, fma(Float64(-4.0 * a), c, fma(t_1, -4.0, fma(t_6, -4.0, fma(Float64(-2.0 * a), c, fma(Float64(Float64(Float64(Float64(t_5 * 0.0) * c) * a) * (b ^ -6.0)), -2.0, fma(Float64(Float64(Float64(t_2 * c) * a) * (b ^ -4.0)), -2.0, fma(Float64(Float64(Float64(t_2 * Float64(c * c)) * Float64(a * a)) * (b ^ -6.0)), -2.0, fma(-2.0, t_1, Float64(Float64(fma(-1.0, t_9, Float64(-0.5 * t_9)) / (b ^ 6.0)) + fma(t_1, 4.0, fma(t_10, 4.0, fma(8.0, t_1, fma(16.0, t_6, Float64(t_10 * 32.0))))))))))))))) * b) / Float64(fma(Float64(-4.0 * a), c, fma(b, b, Float64(b * Float64(b + t_4)))) * Float64(a + a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * 0.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$9 = N[(t$95$8 * 20.0), $MachinePrecision]}, Block[{t$95$10 = N[(t$95$8 * N[Power[b, -6.0], $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], -1.5], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[(t$95$4 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$7 + N[(b * b + N[(N[Sqrt[t$95$7], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$6 * -8.0 + N[(N[(-4.0 * a), $MachinePrecision] * c + N[(t$95$1 * -4.0 + N[(t$95$6 * -4.0 + N[(N[(-2.0 * a), $MachinePrecision] * c + N[(N[(N[(N[(N[(t$95$5 * 0.0), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(t$95$2 * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(t$95$2 * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(-2.0 * t$95$1 + N[(N[(N[(-1.0 * t$95$9 + N[(-0.5 * t$95$9), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * 4.0 + N[(t$95$10 * 4.0 + N[(8.0 * t$95$1 + N[(16.0 * t$95$6 + N[(t$95$10 * 32.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b + N[(b * N[(b + t$95$4), $MachinePrecision]), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 56.3%

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

      13. lift-pow.f64 N/A

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

      14. cube-mult N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 57.3%

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

   5. Applied rewrites 57.3%

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

   if -1.5 < (/.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 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

   4. Taylor expanded in b around inf

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

   6. Applied rewrites 91.2%

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

   8. Applied rewrites 91.2%

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

Alternative 5: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, \sqrt{t\_0} \cdot t\_0\right)}{t\_1 + \mathsf{fma}\left(b, b, \sqrt{t\_1} \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left({b}^{-4} \cdot -2\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right) - c\right) - \left(\frac{20}{{b}^{6} \cdot a} \cdot {\left(a \cdot c\right)}^{4}\right) \cdot 0.25}{b} - \frac{\frac{a}{b \cdot b} \cdot \left(c \cdot c\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (/
       (fma (* (- b) b) b (* (sqrt t_0) t_0))
       (+ t_1 (fma b b (* (sqrt t_1) b))))
      (* 2.0 a))
     (-
      (/
       (-
        (- (* (* (pow b -4.0) -2.0) (* (* (* (* a a) c) c) c)) c)
        (* (* (/ 20.0 (* (pow b 6.0) a)) (pow (* a c) 4.0)) 0.25))
       b)
      (/ (* (/ a (* b b)) (* c c)) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (fma((-b * b), b, (sqrt(t_0) * t_0)) / (t_1 + fma(b, b, (sqrt(t_1) * b)))) / (2.0 * a);
	} else {
		tmp = (((((pow(b, -4.0) * -2.0) * ((((a * a) * c) * c) * c)) - c) - (((20.0 / (pow(b, 6.0) * a)) * pow((a * c), 4.0)) * 0.25)) / b) - (((a / (b * b)) * (c * c)) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = 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)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, Float64(sqrt(t_0) * t_0)) / Float64(t_1 + fma(b, b, Float64(sqrt(t_1) * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64((b ^ -4.0) * -2.0) * Float64(Float64(Float64(Float64(a * a) * c) * c) * c)) - c) - Float64(Float64(Float64(20.0 / Float64((b ^ 6.0) * a)) * (Float64(a * c) ^ 4.0)) * 0.25)) / b) - Float64(Float64(Float64(a / Float64(b * b)) * Float64(c * c)) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], -1.5], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(b * b + N[(N[Sqrt[t$95$1], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(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] - c), $MachinePrecision] - N[(N[(N[(20.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] - N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 56.3%

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

      13. lift-pow.f64 N/A

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

      14. cube-mult N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 57.3%

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

   5. Applied rewrites 57.3%

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

   if -1.5 < (/.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 55.5%

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

   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%

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

   8. Applied rewrites 90.6%

      \[\leadsto \frac{\left(\left({b}^{-4} \cdot -2\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right) - c\right) - \left(\frac{20}{{b}^{6} \cdot a} \cdot {\left(a \cdot c\right)}^{4}\right) \cdot 0.25}{b} - \color{blue}{\frac{\frac{a}{b \cdot b} \cdot \left(c \cdot c\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, \sqrt{t\_0} \cdot t\_0\right)}{t\_1 + \mathsf{fma}\left(b, b, \sqrt{t\_1} \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({b}^{-4} \cdot -2\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right) - c}{b} - \frac{\mathsf{fma}\left(0.25, \frac{20}{{b}^{6} \cdot a} \cdot {\left(a \cdot c\right)}^{4}, \frac{a}{b \cdot b} \cdot \left(c \cdot c\right)\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (/
       (fma (* (- b) b) b (* (sqrt t_0) t_0))
       (+ t_1 (fma b b (* (sqrt t_1) b))))
      (* 2.0 a))
     (-
      (/ (- (* (* (pow b -4.0) -2.0) (* (* (* (* a a) c) c) c)) c) b)
      (/
       (fma
        0.25
        (* (/ 20.0 (* (pow b 6.0) a)) (pow (* a c) 4.0))
        (* (/ a (* b b)) (* c c)))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (fma((-b * b), b, (sqrt(t_0) * t_0)) / (t_1 + fma(b, b, (sqrt(t_1) * b)))) / (2.0 * a);
	} else {
		tmp = ((((pow(b, -4.0) * -2.0) * ((((a * a) * c) * c) * c)) - c) / b) - (fma(0.25, ((20.0 / (pow(b, 6.0) * a)) * pow((a * c), 4.0)), ((a / (b * b)) * (c * c))) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = 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)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, Float64(sqrt(t_0) * t_0)) / Float64(t_1 + fma(b, b, Float64(sqrt(t_1) * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64((b ^ -4.0) * -2.0) * Float64(Float64(Float64(Float64(a * a) * c) * c) * c)) - c) / b) - Float64(fma(0.25, Float64(Float64(20.0 / Float64((b ^ 6.0) * a)) * (Float64(a * c) ^ 4.0)), Float64(Float64(a / Float64(b * b)) * Float64(c * c))) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], -1.5], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(b * b + N[(N[Sqrt[t$95$1], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(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] - c), $MachinePrecision] / b), $MachinePrecision] - N[(N[(0.25 * N[(N[(20.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 56.3%

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

      13. lift-pow.f64 N/A

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

      14. cube-mult N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 57.3%

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

   5. Applied rewrites 57.3%

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

   if -1.5 < (/.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 55.5%

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

   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%

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

   8. Applied rewrites 90.7%

      \[\leadsto \frac{\left({b}^{-4} \cdot -2\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right) - c}{b} - \color{blue}{\frac{\mathsf{fma}\left(0.25, \frac{20}{{b}^{6} \cdot a} \cdot {\left(a \cdot c\right)}^{4}, \frac{a}{b \cdot b} \cdot \left(c \cdot c\right)\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, \sqrt{t\_0} \cdot t\_0\right)}{t\_1 + \mathsf{fma}\left(b, b, \sqrt{t\_1} \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \left(\mathsf{fma}\left(0.25, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (/
       (fma (* (- b) b) b (* (sqrt t_0) t_0))
       (+ t_1 (fma b b (* (sqrt t_1) b))))
      (* 2.0 a))
     (/
      (-
       (- c)
       (-
        (fma
         0.25
         (* (pow (* c a) 4.0) (/ 20.0 (* (pow b 6.0) a)))
         (* (* c c) (/ a (* b b))))
        (* (* (* (* a a) c) (* c c)) (* (pow b -4.0) -2.0))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (fma((-b * b), b, (sqrt(t_0) * t_0)) / (t_1 + fma(b, b, (sqrt(t_1) * b)))) / (2.0 * a);
	} else {
		tmp = (-c - (fma(0.25, (pow((c * a), 4.0) * (20.0 / (pow(b, 6.0) * a))), ((c * c) * (a / (b * b)))) - ((((a * a) * c) * (c * c)) * (pow(b, -4.0) * -2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = 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)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, Float64(sqrt(t_0) * t_0)) / Float64(t_1 + fma(b, b, Float64(sqrt(t_1) * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(fma(0.25, Float64((Float64(c * a) ^ 4.0) * Float64(20.0 / Float64((b ^ 6.0) * a))), Float64(Float64(c * c) * Float64(a / Float64(b * b)))) - Float64(Float64(Float64(Float64(a * a) * c) * Float64(c * c)) * Float64((b ^ -4.0) * -2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], -1.5], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(b * b + N[(N[Sqrt[t$95$1], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(N[(0.25 * N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[Power[b, -4.0], $MachinePrecision] * -2.0), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 56.3%

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

      13. lift-pow.f64 N/A

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

      14. cube-mult N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 57.3%

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

   5. Applied rewrites 57.3%

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

   if -1.5 < (/.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 55.5%

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

   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%

      \[\leadsto \frac{\left(-c\right) - \left(\mathsf{fma}\left(0.25, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, \sqrt{t\_0} \cdot t\_0\right)}{t\_1 + \mathsf{fma}\left(b, b, \sqrt{t\_1} \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left({b}^{-4} \cdot -2\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right) - c\right) - \left(\frac{20}{{b}^{6} \cdot a} \cdot {\left(a \cdot c\right)}^{4}\right) \cdot 0.25\right) - \frac{a}{b \cdot b} \cdot \left(c \cdot c\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (/
       (fma (* (- b) b) b (* (sqrt t_0) t_0))
       (+ t_1 (fma b b (* (sqrt t_1) b))))
      (* 2.0 a))
     (/
      (-
       (-
        (- (* (* (pow b -4.0) -2.0) (* (* (* (* a a) c) c) c)) c)
        (* (* (/ 20.0 (* (pow b 6.0) a)) (pow (* a c) 4.0)) 0.25))
       (* (/ a (* b b)) (* c c)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (fma((-b * b), b, (sqrt(t_0) * t_0)) / (t_1 + fma(b, b, (sqrt(t_1) * b)))) / (2.0 * a);
	} else {
		tmp = (((((pow(b, -4.0) * -2.0) * ((((a * a) * c) * c) * c)) - c) - (((20.0 / (pow(b, 6.0) * a)) * pow((a * c), 4.0)) * 0.25)) - ((a / (b * b)) * (c * c))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = 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)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, Float64(sqrt(t_0) * t_0)) / Float64(t_1 + fma(b, b, Float64(sqrt(t_1) * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64((b ^ -4.0) * -2.0) * Float64(Float64(Float64(Float64(a * a) * c) * c) * c)) - c) - Float64(Float64(Float64(20.0 / Float64((b ^ 6.0) * a)) * (Float64(a * c) ^ 4.0)) * 0.25)) - Float64(Float64(a / Float64(b * b)) * Float64(c * c))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], -1.5], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(b * b + N[(N[Sqrt[t$95$1], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(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] - c), $MachinePrecision] - N[(N[(N[(20.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision] - N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 56.3%

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

      13. lift-pow.f64 N/A

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

      14. cube-mult N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 57.3%

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

   5. Applied rewrites 57.3%

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

   if -1.5 < (/.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 55.5%

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

   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%

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

   8. Applied rewrites 90.7%

      \[\leadsto \frac{\left(\left(\left({b}^{-4} \cdot -2\right) \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right) - c\right) - \left(\frac{20}{{b}^{6} \cdot a} \cdot {\left(a \cdot c\right)}^{4}\right) \cdot 0.25\right) - \frac{a}{b \cdot b} \cdot \left(c \cdot c\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, \sqrt{t\_0} \cdot t\_0\right)}{t\_1 + \mathsf{fma}\left(b, b, \sqrt{t\_1} \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right) - c\right) - \mathsf{fma}\left(0.25, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (/
       (fma (* (- b) b) b (* (sqrt t_0) t_0))
       (+ t_1 (fma b b (* (sqrt t_1) b))))
      (* 2.0 a))
     (/
      (-
       (- (* (* (* (* a a) c) (* c c)) (* (pow b -4.0) -2.0)) c)
       (fma
        0.25
        (* (pow (* c a) 4.0) (/ 20.0 (* (pow b 6.0) a)))
        (* (* c c) (/ a (* b b)))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (fma((-b * b), b, (sqrt(t_0) * t_0)) / (t_1 + fma(b, b, (sqrt(t_1) * b)))) / (2.0 * a);
	} else {
		tmp = ((((((a * a) * c) * (c * c)) * (pow(b, -4.0) * -2.0)) - c) - fma(0.25, (pow((c * a), 4.0) * (20.0 / (pow(b, 6.0) * a))), ((c * c) * (a / (b * b))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = 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)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, Float64(sqrt(t_0) * t_0)) / Float64(t_1 + fma(b, b, Float64(sqrt(t_1) * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * a) * c) * Float64(c * c)) * Float64((b ^ -4.0) * -2.0)) - c) - fma(0.25, Float64((Float64(c * a) ^ 4.0) * Float64(20.0 / Float64((b ^ 6.0) * a))), Float64(Float64(c * c) * Float64(a / Float64(b * b))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], -1.5], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(b * b + N[(N[Sqrt[t$95$1], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[Power[b, -4.0], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] - N[(0.25 * N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 56.3%

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

      13. lift-pow.f64 N/A

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

      14. cube-mult N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 57.3%

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

   5. Applied rewrites 57.3%

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

   if -1.5 < (/.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 55.5%

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

   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%

      \[\leadsto \color{blue}{\frac{\left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right) - c\right) - \mathsf{fma}\left(0.25, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, \sqrt{t\_0} \cdot t\_0\right)}{t\_1 + \mathsf{fma}\left(b, b, \sqrt{t\_1} \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{5}}, -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (/
       (fma (* (- b) b) b (* (sqrt t_0) t_0))
       (+ t_1 (fma b b (* (sqrt t_1) b))))
      (* 2.0 a))
     (fma
      -1.0
      (/ c b)
      (*
       a
       (fma
        -2.0
        (/ (* a (pow c 3.0)) (pow b 5.0))
        (* -1.0 (/ (pow c 2.0) (pow b 3.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (fma((-b * b), b, (sqrt(t_0) * t_0)) / (t_1 + fma(b, b, (sqrt(t_1) * b)))) / (2.0 * a);
	} else {
		tmp = fma(-1.0, (c / b), (a * fma(-2.0, ((a * pow(c, 3.0)) / pow(b, 5.0)), (-1.0 * (pow(c, 2.0) / pow(b, 3.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = 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)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, Float64(sqrt(t_0) * t_0)) / Float64(t_1 + fma(b, b, Float64(sqrt(t_1) * b)))) / Float64(2.0 * a));
	else
		tmp = fma(-1.0, Float64(c / b), Float64(a * fma(-2.0, Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0)), Float64(-1.0 * Float64((c ^ 2.0) / (b ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], -1.5], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(b * b + N[(N[Sqrt[t$95$1], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision] + N[(a * N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 56.3%

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

      13. lift-pow.f64 N/A

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

      14. cube-mult N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 57.3%

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

   5. Applied rewrites 57.3%

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

   if -1.5 < (/.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 55.5%

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

   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%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 87.6%

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

Alternative 11: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, \sqrt{t\_0} \cdot t\_0\right)}{t\_1 + \mathsf{fma}\left(b, b, \sqrt{t\_1} \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (/
       (fma (* (- b) b) b (* (sqrt t_0) t_0))
       (+ t_1 (fma b b (* (sqrt t_1) b))))
      (* 2.0 a))
     (/
      (-
       (*
        a
        (-
         (* -2.0 (/ (* a (pow c 3.0)) (pow b 4.0)))
         (/ (pow c 2.0) (pow b 2.0))))
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (fma((-b * b), b, (sqrt(t_0) * t_0)) / (t_1 + fma(b, b, (sqrt(t_1) * b)))) / (2.0 * a);
	} else {
		tmp = ((a * ((-2.0 * ((a * pow(c, 3.0)) / pow(b, 4.0))) - (pow(c, 2.0) / pow(b, 2.0)))) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = 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)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, Float64(sqrt(t_0) * t_0)) / Float64(t_1 + fma(b, b, Float64(sqrt(t_1) * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 4.0))) - Float64((c ^ 2.0) / (b ^ 2.0)))) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], -1.5], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(b * b + N[(N[Sqrt[t$95$1], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 56.3%

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

      13. lift-pow.f64 N/A

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

      14. cube-mult N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 57.3%

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

   5. Applied rewrites 57.3%

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

   if -1.5 < (/.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 55.5%

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

   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%

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

   7. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   9. Applied rewrites 87.6%

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

Alternative 12: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, \sqrt{t\_0} \cdot t\_0\right)}{t\_1 + \mathsf{fma}\left(b, b, \sqrt{t\_1} \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right) - c\right) - \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (/
       (fma (* (- b) b) b (* (sqrt t_0) t_0))
       (+ t_1 (fma b b (* (sqrt t_1) b))))
      (* 2.0 a))
     (/
      (-
       (- (* (* (* (* a a) c) (* c c)) (* (pow b -4.0) -2.0)) c)
       (/ (* a (pow c 2.0)) (pow b 2.0)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (fma((-b * b), b, (sqrt(t_0) * t_0)) / (t_1 + fma(b, b, (sqrt(t_1) * b)))) / (2.0 * a);
	} else {
		tmp = ((((((a * a) * c) * (c * c)) * (pow(b, -4.0) * -2.0)) - c) - ((a * pow(c, 2.0)) / pow(b, 2.0))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = 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)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, Float64(sqrt(t_0) * t_0)) / Float64(t_1 + fma(b, b, Float64(sqrt(t_1) * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * a) * c) * Float64(c * c)) * Float64((b ^ -4.0) * -2.0)) - c) - Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], -1.5], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(b * b + N[(N[Sqrt[t$95$1], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[Power[b, -4.0], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] - N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 56.3%

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

      13. lift-pow.f64 N/A

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

      14. cube-mult N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 57.3%

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

   5. Applied rewrites 57.3%

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

   if -1.5 < (/.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 55.5%

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

   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%

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

   7. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 87.5%

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

   9. Applied rewrites 87.5%

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

Alternative 13: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(-b\right) \cdot b, b, \sqrt{t\_0} \cdot t\_0\right)}{t\_1 + \mathsf{fma}\left(b, b, \sqrt{t\_1} \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (/
       (fma (* (- b) b) b (* (sqrt t_0) t_0))
       (+ t_1 (fma b b (* (sqrt t_1) b))))
      (* 2.0 a))
     (/
      (*
       c
       (-
        (* c (- (* -2.0 (/ (* (pow a 2.0) c) (pow b 4.0))) (/ a (pow b 2.0))))
        1.0))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (fma((-b * b), b, (sqrt(t_0) * t_0)) / (t_1 + fma(b, b, (sqrt(t_1) * b)))) / (2.0 * a);
	} else {
		tmp = (c * ((c * ((-2.0 * ((pow(a, 2.0) * c) / pow(b, 4.0))) - (a / pow(b, 2.0)))) - 1.0)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = 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)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(Float64(-b) * b), b, Float64(sqrt(t_0) * t_0)) / Float64(t_1 + fma(b, b, Float64(sqrt(t_1) * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * c) / (b ^ 4.0))) - Float64(a / (b ^ 2.0)))) - 1.0)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], -1.5], N[(N[(N[(N[((-b) * b), $MachinePrecision] * b + N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(b * b + N[(N[Sqrt[t$95$1], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow3 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 56.3%

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

      13. lift-pow.f64 N/A

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

      14. cube-mult N/A

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

      15. lift-sqrt.f64 N/A

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

      16. lift-sqrt.f64 N/A

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

      17. rem-square-sqrt N/A

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

      18. lower-*.f64 57.3%

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

   5. Applied rewrites 57.3%

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

   if -1.5 < (/.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 55.5%

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

   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%

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

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   9. Applied rewrites 87.4%

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

Alternative 14: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\sqrt{t\_0} \cdot t\_0 - \left(b \cdot b\right) \cdot b}{t\_1 + \mathsf{fma}\left(b, b, \sqrt{t\_1} \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (/
       (- (* (sqrt t_0) t_0) (* (* b b) b))
       (+ t_1 (fma b b (* (sqrt t_1) b))))
      (* 2.0 a))
     (/
      (*
       c
       (-
        (* c (- (* -2.0 (/ (* (pow a 2.0) c) (pow b 4.0))) (/ a (pow b 2.0))))
        1.0))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (((sqrt(t_0) * t_0) - ((b * b) * b)) / (t_1 + fma(b, b, (sqrt(t_1) * b)))) / (2.0 * a);
	} else {
		tmp = (c * ((c * ((-2.0 * ((pow(a, 2.0) * c) / pow(b, 4.0))) - (a / pow(b, 2.0)))) - 1.0)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = 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)) <= -1.5)
		tmp = Float64(Float64(Float64(Float64(sqrt(t_0) * t_0) - Float64(Float64(b * b) * b)) / Float64(t_1 + fma(b, b, Float64(sqrt(t_1) * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * c) / (b ^ 4.0))) - Float64(a / (b ^ 2.0)))) - 1.0)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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], -1.5], N[(N[(N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(b * b + N[(N[Sqrt[t$95$1], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. flip3-- N/A

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

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

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

   3. Applied rewrites 55.4%

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

      1. lift-pow.f64 N/A

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

      2. cube-mult N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. lower-*.f64 56.4%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 56.4%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 56.4%

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

      13. lift-pow.f64 N/A

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

      14. unpow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 56.5%

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

   5. Applied rewrites 56.5%

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

   if -1.5 < (/.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 55.5%

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

   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%

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

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   9. Applied rewrites 87.4%

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

Alternative 15: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{t\_1 \cdot t\_0 - \left(b \cdot b\right) \cdot b}{a} \cdot \frac{\frac{1}{\mathsf{fma}\left(b, b + \left(t\_1 + b\right), \left(-4 \cdot c\right) \cdot a\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (*
      (/ (- (* t_1 t_0) (* (* b b) b)) a)
      (/ (/ 1.0 (fma b (+ b (+ t_1 b)) (* (* -4.0 c) a))) 2.0))
     (/
      (*
       c
       (-
        (* c (- (* -2.0 (/ (* (pow a 2.0) c) (pow b 4.0))) (/ a (pow b 2.0))))
        1.0))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = (((t_1 * t_0) - ((b * b) * b)) / a) * ((1.0 / fma(b, (b + (t_1 + b)), ((-4.0 * c) * a))) / 2.0);
	} else {
		tmp = (c * ((c * ((-2.0 * ((pow(a, 2.0) * c) / pow(b, 4.0))) - (a / pow(b, 2.0)))) - 1.0)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.5)
		tmp = Float64(Float64(Float64(Float64(t_1 * t_0) - Float64(Float64(b * b) * b)) / a) * Float64(Float64(1.0 / fma(b, Float64(b + Float64(t_1 + b)), Float64(Float64(-4.0 * c) * a))) / 2.0));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * c) / (b ^ 4.0))) - Float64(a / (b ^ 2.0)))) - 1.0)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$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], -1.5], N[(N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * N[(N[(1.0 / N[(b * N[(b + N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 54.6%

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

   5. Applied rewrites 56.5%

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

   if -1.5 < (/.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 55.5%

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

   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%

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

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   9. Applied rewrites 87.4%

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

Alternative 16: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\left(t\_1 \cdot t\_0 - \left(b \cdot b\right) \cdot b\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(b, b + \left(t\_1 + b\right), \left(-4 \cdot c\right) \cdot a\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (*
      (- (* t_1 t_0) (* (* b b) b))
      (/ (/ 1.0 (fma b (+ b (+ t_1 b)) (* (* -4.0 c) a))) (+ a a)))
     (/
      (*
       c
       (-
        (* c (- (* -2.0 (/ (* (pow a 2.0) c) (pow b 4.0))) (/ a (pow b 2.0))))
        1.0))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = ((t_1 * t_0) - ((b * b) * b)) * ((1.0 / fma(b, (b + (t_1 + b)), ((-4.0 * c) * a))) / (a + a));
	} else {
		tmp = (c * ((c * ((-2.0 * ((pow(a, 2.0) * c) / pow(b, 4.0))) - (a / pow(b, 2.0)))) - 1.0)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.5)
		tmp = Float64(Float64(Float64(t_1 * t_0) - Float64(Float64(b * b) * b)) * Float64(Float64(1.0 / fma(b, Float64(b + Float64(t_1 + b)), Float64(Float64(-4.0 * c) * a))) / Float64(a + a)));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * c) / (b ^ 4.0))) - Float64(a / (b ^ 2.0)))) - 1.0)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$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], -1.5], N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(b * N[(b + N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 54.6%

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

   5. Applied rewrites 56.5%

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

   if -1.5 < (/.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 55.5%

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

   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%

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

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   9. Applied rewrites 87.4%

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

Alternative 17: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{t\_1 \cdot t\_0 - \left(b \cdot b\right) \cdot b}{\left(a + a\right) \cdot \mathsf{fma}\left(b, b + \left(t\_1 + b\right), \left(-4 \cdot c\right) \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.5)
     (/
      (- (* t_1 t_0) (* (* b b) b))
      (* (+ a a) (fma b (+ b (+ t_1 b)) (* (* -4.0 c) a))))
     (/
      (*
       c
       (-
        (* c (- (* -2.0 (/ (* (pow a 2.0) c) (pow b 4.0))) (/ a (pow b 2.0))))
        1.0))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.5) {
		tmp = ((t_1 * t_0) - ((b * b) * b)) / ((a + a) * fma(b, (b + (t_1 + b)), ((-4.0 * c) * a)));
	} else {
		tmp = (c * ((c * ((-2.0 * ((pow(a, 2.0) * c) / pow(b, 4.0))) - (a / pow(b, 2.0)))) - 1.0)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.5)
		tmp = Float64(Float64(Float64(t_1 * t_0) - Float64(Float64(b * b) * b)) / Float64(Float64(a + a) * fma(b, Float64(b + Float64(t_1 + b)), Float64(Float64(-4.0 * c) * a))));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * c) / (b ^ 4.0))) - Float64(a / (b ^ 2.0)))) - 1.0)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$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], -1.5], N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(N[(a + a), $MachinePrecision] * N[(b * N[(b + N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $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)) < -1.5

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 54.6%

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

   5. Applied rewrites 56.5%

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

   if -1.5 < (/.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 55.5%

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

   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%

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

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   9. Applied rewrites 87.4%

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

Alternative 18: 85.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \left(b \cdot b\right) \cdot b\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.65:\\ \;\;\;\;\frac{t\_1 \cdot t\_0 - t\_2}{\left(a + a\right) \cdot \mathsf{fma}\left(b, b + \left(t\_1 + b\right), \left(-4 \cdot c\right) \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} + \left(-\frac{\left(c \cdot c\right) \cdot a}{t\_2}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (sqrt t_0)) (t_2 (* (* b b) b)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.65)
     (/
      (- (* t_1 t_0) t_2)
      (* (+ a a) (fma b (+ b (+ t_1 b)) (* (* -4.0 c) a))))
     (+ (/ (- c) b) (- (/ (* (* c c) a) t_2))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = sqrt(t_0);
	double t_2 = (b * b) * b;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.65) {
		tmp = ((t_1 * t_0) - t_2) / ((a + a) * fma(b, (b + (t_1 + b)), ((-4.0 * c) * a)));
	} else {
		tmp = (-c / b) + -(((c * c) * a) / t_2);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = sqrt(t_0)
	t_2 = Float64(Float64(b * 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)) <= -0.65)
		tmp = Float64(Float64(Float64(t_1 * t_0) - t_2) / Float64(Float64(a + a) * fma(b, Float64(b + Float64(t_1 + b)), Float64(Float64(-4.0 * c) * a))));
	else
		tmp = Float64(Float64(Float64(-c) / b) + Float64(-Float64(Float64(Float64(c * c) * a) / t_2)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(b * b), $MachinePrecision] * b), $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], -0.65], N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] - t$95$2), $MachinePrecision] / N[(N[(a + a), $MachinePrecision] * N[(b * N[(b + N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-c) / b), $MachinePrecision] + (-N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / t$95$2), $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)) < -0.650000000000000022

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 54.6%

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

   5. Applied rewrites 56.5%

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

   if -0.650000000000000022 < (/.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 55.5%

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

   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%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.4%

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

   7. Applied rewrites 81.4%

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

      1. lift-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-/.f64 81.4%

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 81.4%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.4%

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 81.4%

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

      17. lift-pow.f64 N/A

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

      18. unpow3 N/A

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

      19. lift-*.f64 N/A

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

      20. lower-*.f64 81.4%

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

   9. Applied rewrites 81.4%

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

Alternative 19: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.65:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} + \left(-\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}\right)\\ \end{array} \]

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)) -0.65)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (+ (/ (- c) b) (- (/ (* (* c c) a) (* (* b b) 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)) <= -0.65) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = (-c / b) + -(((c * c) * a) / ((b * b) * 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)) <= -0.65)
		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(c * c) * a) / Float64(Float64(b * b) * 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], -0.65], 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[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * 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)) < -0.650000000000000022

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 57.0%

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

   if -0.650000000000000022 < (/.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 55.5%

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

   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%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.4%

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

   7. Applied rewrites 81.4%

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

      1. lift-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-/.f64 81.4%

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 81.4%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.4%

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 81.4%

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

      17. lift-pow.f64 N/A

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

      18. unpow3 N/A

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

      19. lift-*.f64 N/A

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

      20. lower-*.f64 81.4%

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

   9. Applied rewrites 81.4%

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

Alternative 20: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.65:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b} + \left(-\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.65)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (+ (/ (- c) b) (- (/ (* (* c c) a) (* (* b b) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.65) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = (-c / b) + -(((c * c) * a) / ((b * b) * 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)) <= -0.65)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) / b) + Float64(-Float64(Float64(Float64(c * c) * a) / Float64(Float64(b * b) * 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], -0.65], 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[((-c) / b), $MachinePrecision] + (-N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * 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)) < -0.650000000000000022

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 55.5%

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

   3. Applied rewrites 55.5%

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

   if -0.650000000000000022 < (/.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 55.5%

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

   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%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.4%

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

   7. Applied rewrites 81.4%

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

      1. lift-fma.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-frac-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-/.f64 81.4%

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

      8. lift-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 81.4%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.4%

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 81.4%

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

      17. lift-pow.f64 N/A

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

      18. unpow3 N/A

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

      19. lift-*.f64 N/A

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

      20. lower-*.f64 81.4%

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

   9. Applied rewrites 81.4%

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

Alternative 21: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -7 \cdot 10^{-6}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -7e-6)
   (* (- (sqrt (fma (* c -4.0) a (* b b))) b) (/ 0.5 a))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -7e-6) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = -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)) <= -7e-6)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -7e-6], N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / 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)) < -6.99999999999999989e-6

   1. Initial program 55.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. mult-flip N/A

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

      3. lower-*.f64 N/A

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

   3. Applied rewrites 55.5%

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

   if -6.99999999999999989e-6 < (/.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 55.5%

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

   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 64.4%

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

   4. Applied rewrites 64.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 64.4%

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

   6. Applied rewrites 64.4%

      \[\leadsto \frac{-c}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -7e-6)
   (/ (- (sqrt (fma (* c -4.0) a (* b b))) b) (+ a a))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -7e-6) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) / (a + a);
	} else {
		tmp = -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)) <= -7e-6)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -7e-6], 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[((-c) / 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)) < -6.99999999999999989e-6

   1. Initial program 55.5%

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

   3. Applied rewrites 55.5%

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

   if -6.99999999999999989e-6 < (/.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 55.5%

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

   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 64.4%

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

   4. Applied rewrites 64.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 64.4%

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

   6. Applied rewrites 64.4%

      \[\leadsto \frac{-c}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 64.4% 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 55.5%

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

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 64.4%

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

4. Applied rewrites 64.4%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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 64.4%

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

6. Applied rewrites 64.4%

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

Reproduce

herbie shell --seed 2025183 
(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)))