Quadratic roots, narrow range

Percentage Accurate: 55.8% → 92.0%

Time: 10.8s

Alternatives: 21

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: 55.8% 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: 92.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := {\left(a \cdot c\right)}^{4}\\ t_1 := \left(c \cdot c\right) \cdot \left(a \cdot a\right)\\ t_2 := \frac{t\_1}{b \cdot b}\\ t_3 := t\_1 \cdot 0\\ t_4 := \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ t_5 := t\_4 \cdot {b}^{-4}\\ t_6 := t\_0 \cdot {b}^{-6}\\ t_7 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_8 := t\_0 \cdot 20\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -70:\\ \;\;\;\;\frac{\frac{t\_7 - b \cdot b}{\sqrt{t\_7} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_5, -8, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(t\_2, -4, \mathsf{fma}\left(t\_5, -4, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(t\_4 \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(\left(\left(t\_3 \cdot c\right) \cdot a\right) \cdot {b}^{-4}, -2, \mathsf{fma}\left(\left(\left(t\_3 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, t\_2, \frac{\mathsf{fma}\left(-1, t\_8, -0.5 \cdot t\_8\right)}{{b}^{6}} + \mathsf{fma}\left(t\_2, 4, \mathsf{fma}\left(t\_6, 4, \mathsf{fma}\left(8, t\_2, \mathsf{fma}\left(16, t\_5, t\_6 \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{t\_7 + {b}^{2} \cdot \left(2 + \mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{a \cdot c}{{b}^{2}}, -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)}}{2 \cdot a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* a c) 4.0))
        (t_1 (* (* c c) (* a a)))
        (t_2 (/ t_1 (* b b)))
        (t_3 (* t_1 0.0))
        (t_4 (* (* (* c c) c) (* (* a a) a)))
        (t_5 (* t_4 (pow b -4.0)))
        (t_6 (* t_0 (pow b -6.0)))
        (t_7 (fma (* c -4.0) a (* b b)))
        (t_8 (* t_0 20.0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -70.0)
     (/ (/ (- t_7 (* b b)) (+ (sqrt t_7) b)) (* 2.0 a))
     (/
      (/
       (*
        (fma
         t_5
         -8.0
         (fma
          (* -4.0 a)
          c
          (fma
           t_2
           -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_3 c) a) (pow b -4.0))
               -2.0
               (fma
                (* (* (* t_3 (* c c)) (* a a)) (pow b -6.0))
                -2.0
                (fma
                 -2.0
                 t_2
                 (+
                  (/ (fma -1.0 t_8 (* -0.5 t_8)) (pow b 6.0))
                  (fma
                   t_2
                   4.0
                   (fma
                    t_6
                    4.0
                    (fma 8.0 t_2 (fma 16.0 t_5 (* t_6 32.0)))))))))))))))
        b)
       (+
        t_7
        (*
         (pow b 2.0)
         (+
          2.0
          (fma
           -4.0
           (/ (* (pow a 3.0) (pow c 3.0)) (pow b 6.0))
           (fma
            -2.0
            (/ (* a c) (pow b 2.0))
            (* -2.0 (/ (* (pow a 2.0) (pow c 2.0)) (pow b 4.0)))))))))
      (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = pow((a * c), 4.0);
	double t_1 = (c * c) * (a * a);
	double t_2 = t_1 / (b * b);
	double t_3 = t_1 * 0.0;
	double t_4 = ((c * c) * c) * ((a * a) * a);
	double t_5 = t_4 * pow(b, -4.0);
	double t_6 = t_0 * pow(b, -6.0);
	double t_7 = fma((c * -4.0), a, (b * b));
	double t_8 = t_0 * 20.0;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -70.0) {
		tmp = ((t_7 - (b * b)) / (sqrt(t_7) + b)) / (2.0 * a);
	} else {
		tmp = ((fma(t_5, -8.0, fma((-4.0 * a), c, fma(t_2, -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_3 * c) * a) * pow(b, -4.0)), -2.0, fma((((t_3 * (c * c)) * (a * a)) * pow(b, -6.0)), -2.0, fma(-2.0, t_2, ((fma(-1.0, t_8, (-0.5 * t_8)) / pow(b, 6.0)) + fma(t_2, 4.0, fma(t_6, 4.0, fma(8.0, t_2, fma(16.0, t_5, (t_6 * 32.0))))))))))))))) * b) / (t_7 + (pow(b, 2.0) * (2.0 + fma(-4.0, ((pow(a, 3.0) * pow(c, 3.0)) / pow(b, 6.0)), fma(-2.0, ((a * c) / pow(b, 2.0)), (-2.0 * ((pow(a, 2.0) * pow(c, 2.0)) / pow(b, 4.0))))))))) / (2.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * c) ^ 4.0
	t_1 = Float64(Float64(c * c) * Float64(a * a))
	t_2 = Float64(t_1 / Float64(b * b))
	t_3 = Float64(t_1 * 0.0)
	t_4 = Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) * a))
	t_5 = Float64(t_4 * (b ^ -4.0))
	t_6 = Float64(t_0 * (b ^ -6.0))
	t_7 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_8 = Float64(t_0 * 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)) <= -70.0)
		tmp = Float64(Float64(Float64(t_7 - Float64(b * b)) / Float64(sqrt(t_7) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(fma(t_5, -8.0, fma(Float64(-4.0 * a), c, fma(t_2, -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_3 * c) * a) * (b ^ -4.0)), -2.0, fma(Float64(Float64(Float64(t_3 * Float64(c * c)) * Float64(a * a)) * (b ^ -6.0)), -2.0, fma(-2.0, t_2, Float64(Float64(fma(-1.0, t_8, Float64(-0.5 * t_8)) / (b ^ 6.0)) + fma(t_2, 4.0, fma(t_6, 4.0, fma(8.0, t_2, fma(16.0, t_5, Float64(t_6 * 32.0))))))))))))))) * b) / Float64(t_7 + Float64((b ^ 2.0) * Float64(2.0 + fma(-4.0, Float64(Float64((a ^ 3.0) * (c ^ 3.0)) / (b ^ 6.0)), fma(-2.0, Float64(Float64(a * c) / (b ^ 2.0)), Float64(-2.0 * Float64(Float64((a ^ 2.0) * (c ^ 2.0)) / (b ^ 4.0))))))))) / Float64(2.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * 0.0), $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[(t$95$0 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$0 * 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], -70.0], N[(N[(N[(t$95$7 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$7], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$5 * -8.0 + N[(N[(-4.0 * a), $MachinePrecision] * c + N[(t$95$2 * -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$3 * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(t$95$3 * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(-2.0 * t$95$2 + 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$2 * 4.0 + N[(t$95$6 * 4.0 + N[(8.0 * t$95$2 + N[(16.0 * t$95$5 + N[(t$95$6 * 32.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(t$95$7 + N[(N[Power[b, 2.0], $MachinePrecision] * N[(2.0 + N[(-4.0 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * 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)) < -70

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -70 < (/.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 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 55.7%

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

   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 91.0%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

   9. Applied rewrites 91.1%

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

   11. Applied rewrites 91.1%

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

Alternative 2: 92.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := {\left(a \cdot c\right)}^{4}\\ t_1 := \left(c \cdot c\right) \cdot \left(a \cdot a\right)\\ t_2 := \frac{t\_1}{b \cdot b}\\ t_3 := t\_1 \cdot 0\\ t_4 := \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ t_5 := t\_4 \cdot {b}^{-4}\\ t_6 := t\_0 \cdot {b}^{-6}\\ t_7 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_8 := t\_0 \cdot 20\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -70:\\ \;\;\;\;\frac{\frac{t\_7 - b \cdot b}{\sqrt{t\_7} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_5, -8, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(t\_2, -4, \mathsf{fma}\left(t\_5, -4, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(t\_4 \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(\left(\left(t\_3 \cdot c\right) \cdot a\right) \cdot {b}^{-4}, -2, \mathsf{fma}\left(\left(\left(t\_3 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, t\_2, \frac{\mathsf{fma}\left(-1, t\_8, -0.5 \cdot t\_8\right)}{{b}^{6}} + \mathsf{fma}\left(t\_2, 4, \mathsf{fma}\left(t\_6, 4, \mathsf{fma}\left(8, t\_2, \mathsf{fma}\left(16, t\_5, t\_6 \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{a} \cdot \frac{\frac{1}{\mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(b, b, \left(\mathsf{fma}\left(t\_4 \cdot {b}^{-6}, -4, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, t\_1 \cdot {b}^{-4}\right)\right) + 2\right) \cdot \left(b \cdot b\right)\right)\right)}}{2}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* a c) 4.0))
        (t_1 (* (* c c) (* a a)))
        (t_2 (/ t_1 (* b b)))
        (t_3 (* t_1 0.0))
        (t_4 (* (* (* c c) c) (* (* a a) a)))
        (t_5 (* t_4 (pow b -4.0)))
        (t_6 (* t_0 (pow b -6.0)))
        (t_7 (fma (* c -4.0) a (* b b)))
        (t_8 (* t_0 20.0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -70.0)
     (/ (/ (- t_7 (* b b)) (+ (sqrt t_7) b)) (* 2.0 a))
     (*
      (/
       (*
        (fma
         t_5
         -8.0
         (fma
          (* -4.0 a)
          c
          (fma
           t_2
           -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_3 c) a) (pow b -4.0))
               -2.0
               (fma
                (* (* (* t_3 (* c c)) (* a a)) (pow b -6.0))
                -2.0
                (fma
                 -2.0
                 t_2
                 (+
                  (/ (fma -1.0 t_8 (* -0.5 t_8)) (pow b 6.0))
                  (fma
                   t_2
                   4.0
                   (fma
                    t_6
                    4.0
                    (fma 8.0 t_2 (fma 16.0 t_5 (* t_6 32.0)))))))))))))))
        b)
       a)
      (/
       (/
        1.0
        (fma
         (* -4.0 a)
         c
         (fma
          b
          b
          (*
           (+
            (fma
             (* t_4 (pow b -6.0))
             -4.0
             (* -2.0 (fma a (/ c (* b b)) (* t_1 (pow b -4.0)))))
            2.0)
           (* b b)))))
       2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = pow((a * c), 4.0);
	double t_1 = (c * c) * (a * a);
	double t_2 = t_1 / (b * b);
	double t_3 = t_1 * 0.0;
	double t_4 = ((c * c) * c) * ((a * a) * a);
	double t_5 = t_4 * pow(b, -4.0);
	double t_6 = t_0 * pow(b, -6.0);
	double t_7 = fma((c * -4.0), a, (b * b));
	double t_8 = t_0 * 20.0;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -70.0) {
		tmp = ((t_7 - (b * b)) / (sqrt(t_7) + b)) / (2.0 * a);
	} else {
		tmp = ((fma(t_5, -8.0, fma((-4.0 * a), c, fma(t_2, -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_3 * c) * a) * pow(b, -4.0)), -2.0, fma((((t_3 * (c * c)) * (a * a)) * pow(b, -6.0)), -2.0, fma(-2.0, t_2, ((fma(-1.0, t_8, (-0.5 * t_8)) / pow(b, 6.0)) + fma(t_2, 4.0, fma(t_6, 4.0, fma(8.0, t_2, fma(16.0, t_5, (t_6 * 32.0))))))))))))))) * b) / a) * ((1.0 / fma((-4.0 * a), c, fma(b, b, ((fma((t_4 * pow(b, -6.0)), -4.0, (-2.0 * fma(a, (c / (b * b)), (t_1 * pow(b, -4.0))))) + 2.0) * (b * b))))) / 2.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * c) ^ 4.0
	t_1 = Float64(Float64(c * c) * Float64(a * a))
	t_2 = Float64(t_1 / Float64(b * b))
	t_3 = Float64(t_1 * 0.0)
	t_4 = Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) * a))
	t_5 = Float64(t_4 * (b ^ -4.0))
	t_6 = Float64(t_0 * (b ^ -6.0))
	t_7 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_8 = Float64(t_0 * 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)) <= -70.0)
		tmp = Float64(Float64(Float64(t_7 - Float64(b * b)) / Float64(sqrt(t_7) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(fma(t_5, -8.0, fma(Float64(-4.0 * a), c, fma(t_2, -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_3 * c) * a) * (b ^ -4.0)), -2.0, fma(Float64(Float64(Float64(t_3 * Float64(c * c)) * Float64(a * a)) * (b ^ -6.0)), -2.0, fma(-2.0, t_2, Float64(Float64(fma(-1.0, t_8, Float64(-0.5 * t_8)) / (b ^ 6.0)) + fma(t_2, 4.0, fma(t_6, 4.0, fma(8.0, t_2, fma(16.0, t_5, Float64(t_6 * 32.0))))))))))))))) * b) / a) * Float64(Float64(1.0 / fma(Float64(-4.0 * a), c, fma(b, b, Float64(Float64(fma(Float64(t_4 * (b ^ -6.0)), -4.0, Float64(-2.0 * fma(a, Float64(c / Float64(b * b)), Float64(t_1 * (b ^ -4.0))))) + 2.0) * Float64(b * b))))) / 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * 0.0), $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[(t$95$0 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$0 * 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], -70.0], N[(N[(N[(t$95$7 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$7], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$5 * -8.0 + N[(N[(-4.0 * a), $MachinePrecision] * c + N[(t$95$2 * -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$3 * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(t$95$3 * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(-2.0 * t$95$2 + 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$2 * 4.0 + N[(t$95$6 * 4.0 + N[(8.0 * t$95$2 + N[(16.0 * t$95$5 + N[(t$95$6 * 32.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / a), $MachinePrecision] * N[(N[(1.0 / N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * 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$1 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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)) < -70

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -70 < (/.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 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 55.7%

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

   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 91.0%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

   9. Applied rewrites 91.1%

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

   11. Applied rewrites 91.1%

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

Alternative 3: 92.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := {\left(a \cdot c\right)}^{4}\\ t_1 := \left(c \cdot c\right) \cdot \left(a \cdot a\right)\\ t_2 := \frac{t\_1}{b \cdot b}\\ t_3 := t\_1 \cdot 0\\ t_4 := \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ t_5 := t\_4 \cdot {b}^{-4}\\ t_6 := t\_0 \cdot {b}^{-6}\\ t_7 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_8 := t\_0 \cdot 20\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -70:\\ \;\;\;\;\frac{\frac{t\_7 - b \cdot b}{\sqrt{t\_7} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_5, -8, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(t\_2, -4, \mathsf{fma}\left(t\_5, -4, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(t\_4 \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(\left(\left(t\_3 \cdot c\right) \cdot a\right) \cdot {b}^{-4}, -2, \mathsf{fma}\left(\left(\left(t\_3 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, t\_2, \frac{\mathsf{fma}\left(-1, t\_8, -0.5 \cdot t\_8\right)}{{b}^{6}} + \mathsf{fma}\left(t\_2, 4, \mathsf{fma}\left(t\_6, 4, \mathsf{fma}\left(8, t\_2, \mathsf{fma}\left(16, t\_5, t\_6 \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(b, b, \left(\mathsf{fma}\left(t\_4 \cdot {b}^{-6}, -4, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, t\_1 \cdot {b}^{-4}\right)\right) + 2\right) \cdot \left(b \cdot b\right)\right)\right)}}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* a c) 4.0))
        (t_1 (* (* c c) (* a a)))
        (t_2 (/ t_1 (* b b)))
        (t_3 (* t_1 0.0))
        (t_4 (* (* (* c c) c) (* (* a a) a)))
        (t_5 (* t_4 (pow b -4.0)))
        (t_6 (* t_0 (pow b -6.0)))
        (t_7 (fma (* c -4.0) a (* b b)))
        (t_8 (* t_0 20.0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -70.0)
     (/ (/ (- t_7 (* b b)) (+ (sqrt t_7) b)) (* 2.0 a))
     (/
      (/
       (*
        (fma
         t_5
         -8.0
         (fma
          (* -4.0 a)
          c
          (fma
           t_2
           -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_3 c) a) (pow b -4.0))
               -2.0
               (fma
                (* (* (* t_3 (* c c)) (* a a)) (pow b -6.0))
                -2.0
                (fma
                 -2.0
                 t_2
                 (+
                  (/ (fma -1.0 t_8 (* -0.5 t_8)) (pow b 6.0))
                  (fma
                   t_2
                   4.0
                   (fma
                    t_6
                    4.0
                    (fma 8.0 t_2 (fma 16.0 t_5 (* t_6 32.0)))))))))))))))
        b)
       (fma
        (* -4.0 a)
        c
        (fma
         b
         b
         (*
          (+
           (fma
            (* t_4 (pow b -6.0))
            -4.0
            (* -2.0 (fma a (/ c (* b b)) (* t_1 (pow b -4.0)))))
           2.0)
          (* b b)))))
      (+ a a)))))

C

double code(double a, double b, double c) {
	double t_0 = pow((a * c), 4.0);
	double t_1 = (c * c) * (a * a);
	double t_2 = t_1 / (b * b);
	double t_3 = t_1 * 0.0;
	double t_4 = ((c * c) * c) * ((a * a) * a);
	double t_5 = t_4 * pow(b, -4.0);
	double t_6 = t_0 * pow(b, -6.0);
	double t_7 = fma((c * -4.0), a, (b * b));
	double t_8 = t_0 * 20.0;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -70.0) {
		tmp = ((t_7 - (b * b)) / (sqrt(t_7) + b)) / (2.0 * a);
	} else {
		tmp = ((fma(t_5, -8.0, fma((-4.0 * a), c, fma(t_2, -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_3 * c) * a) * pow(b, -4.0)), -2.0, fma((((t_3 * (c * c)) * (a * a)) * pow(b, -6.0)), -2.0, fma(-2.0, t_2, ((fma(-1.0, t_8, (-0.5 * t_8)) / pow(b, 6.0)) + fma(t_2, 4.0, fma(t_6, 4.0, fma(8.0, t_2, fma(16.0, t_5, (t_6 * 32.0))))))))))))))) * b) / fma((-4.0 * a), c, fma(b, b, ((fma((t_4 * pow(b, -6.0)), -4.0, (-2.0 * fma(a, (c / (b * b)), (t_1 * pow(b, -4.0))))) + 2.0) * (b * b))))) / (a + a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * c) ^ 4.0
	t_1 = Float64(Float64(c * c) * Float64(a * a))
	t_2 = Float64(t_1 / Float64(b * b))
	t_3 = Float64(t_1 * 0.0)
	t_4 = Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) * a))
	t_5 = Float64(t_4 * (b ^ -4.0))
	t_6 = Float64(t_0 * (b ^ -6.0))
	t_7 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_8 = Float64(t_0 * 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)) <= -70.0)
		tmp = Float64(Float64(Float64(t_7 - Float64(b * b)) / Float64(sqrt(t_7) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(fma(t_5, -8.0, fma(Float64(-4.0 * a), c, fma(t_2, -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_3 * c) * a) * (b ^ -4.0)), -2.0, fma(Float64(Float64(Float64(t_3 * Float64(c * c)) * Float64(a * a)) * (b ^ -6.0)), -2.0, fma(-2.0, t_2, Float64(Float64(fma(-1.0, t_8, Float64(-0.5 * t_8)) / (b ^ 6.0)) + fma(t_2, 4.0, fma(t_6, 4.0, fma(8.0, t_2, fma(16.0, t_5, Float64(t_6 * 32.0))))))))))))))) * b) / fma(Float64(-4.0 * a), c, fma(b, b, Float64(Float64(fma(Float64(t_4 * (b ^ -6.0)), -4.0, Float64(-2.0 * fma(a, Float64(c / Float64(b * b)), Float64(t_1 * (b ^ -4.0))))) + 2.0) * Float64(b * b))))) / Float64(a + a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * 0.0), $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[(t$95$0 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$0 * 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], -70.0], N[(N[(N[(t$95$7 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$7], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$5 * -8.0 + N[(N[(-4.0 * a), $MachinePrecision] * c + N[(t$95$2 * -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$3 * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(t$95$3 * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(-2.0 * t$95$2 + 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$2 * 4.0 + N[(t$95$6 * 4.0 + N[(8.0 * t$95$2 + N[(16.0 * t$95$5 + N[(t$95$6 * 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[(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$1 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -70

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -70 < (/.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 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 55.7%

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

   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 91.0%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

   9. Applied rewrites 91.1%

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

   11. Applied rewrites 91.1%

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

Alternative 4: 92.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \left(c \cdot c\right) \cdot \left(a \cdot a\right)\\ t_1 := \frac{t\_0}{b \cdot b}\\ t_2 := t\_0 \cdot 0\\ t_3 := \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ t_4 := t\_3 \cdot {b}^{-4}\\ t_5 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_6 := {\left(a \cdot c\right)}^{4}\\ t_7 := t\_6 \cdot 20\\ t_8 := t\_6 \cdot {b}^{-6}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -70:\\ \;\;\;\;\frac{\frac{t\_5 - b \cdot b}{\sqrt{t\_5} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_4, -8, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(t\_1, -4, \mathsf{fma}\left(t\_4, -4, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(t\_3 \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(\left(\left(t\_2 \cdot c\right) \cdot a\right) \cdot {b}^{-4}, -2, \mathsf{fma}\left(\left(\left(t\_2 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, t\_1, \frac{\mathsf{fma}\left(-1, t\_7, -0.5 \cdot t\_7\right)}{{b}^{6}} + \mathsf{fma}\left(t\_1, 4, \mathsf{fma}\left(t\_8, 4, \mathsf{fma}\left(8, t\_1, \mathsf{fma}\left(16, t\_4, t\_8 \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(b, b, \left(\mathsf{fma}\left(t\_3 \cdot {b}^{-6}, -4, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, t\_0 \cdot {b}^{-4}\right)\right) + 2\right) \cdot \left(b \cdot b\right)\right)\right) \cdot \left(a + a\right)}\\ \end{array} \]

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 (* (* (* c c) c) (* (* a a) a)))
        (t_4 (* t_3 (pow b -4.0)))
        (t_5 (fma (* c -4.0) a (* b b)))
        (t_6 (pow (* a c) 4.0))
        (t_7 (* t_6 20.0))
        (t_8 (* t_6 (pow b -6.0))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -70.0)
     (/ (/ (- t_5 (* b b)) (+ (sqrt t_5) b)) (* 2.0 a))
     (/
      (*
       (fma
        t_4
        -8.0
        (fma
         (* -4.0 a)
         c
         (fma
          t_1
          -4.0
          (fma
           t_4
           -4.0
           (fma
            (* -2.0 a)
            c
            (fma
             (* (* (* (* t_3 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_7 (* -0.5 t_7)) (pow b 6.0))
                 (fma
                  t_1
                  4.0
                  (fma
                   t_8
                   4.0
                   (fma 8.0 t_1 (fma 16.0 t_4 (* t_8 32.0)))))))))))))))
       b)
      (*
       (fma
        (* -4.0 a)
        c
        (fma
         b
         b
         (*
          (+
           (fma
            (* t_3 (pow b -6.0))
            -4.0
            (* -2.0 (fma a (/ c (* b b)) (* t_0 (pow b -4.0)))))
           2.0)
          (* b b))))
       (+ 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 = ((c * c) * c) * ((a * a) * a);
	double t_4 = t_3 * pow(b, -4.0);
	double t_5 = fma((c * -4.0), a, (b * b));
	double t_6 = pow((a * c), 4.0);
	double t_7 = t_6 * 20.0;
	double t_8 = t_6 * pow(b, -6.0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -70.0) {
		tmp = ((t_5 - (b * b)) / (sqrt(t_5) + b)) / (2.0 * a);
	} else {
		tmp = (fma(t_4, -8.0, fma((-4.0 * a), c, fma(t_1, -4.0, fma(t_4, -4.0, fma((-2.0 * a), c, fma(((((t_3 * 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_7, (-0.5 * t_7)) / pow(b, 6.0)) + fma(t_1, 4.0, fma(t_8, 4.0, fma(8.0, t_1, fma(16.0, t_4, (t_8 * 32.0))))))))))))))) * b) / (fma((-4.0 * a), c, fma(b, b, ((fma((t_3 * pow(b, -6.0)), -4.0, (-2.0 * fma(a, (c / (b * b)), (t_0 * pow(b, -4.0))))) + 2.0) * (b * b)))) * (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 = Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) * a))
	t_4 = Float64(t_3 * (b ^ -4.0))
	t_5 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_6 = Float64(a * c) ^ 4.0
	t_7 = Float64(t_6 * 20.0)
	t_8 = Float64(t_6 * (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)) <= -70.0)
		tmp = Float64(Float64(Float64(t_5 - Float64(b * b)) / Float64(sqrt(t_5) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(t_4, -8.0, fma(Float64(-4.0 * a), c, fma(t_1, -4.0, fma(t_4, -4.0, fma(Float64(-2.0 * a), c, fma(Float64(Float64(Float64(Float64(t_3 * 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_7, Float64(-0.5 * t_7)) / (b ^ 6.0)) + fma(t_1, 4.0, fma(t_8, 4.0, fma(8.0, t_1, fma(16.0, t_4, Float64(t_8 * 32.0))))))))))))))) * b) / Float64(fma(Float64(-4.0 * a), c, fma(b, b, Float64(Float64(fma(Float64(t_3 * (b ^ -6.0)), -4.0, Float64(-2.0 * fma(a, Float64(c / Float64(b * b)), Float64(t_0 * (b ^ -4.0))))) + 2.0) * Float64(b * b)))) * 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[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$7 = N[(t$95$6 * 20.0), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$6 * 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], -70.0], N[(N[(N[(t$95$5 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$5], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$4 * -8.0 + N[(N[(-4.0 * a), $MachinePrecision] * c + N[(t$95$1 * -4.0 + N[(t$95$4 * -4.0 + N[(N[(-2.0 * a), $MachinePrecision] * c + N[(N[(N[(N[(N[(t$95$3 * 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$7 + N[(-0.5 * t$95$7), $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$4 + 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[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b + N[(N[(N[(N[(t$95$3 * 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] + 2.0), $MachinePrecision] * N[(b * b), $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)) < -70

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -70 < (/.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 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 55.7%

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

   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 91.0%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

   9. Applied rewrites 91.1%

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

   11. Applied rewrites 91.1%

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

Alternative 5: 91.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ t_1 := \left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ t_2 := \frac{t\_0}{t\_1}\\ t_3 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_4 := {\left(a \cdot c\right)}^{4}\\ t_5 := t\_4 \cdot {b}^{-6}\\ t_6 := t\_4 \cdot 20\\ t_7 := \sqrt{t\_3}\\ t_8 := \left(c \cdot c\right) \cdot \left(a \cdot a\right)\\ t_9 := \frac{t\_8}{b \cdot b}\\ t_10 := t\_8 \cdot 0\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -70:\\ \;\;\;\;\frac{\frac{t\_3 - b \cdot b}{t\_7 + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t\_2 \cdot -8, b, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(t\_9, -4, \mathsf{fma}\left(t\_2, -4, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(t\_0 \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(a \cdot \frac{t\_10 \cdot c}{t\_1}, -2, \mathsf{fma}\left(\left(\left(\left(c \cdot c\right) \cdot t\_10\right) \cdot \left(a \cdot a\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, t\_9, \frac{\mathsf{fma}\left(-1, t\_6, -0.5 \cdot t\_6\right)}{{b}^{6}} + \mathsf{fma}\left(t\_9, 4, \mathsf{fma}\left(t\_5, 4, \mathsf{fma}\left(8, t\_9, \mathsf{fma}\left(16, t\_2, t\_5 \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b\right)}{t\_3 + \mathsf{fma}\left(b, b, t\_7 \cdot b\right)}}{2 \cdot a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* (* c c) c) (* (* a a) a)))
        (t_1 (* (* (* b b) b) b))
        (t_2 (/ t_0 t_1))
        (t_3 (fma (* c -4.0) a (* b b)))
        (t_4 (pow (* a c) 4.0))
        (t_5 (* t_4 (pow b -6.0)))
        (t_6 (* t_4 20.0))
        (t_7 (sqrt t_3))
        (t_8 (* (* c c) (* a a)))
        (t_9 (/ t_8 (* b b)))
        (t_10 (* t_8 0.0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -70.0)
     (/ (/ (- t_3 (* b b)) (+ t_7 b)) (* 2.0 a))
     (/
      (/
       (fma
        (* t_2 -8.0)
        b
        (*
         (fma
          (* -4.0 a)
          c
          (fma
           t_9
           -4.0
           (fma
            t_2
            -4.0
            (fma
             (* -2.0 a)
             c
             (fma
              (* (* (* (* t_0 0.0) c) a) (pow b -6.0))
              -2.0
              (fma
               (* a (/ (* t_10 c) t_1))
               -2.0
               (fma
                (* (* (* (* c c) t_10) (* a a)) (pow b -6.0))
                -2.0
                (fma
                 -2.0
                 t_9
                 (+
                  (/ (fma -1.0 t_6 (* -0.5 t_6)) (pow b 6.0))
                  (fma
                   t_9
                   4.0
                   (fma
                    t_5
                    4.0
                    (fma 8.0 t_9 (fma 16.0 t_2 (* t_5 32.0))))))))))))))
         b))
       (+ t_3 (fma b b (* t_7 b))))
      (* 2.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = ((c * c) * c) * ((a * a) * a);
	double t_1 = ((b * b) * b) * b;
	double t_2 = t_0 / t_1;
	double t_3 = fma((c * -4.0), a, (b * b));
	double t_4 = pow((a * c), 4.0);
	double t_5 = t_4 * pow(b, -6.0);
	double t_6 = t_4 * 20.0;
	double t_7 = sqrt(t_3);
	double t_8 = (c * c) * (a * a);
	double t_9 = t_8 / (b * b);
	double t_10 = t_8 * 0.0;
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -70.0) {
		tmp = ((t_3 - (b * b)) / (t_7 + b)) / (2.0 * a);
	} else {
		tmp = (fma((t_2 * -8.0), b, (fma((-4.0 * a), c, fma(t_9, -4.0, fma(t_2, -4.0, fma((-2.0 * a), c, fma(((((t_0 * 0.0) * c) * a) * pow(b, -6.0)), -2.0, fma((a * ((t_10 * c) / t_1)), -2.0, fma(((((c * c) * t_10) * (a * a)) * pow(b, -6.0)), -2.0, fma(-2.0, t_9, ((fma(-1.0, t_6, (-0.5 * t_6)) / pow(b, 6.0)) + fma(t_9, 4.0, fma(t_5, 4.0, fma(8.0, t_9, fma(16.0, t_2, (t_5 * 32.0)))))))))))))) * b)) / (t_3 + fma(b, b, (t_7 * b)))) / (2.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) * a))
	t_1 = Float64(Float64(Float64(b * b) * b) * b)
	t_2 = Float64(t_0 / t_1)
	t_3 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_4 = Float64(a * c) ^ 4.0
	t_5 = Float64(t_4 * (b ^ -6.0))
	t_6 = Float64(t_4 * 20.0)
	t_7 = sqrt(t_3)
	t_8 = Float64(Float64(c * c) * Float64(a * a))
	t_9 = Float64(t_8 / Float64(b * b))
	t_10 = Float64(t_8 * 0.0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -70.0)
		tmp = Float64(Float64(Float64(t_3 - Float64(b * b)) / Float64(t_7 + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(t_2 * -8.0), b, Float64(fma(Float64(-4.0 * a), c, fma(t_9, -4.0, fma(t_2, -4.0, fma(Float64(-2.0 * a), c, fma(Float64(Float64(Float64(Float64(t_0 * 0.0) * c) * a) * (b ^ -6.0)), -2.0, fma(Float64(a * Float64(Float64(t_10 * c) / t_1)), -2.0, fma(Float64(Float64(Float64(Float64(c * c) * t_10) * Float64(a * a)) * (b ^ -6.0)), -2.0, fma(-2.0, t_9, Float64(Float64(fma(-1.0, t_6, Float64(-0.5 * t_6)) / (b ^ 6.0)) + fma(t_9, 4.0, fma(t_5, 4.0, fma(8.0, t_9, fma(16.0, t_2, Float64(t_5 * 32.0)))))))))))))) * b)) / Float64(t_3 + fma(b, b, Float64(t_7 * b)))) / Float64(2.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * -4.0), $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[(t$95$4 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 * 20.0), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[t$95$3], $MachinePrecision]}, Block[{t$95$8 = N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$8 / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(t$95$8 * 0.0), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -70.0], N[(N[(N[(t$95$3 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(t$95$7 + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$2 * -8.0), $MachinePrecision] * b + N[(N[(N[(-4.0 * a), $MachinePrecision] * c + N[(t$95$9 * -4.0 + N[(t$95$2 * -4.0 + N[(N[(-2.0 * a), $MachinePrecision] * c + N[(N[(N[(N[(N[(t$95$0 * 0.0), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(a * N[(N[(t$95$10 * c), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(N[(c * c), $MachinePrecision] * t$95$10), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(-2.0 * t$95$9 + N[(N[(N[(-1.0 * t$95$6 + N[(-0.5 * t$95$6), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$9 * 4.0 + N[(t$95$5 * 4.0 + N[(8.0 * t$95$9 + N[(16.0 * t$95$2 + N[(t$95$5 * 32.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 + N[(b * b + N[(t$95$7 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * 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)) < -70

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -70 < (/.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 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 55.7%

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

   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 91.0%

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

   8. Applied rewrites 91.0%

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

Alternative 6: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}\\ t_1 := {a}^{4} \cdot {c}^{4}\\ t_2 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_3 := \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -70:\\ \;\;\;\;\frac{\frac{t\_2 - b \cdot b}{\sqrt{t\_2} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-4, a \cdot c, \mathsf{fma}\left(-4, t\_0, \mathsf{fma}\left(-2, a \cdot c, \mathsf{fma}\left(-2, t\_3, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(4, t\_1, 16 \cdot t\_1\right)}{{b}^{6}}, \mathsf{fma}\left(8, t\_3, \mathsf{fma}\left(8, t\_0, 16 \cdot \frac{t\_1}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right) \cdot \left(a + a\right)}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* (pow a 3.0) (pow c 3.0)) (pow b 4.0)))
        (t_1 (* (pow a 4.0) (pow c 4.0)))
        (t_2 (fma (* c -4.0) a (* b b)))
        (t_3 (/ (* (pow a 2.0) (pow c 2.0)) (pow b 2.0))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -70.0)
     (/ (/ (- t_2 (* b b)) (+ (sqrt t_2) b)) (* 2.0 a))
     (/
      (*
       b
       (fma
        -4.0
        (* a c)
        (fma
         -4.0
         t_0
         (fma
          -2.0
          (* a c)
          (fma
           -2.0
           t_3
           (fma
            -0.5
            (/ (fma 4.0 t_1 (* 16.0 t_1)) (pow b 6.0))
            (fma 8.0 t_3 (fma 8.0 t_0 (* 16.0 (/ t_1 (pow b 6.0)))))))))))
      (*
       (fma b (+ b (+ (sqrt (fma (* -4.0 c) a (* b b))) b)) (* (* -4.0 c) a))
       (+ a a))))))

C

double code(double a, double b, double c) {
	double t_0 = (pow(a, 3.0) * pow(c, 3.0)) / pow(b, 4.0);
	double t_1 = pow(a, 4.0) * pow(c, 4.0);
	double t_2 = fma((c * -4.0), a, (b * b));
	double t_3 = (pow(a, 2.0) * pow(c, 2.0)) / pow(b, 2.0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -70.0) {
		tmp = ((t_2 - (b * b)) / (sqrt(t_2) + b)) / (2.0 * a);
	} else {
		tmp = (b * fma(-4.0, (a * c), fma(-4.0, t_0, fma(-2.0, (a * c), fma(-2.0, t_3, fma(-0.5, (fma(4.0, t_1, (16.0 * t_1)) / pow(b, 6.0)), fma(8.0, t_3, fma(8.0, t_0, (16.0 * (t_1 / pow(b, 6.0))))))))))) / (fma(b, (b + (sqrt(fma((-4.0 * c), a, (b * b))) + b)), ((-4.0 * c) * a)) * (a + a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64((a ^ 3.0) * (c ^ 3.0)) / (b ^ 4.0))
	t_1 = Float64((a ^ 4.0) * (c ^ 4.0))
	t_2 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_3 = Float64(Float64((a ^ 2.0) * (c ^ 2.0)) / (b ^ 2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -70.0)
		tmp = Float64(Float64(Float64(t_2 - Float64(b * b)) / Float64(sqrt(t_2) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(b * fma(-4.0, Float64(a * c), fma(-4.0, t_0, fma(-2.0, Float64(a * c), fma(-2.0, t_3, fma(-0.5, Float64(fma(4.0, t_1, Float64(16.0 * t_1)) / (b ^ 6.0)), fma(8.0, t_3, fma(8.0, t_0, Float64(16.0 * Float64(t_1 / (b ^ 6.0))))))))))) / Float64(fma(b, Float64(b + Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b)), Float64(Float64(-4.0 * c) * a)) * Float64(a + a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $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], -70.0], N[(N[(N[(t$95$2 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$2], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(-4.0 * N[(a * c), $MachinePrecision] + N[(-4.0 * t$95$0 + N[(-2.0 * N[(a * c), $MachinePrecision] + N[(-2.0 * t$95$3 + N[(-0.5 * N[(N[(4.0 * t$95$1 + N[(16.0 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(8.0 * t$95$3 + N[(8.0 * t$95$0 + N[(16.0 * N[(t$95$1 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * N[(b + N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a), $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)) < -70

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -70 < (/.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 55.8%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

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

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

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

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

      6. remove-double-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-unsound-log.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 56.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 91.0%

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

Alternative 7: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := a \cdot {c}^{2}\\ t_1 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_2 := {a}^{2} \cdot {c}^{3}\\ t_3 := {a}^{4} \cdot {c}^{4}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -70:\\ \;\;\;\;\frac{\frac{t\_1 - b \cdot b}{\sqrt{t\_1} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-4, a \cdot c, -2 \cdot \left(a \cdot c\right)\right)}{a}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-4, t\_2, 8 \cdot t\_2\right)}{{b}^{4}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, t\_0, 8 \cdot t\_0\right)}{{b}^{2}}, 0.5 \cdot \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(4, t\_3, 16 \cdot t\_3\right)}{a}, 16 \cdot \left({a}^{3} \cdot {c}^{4}\right)\right)}{{b}^{6}}\right)\right)\right)}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (pow c 2.0)))
        (t_1 (fma (* c -4.0) a (* b b)))
        (t_2 (* (pow a 2.0) (pow c 3.0)))
        (t_3 (* (pow a 4.0) (pow c 4.0))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -70.0)
     (/ (/ (- t_1 (* b b)) (+ (sqrt t_1) b)) (* 2.0 a))
     (/
      (*
       b
       (fma
        0.5
        (/ (fma -4.0 (* a c) (* -2.0 (* a c))) a)
        (fma
         0.5
         (/ (fma -4.0 t_2 (* 8.0 t_2)) (pow b 4.0))
         (fma
          0.5
          (/ (fma -2.0 t_0 (* 8.0 t_0)) (pow b 2.0))
          (*
           0.5
           (/
            (fma
             -0.5
             (/ (fma 4.0 t_3 (* 16.0 t_3)) a)
             (* 16.0 (* (pow a 3.0) (pow c 4.0))))
            (pow b 6.0)))))))
      (fma
       b
       (+ b (+ (sqrt (fma (* -4.0 c) a (* b b))) b))
       (* (* -4.0 c) a))))))

C

double code(double a, double b, double c) {
	double t_0 = a * pow(c, 2.0);
	double t_1 = fma((c * -4.0), a, (b * b));
	double t_2 = pow(a, 2.0) * pow(c, 3.0);
	double t_3 = pow(a, 4.0) * pow(c, 4.0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -70.0) {
		tmp = ((t_1 - (b * b)) / (sqrt(t_1) + b)) / (2.0 * a);
	} else {
		tmp = (b * fma(0.5, (fma(-4.0, (a * c), (-2.0 * (a * c))) / a), fma(0.5, (fma(-4.0, t_2, (8.0 * t_2)) / pow(b, 4.0)), fma(0.5, (fma(-2.0, t_0, (8.0 * t_0)) / pow(b, 2.0)), (0.5 * (fma(-0.5, (fma(4.0, t_3, (16.0 * t_3)) / a), (16.0 * (pow(a, 3.0) * pow(c, 4.0)))) / pow(b, 6.0))))))) / fma(b, (b + (sqrt(fma((-4.0 * c), a, (b * b))) + b)), ((-4.0 * c) * a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * (c ^ 2.0))
	t_1 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_2 = Float64((a ^ 2.0) * (c ^ 3.0))
	t_3 = Float64((a ^ 4.0) * (c ^ 4.0))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -70.0)
		tmp = Float64(Float64(Float64(t_1 - Float64(b * b)) / Float64(sqrt(t_1) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(b * fma(0.5, Float64(fma(-4.0, Float64(a * c), Float64(-2.0 * Float64(a * c))) / a), fma(0.5, Float64(fma(-4.0, t_2, Float64(8.0 * t_2)) / (b ^ 4.0)), fma(0.5, Float64(fma(-2.0, t_0, Float64(8.0 * t_0)) / (b ^ 2.0)), Float64(0.5 * Float64(fma(-0.5, Float64(fma(4.0, t_3, Float64(16.0 * t_3)) / a), Float64(16.0 * Float64((a ^ 3.0) * (c ^ 4.0)))) / (b ^ 6.0))))))) / fma(b, Float64(b + Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b)), Float64(Float64(-4.0 * c) * a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $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], -70.0], N[(N[(N[(t$95$1 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$1], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(0.5 * N[(N[(-4.0 * N[(a * c), $MachinePrecision] + N[(-2.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] + N[(0.5 * N[(N[(-4.0 * t$95$2 + N[(8.0 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(-2.0 * t$95$0 + N[(8.0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(-0.5 * N[(N[(4.0 * t$95$3 + N[(16.0 * t$95$3), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] + N[(16.0 * N[(N[Power[a, 3.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b + N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * 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)) < -70

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -70 < (/.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 55.8%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

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

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

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

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

      6. remove-double-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-unsound-log.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 56.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 90.9%

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

Alternative 8: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -8:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \left(\mathsf{fma}\left(0.25, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right) \cdot -2\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -8.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) 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))))
        (* (* (* (* c c) c) (/ (* a a) (* (* b b) (* b b)))) -2.0)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -8.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + 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)))) - ((((c * c) * c) * ((a * a) / ((b * b) * (b * b)))) * -2.0))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -8.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) - Float64(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(c * c) * c) * Float64(Float64(a * a) / Float64(Float64(b * b) * Float64(b * b)))) * -2.0))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -8.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(N[(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[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -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)) < -8

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -8 < (/.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 55.8%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.4%

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

   6. Applied rewrites 90.4%

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

Alternative 9: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -8:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a \cdot a}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} \cdot \left(\left(c \cdot c\right) \cdot c\right), -2, {\left(a \cdot c\right)}^{4} \cdot \left(\frac{20}{{b}^{6} \cdot a} \cdot -0.25\right) - \mathsf{fma}\left(\frac{a}{b \cdot b}, c \cdot c, c\right)\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -8.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (/
      (fma
       (* (/ (* a a) (* (* (* b b) b) b)) (* (* c c) c))
       -2.0
       (-
        (* (pow (* a c) 4.0) (* (/ 20.0 (* (pow b 6.0) a)) -0.25))
        (fma (/ a (* b b)) (* c c) c)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -8.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = fma((((a * a) / (((b * b) * b) * b)) * ((c * c) * c)), -2.0, ((pow((a * c), 4.0) * ((20.0 / (pow(b, 6.0) * a)) * -0.25)) - fma((a / (b * b)), (c * c), c))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -8.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(a * a) / Float64(Float64(Float64(b * b) * b) * b)) * Float64(Float64(c * c) * c)), -2.0, Float64(Float64((Float64(a * c) ^ 4.0) * Float64(Float64(20.0 / Float64((b ^ 6.0) * a)) * -0.25)) - fma(Float64(a / Float64(b * b)), Float64(c * c), c))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -8.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(N[(20.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] - N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision] + 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)) < -8

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -8 < (/.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 55.8%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.4%

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

   6. Applied rewrites 90.4%

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

   8. Applied rewrites 90.4%

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

Alternative 10: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -8:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, c, a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{4}}, -1 \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -8.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (/
      (fma
       -1.0
       c
       (*
        a
        (fma
         -2.0
         (/ (* a (pow c 3.0)) (pow b 4.0))
         (* -1.0 (/ (pow c 2.0) (pow b 2.0))))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -8.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = fma(-1.0, c, (a * fma(-2.0, ((a * pow(c, 3.0)) / pow(b, 4.0)), (-1.0 * (pow(c, 2.0) / pow(b, 2.0)))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -8.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(fma(-1.0, c, Float64(a * fma(-2.0, Float64(Float64(a * (c ^ 3.0)) / (b ^ 4.0)), Float64(-1.0 * Float64((c ^ 2.0) / (b ^ 2.0)))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -8.0], 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[(-1.0 * c + N[(a * N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $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)) < -8

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -8 < (/.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 55.8%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-pow.f64 87.3%

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

   7. Applied rewrites 87.3%

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

Alternative 11: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -8:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{5}}, -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -8.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) 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((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -8.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + 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(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)) <= -8.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + 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[(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], -8.0], 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[(-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)) < -8

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -8 < (/.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 55.8%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.4%

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

   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 87.3%

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

Alternative 12: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -8:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{-2}{b}, \mathsf{fma}\left(-1, \frac{c}{b}, -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right)\\ \end{array} \]

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)) -8.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (fma
      (* (* (* c c) c) (/ (* a a) (* (* b b) (* b b))))
      (/ -2.0 b)
      (fma -1.0 (/ c b) (* -1.0 (/ (* a (pow c 2.0)) (pow b 3.0))))))))

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)) <= -8.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = fma((((c * c) * c) * ((a * a) / ((b * b) * (b * b)))), (-2.0 / b), fma(-1.0, (c / b), (-1.0 * ((a * pow(c, 2.0)) / pow(b, 3.0)))));
	}
	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)) <= -8.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = fma(Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) / Float64(Float64(b * b) * Float64(b * b)))), Float64(-2.0 / b), fma(-1.0, Float64(c / b), Float64(-1.0 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	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], -8.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-2.0 / b), $MachinePrecision] + N[(-1.0 * N[(c / b), $MachinePrecision] + N[(-1.0 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $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)) < -8

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -8 < (/.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 55.8%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.4%

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

   6. Applied rewrites 90.4%

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

   7. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 87.3%

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

   9. Applied rewrites 87.3%

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

Alternative 13: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -8:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} - \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -8.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (*
      c
      (-
       (* c (- (* -2.0 (/ (* (pow a 2.0) c) (pow b 5.0))) (/ a (pow b 3.0))))
       (/ 1.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -8.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = c * ((c * ((-2.0 * ((pow(a, 2.0) * c) / pow(b, 5.0))) - (a / pow(b, 3.0)))) - (1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -8.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * c) / (b ^ 5.0))) - Float64(a / (b ^ 3.0)))) - Float64(1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -8.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -8

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -8 < (/.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 55.8%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.4%

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

   6. Applied rewrites 90.4%

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

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   9. Applied rewrites 87.1%

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

Alternative 14: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -8:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{-2}{b}, \frac{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{2}} - 1\right)}{b}\right)\\ \end{array} \]

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)) -8.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (fma
      (* (* (* c c) c) (/ (* a a) (* (* b b) (* b b))))
      (/ -2.0 b)
      (/ (* c (- (* -1.0 (/ (* a c) (pow b 2.0))) 1.0)) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -8.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = fma((((c * c) * c) * ((a * a) / ((b * b) * (b * b)))), (-2.0 / b), ((c * ((-1.0 * ((a * c) / pow(b, 2.0))) - 1.0)) / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -8.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = fma(Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) / Float64(Float64(b * b) * Float64(b * b)))), Float64(-2.0 / b), Float64(Float64(c * Float64(Float64(-1.0 * Float64(Float64(a * c) / (b ^ 2.0))) - 1.0)) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -8.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-2.0 / b), $MachinePrecision] + N[(N[(c * N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $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)) < -8

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -8 < (/.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 55.8%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.4%

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

   6. Applied rewrites 90.4%

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

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 87.1%

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

   9. Applied rewrites 87.1%

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

Alternative 15: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00015:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} - c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.00015)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (/ (- (* -1.0 (/ (* a (pow c 2.0)) (pow b 2.0))) c) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.00015) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = ((-1.0 * ((a * pow(c, 2.0)) / pow(b, 2.0))) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.00015)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-1.0 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.00015], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $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)) < -1.4999999999999999e-4

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -1.4999999999999999e-4 < (/.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 55.8%

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

      1. lift-/.f64 N/A

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

      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 54.9%

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

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.0%

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

   6. Applied rewrites 81.0%

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

Alternative 16: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00015:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.00015)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (* c (- (* -1.0 (/ (* a c) (pow b 3.0))) (/ 1.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.00015) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = c * ((-1.0 * ((a * c) / pow(b, 3.0))) - (1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.00015)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(c * Float64(Float64(-1.0 * Float64(Float64(a * c) / (b ^ 3.0))) - Float64(1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.00015], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.4999999999999999e-4

   1. Initial program 55.8%

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

      1. lift-+.f64 N/A

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

      2. +-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 57.3%

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

   if -1.4999999999999999e-4 < (/.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 55.8%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.4%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 80.9%

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

   7. Applied rewrites 80.9%

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

Alternative 17: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00015:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.00015)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (* c (- (* -1.0 (/ (* a c) (pow b 3.0))) (/ 1.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.00015) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = c * ((-1.0 * ((a * c) / pow(b, 3.0))) - (1.0 / 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)) <= -0.00015)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(c * Float64(Float64(-1.0 * Float64(Float64(a * c) / (b ^ 3.0))) - Float64(1.0 / 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], -0.00015], 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[(c * N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.4999999999999999e-4

   1. Initial program 55.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 55.9%

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

   3. Applied rewrites 55.9%

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

   if -1.4999999999999999e-4 < (/.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 55.8%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.4%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 80.9%

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

   7. Applied rewrites 80.9%

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

Alternative 18: 84.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00015:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{2}} - 1\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.00015)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (/ (* c (- (* -1.0 (/ (* a c) (pow b 2.0))) 1.0)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.00015) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = (c * ((-1.0 * ((a * c) / pow(b, 2.0))) - 1.0)) / 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)) <= -0.00015)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(-1.0 * Float64(Float64(a * c) / (b ^ 2.0))) - 1.0)) / 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], -0.00015], 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 * N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $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)) < -1.4999999999999999e-4

   1. Initial program 55.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 55.9%

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

   3. Applied rewrites 55.9%

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

   if -1.4999999999999999e-4 < (/.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 55.8%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.4%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 80.9%

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

   7. Applied rewrites 80.9%

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

Alternative 19: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.7e-5)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 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)) <= -1.7e-5) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * 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)) <= -1.7e-5)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * 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], -1.7e-5], 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[((-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)) < -1.7e-5

   1. Initial program 55.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 55.9%

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

   3. Applied rewrites 55.9%

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

   if -1.7e-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 55.8%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.0%

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

   4. Applied rewrites 64.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 64.0%

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

   6. Applied rewrites 64.0%

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

Alternative 20: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1.7e-5)
   (/ (- (sqrt (fma (* a c) -4.0 (* 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)) <= -1.7e-5) {
		tmp = (sqrt(fma((a * c), -4.0, (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)) <= -1.7e-5)
		tmp = Float64(Float64(sqrt(fma(Float64(a * c), -4.0, 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], -1.7e-5], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + 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)) < -1.7e-5

   1. Initial program 55.8%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

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

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

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

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

      6. remove-double-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-unsound-log.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg 52.9%

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

   3. Applied rewrites 52.9%

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

   5. Applied rewrites 56.7%

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

   7. Applied rewrites 55.8%

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

   if -1.7e-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 55.8%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.0%

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

   4. Applied rewrites 64.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 64.0%

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

   6. Applied rewrites 64.0%

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

Alternative 21: 64.0% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = -c / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 64.0%

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

4. Applied rewrites 64.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 64.0%

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

6. Applied rewrites 64.0%

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

Reproduce

herbie shell --seed 2025191 
(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)))