Quadratic roots, narrow range

Percentage Accurate: 55.7% → 92.3%

Time: 13.0s

Alternatives: 15

Speedup: 3.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

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

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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

Initial Program: 55.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

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

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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

Alternative 1: 92.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot c\right)}^{3}\\ t_1 := \left(a \cdot c\right) \cdot -12\\ t_2 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ t_3 := \left(c \cdot c\right) \cdot 12\\ t_4 := \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {t\_1}^{2}\right)\\ t_5 := \frac{c}{b \cdot b}\\ \mathbf{if}\;b \leq 0.86:\\ \;\;\;\;\frac{b \cdot b - t\_2}{\left(\left(-b\right) - \sqrt{t\_2}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left({t\_3}^{2}, 0.25, \left(\mathsf{fma}\left(-64, {c}^{3}, \left(t\_3 \cdot c\right) \cdot 6\right) \cdot c\right) \cdot -6\right) \cdot {a}^{4}}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-64, t\_0, -0.5 \cdot \left(t\_1 \cdot t\_4\right)\right)}{{b}^{4}} + \frac{t\_4}{b \cdot b}, 0.5 \cdot t\_1\right)\right)}{\mathsf{fma}\left(b, b, \left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(-4, a \cdot t\_5, \mathsf{fma}\left(-2, \mathsf{fma}\left(a, t\_5, \left(a \cdot a\right) \cdot \frac{c \cdot c}{{b}^{4}}\right), \frac{-4 \cdot t\_0}{{b}^{6}}\right)\right) + 2\right)\right) \cdot \left(2 \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* a c) 3.0))
        (t_1 (* (* a c) -12.0))
        (t_2 (fma (* -4.0 a) c (* b b)))
        (t_3 (* (* c c) 12.0))
        (t_4 (fma (* (* a a) (* c c)) 48.0 (* -0.25 (pow t_1 2.0))))
        (t_5 (/ c (* b b))))
   (if (<= b 0.86)
     (/ (- (* b b) t_2) (* (- (- b) (sqrt t_2)) (* 2.0 a)))
     (/
      (*
       b
       (fma
        -0.5
        (/
         (*
          (fma
           (pow t_3 2.0)
           0.25
           (* (* (fma -64.0 (pow c 3.0) (* (* t_3 c) 6.0)) c) -6.0))
          (pow a 4.0))
         (pow b 6.0))
        (fma
         0.5
         (+
          (/ (fma -64.0 t_0 (* -0.5 (* t_1 t_4))) (pow b 4.0))
          (/ t_4 (* b b)))
         (* 0.5 t_1))))
      (*
       (fma
        b
        b
        (*
         (* b b)
         (+
          (fma
           -4.0
           (* a t_5)
           (fma
            -2.0
            (fma a t_5 (* (* a a) (/ (* c c) (pow b 4.0))))
            (/ (* -4.0 t_0) (pow b 6.0))))
          2.0)))
       (* 2.0 a))))))

C

double code(double a, double b, double c) {
	double t_0 = pow((a * c), 3.0);
	double t_1 = (a * c) * -12.0;
	double t_2 = fma((-4.0 * a), c, (b * b));
	double t_3 = (c * c) * 12.0;
	double t_4 = fma(((a * a) * (c * c)), 48.0, (-0.25 * pow(t_1, 2.0)));
	double t_5 = c / (b * b);
	double tmp;
	if (b <= 0.86) {
		tmp = ((b * b) - t_2) / ((-b - sqrt(t_2)) * (2.0 * a));
	} else {
		tmp = (b * fma(-0.5, ((fma(pow(t_3, 2.0), 0.25, ((fma(-64.0, pow(c, 3.0), ((t_3 * c) * 6.0)) * c) * -6.0)) * pow(a, 4.0)) / pow(b, 6.0)), fma(0.5, ((fma(-64.0, t_0, (-0.5 * (t_1 * t_4))) / pow(b, 4.0)) + (t_4 / (b * b))), (0.5 * t_1)))) / (fma(b, b, ((b * b) * (fma(-4.0, (a * t_5), fma(-2.0, fma(a, t_5, ((a * a) * ((c * c) / pow(b, 4.0)))), ((-4.0 * t_0) / pow(b, 6.0)))) + 2.0))) * (2.0 * a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * c) ^ 3.0
	t_1 = Float64(Float64(a * c) * -12.0)
	t_2 = fma(Float64(-4.0 * a), c, Float64(b * b))
	t_3 = Float64(Float64(c * c) * 12.0)
	t_4 = fma(Float64(Float64(a * a) * Float64(c * c)), 48.0, Float64(-0.25 * (t_1 ^ 2.0)))
	t_5 = Float64(c / Float64(b * b))
	tmp = 0.0
	if (b <= 0.86)
		tmp = Float64(Float64(Float64(b * b) - t_2) / Float64(Float64(Float64(-b) - sqrt(t_2)) * Float64(2.0 * a)));
	else
		tmp = Float64(Float64(b * fma(-0.5, Float64(Float64(fma((t_3 ^ 2.0), 0.25, Float64(Float64(fma(-64.0, (c ^ 3.0), Float64(Float64(t_3 * c) * 6.0)) * c) * -6.0)) * (a ^ 4.0)) / (b ^ 6.0)), fma(0.5, Float64(Float64(fma(-64.0, t_0, Float64(-0.5 * Float64(t_1 * t_4))) / (b ^ 4.0)) + Float64(t_4 / Float64(b * b))), Float64(0.5 * t_1)))) / Float64(fma(b, b, Float64(Float64(b * b) * Float64(fma(-4.0, Float64(a * t_5), fma(-2.0, fma(a, t_5, Float64(Float64(a * a) * Float64(Float64(c * c) / (b ^ 4.0)))), Float64(Float64(-4.0 * t_0) / (b ^ 6.0)))) + 2.0))) * Float64(2.0 * a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * c), $MachinePrecision] * -12.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * c), $MachinePrecision] * 12.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * 48.0 + N[(-0.25 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.86], N[(N[(N[(b * b), $MachinePrecision] - t$95$2), $MachinePrecision] / N[(N[((-b) - N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(-0.5 * N[(N[(N[(N[Power[t$95$3, 2.0], $MachinePrecision] * 0.25 + N[(N[(N[(-64.0 * N[Power[c, 3.0], $MachinePrecision] + N[(N[(t$95$3 * c), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(N[(-64.0 * t$95$0 + N[(-0.5 * N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b + N[(N[(b * b), $MachinePrecision] * N[(N[(-4.0 * N[(a * t$95$5), $MachinePrecision] + N[(-2.0 * N[(a * t$95$5 + N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * t$95$0), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.859999999999999987

   1. Initial program 83.8%

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

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 85.9%

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

   if 0.859999999999999987 < b

   1. Initial program 53.2%

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

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5} + {\left(-b\right)}^{3}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{2}}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

   7. Applied rewrites 94.8%

      \[\leadsto \frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -12\right), \mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

   8. Taylor expanded in b around inf

      \[\leadsto \frac{b \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -12\right), \mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, \frac{-1}{2} \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right), \frac{1}{4} \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, \frac{-1}{2} \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{b \cdot b}, \frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right)\right)}{\mathsf{fma}\left(b, b, \color{blue}{{b}^{2} \cdot \left(2 + \left(-4 \cdot \frac{a \cdot c}{{b}^{2}} + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)}\right) \cdot \left(2 \cdot a\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -12\right), \mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, \frac{-1}{2} \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right), \frac{1}{4} \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, \frac{-1}{2} \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{b \cdot b}, \frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right)\right)}{\mathsf{fma}\left(b, b, \color{blue}{{b}^{2} \cdot \left(2 + \left(-4 \cdot \frac{a \cdot c}{{b}^{2}} + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)}\right) \cdot \left(2 \cdot a\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -12\right), \mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, \frac{-1}{2} \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right), \frac{1}{4} \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, \frac{-1}{2} \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{b \cdot b}, \frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right)\right)}{\mathsf{fma}\left(b, b, \color{blue}{\left(b \cdot b\right)} \cdot \left(2 + \left(-4 \cdot \frac{a \cdot c}{{b}^{2}} + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right) \cdot \left(2 \cdot a\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -12\right), \mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, \frac{-1}{2} \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right), \frac{1}{4} \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, \frac{-1}{2} \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{b \cdot b}, \frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right)\right)}{\mathsf{fma}\left(b, b, \color{blue}{\left(b \cdot b\right)} \cdot \left(2 + \left(-4 \cdot \frac{a \cdot c}{{b}^{2}} + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)\right) \cdot \left(2 \cdot a\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -12\right), \mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, \frac{-1}{2} \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right), \frac{1}{4} \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, \frac{-1}{2} \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{b \cdot b}, \frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right)\right)}{\mathsf{fma}\left(b, b, \left(b \cdot b\right) \cdot \color{blue}{\left(\left(-4 \cdot \frac{a \cdot c}{{b}^{2}} + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right) + 2\right)}\right) \cdot \left(2 \cdot a\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -12\right), \mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, \frac{-1}{2} \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right), \frac{1}{4} \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, \frac{-1}{2} \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{b \cdot b}, \frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right)\right)}{\mathsf{fma}\left(b, b, \left(b \cdot b\right) \cdot \color{blue}{\left(\left(-4 \cdot \frac{a \cdot c}{{b}^{2}} + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right) + 2\right)}\right) \cdot \left(2 \cdot a\right)} \]

   10. Applied rewrites 94.9%

       \[\leadsto \frac{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -12\right), \mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right)\right)}{\mathsf{fma}\left(b, b, \color{blue}{\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(-4, a \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(-2, \mathsf{fma}\left(a, \frac{c}{b \cdot b}, \left(a \cdot a\right) \cdot \frac{c \cdot c}{{b}^{4}}\right), \frac{-4 \cdot {\left(a \cdot c\right)}^{3}}{{b}^{6}}\right)\right) + 2\right)}\right) \cdot \left(2 \cdot a\right)} \]

   11. Taylor expanded in a around 0

       \[\leadsto \frac{b \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{{a}^{4} \cdot \left(-6 \cdot \left(c \cdot \left(-64 \cdot {c}^{3} + 6 \cdot \left(c \cdot \left(-36 \cdot {c}^{2} + 48 \cdot {c}^{2}\right)\right)\right)\right) + \frac{1}{4} \cdot {\left(-36 \cdot {c}^{2} + 48 \cdot {c}^{2}\right)}^{2}\right)}{{\color{blue}{b}}^{6}}, \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, \frac{-1}{2} \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, \frac{-1}{4} \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{b \cdot b}, \frac{1}{2} \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right)\right)}{\mathsf{fma}\left(b, b, \left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(-4, a \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(-2, \mathsf{fma}\left(a, \frac{c}{b \cdot b}, \left(a \cdot a\right) \cdot \frac{c \cdot c}{{b}^{4}}\right), \frac{-4 \cdot {\left(a \cdot c\right)}^{3}}{{b}^{6}}\right)\right) + 2\right)\right) \cdot \left(2 \cdot a\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 94.9%

       \[\leadsto \frac{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left({\left(\left(c \cdot c\right) \cdot 12\right)}^{2}, 0.25, \left(\mathsf{fma}\left(-64, {c}^{3}, \left(\left(\left(c \cdot c\right) \cdot 12\right) \cdot c\right) \cdot 6\right) \cdot c\right) \cdot -6\right) \cdot {a}^{4}}{{\color{blue}{b}}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right)\right)}{\mathsf{fma}\left(b, b, \left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(-4, a \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(-2, \mathsf{fma}\left(a, \frac{c}{b \cdot b}, \left(a \cdot a\right) \cdot \frac{c \cdot c}{{b}^{4}}\right), \frac{-4 \cdot {\left(a \cdot c\right)}^{3}}{{b}^{6}}\right)\right) + 2\right)\right) \cdot \left(2 \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot c\right) \cdot 12\\ t_1 := \mathsf{fma}\left(-64, {c}^{3}, 6 \cdot \left(c \cdot t\_0\right)\right)\\ t_2 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ t_3 := \sqrt{t\_2}\\ \mathbf{if}\;b \leq 0.86:\\ \;\;\;\;\frac{b \cdot b - t\_2}{\left(\left(-b\right) - t\_3\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \left(a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-0.5, \frac{a \cdot \mathsf{fma}\left(0.25, {t\_0}^{2}, -6 \cdot \left(c \cdot t\_1\right)\right)}{{b}^{6}}, 0.5 \cdot \frac{t\_1}{{b}^{4}}\right), 0.5 \cdot \frac{t\_0}{b \cdot b}\right), -6 \cdot c\right)\right)}{\mathsf{fma}\left(b, b, t\_2 + b \cdot t\_3\right) \cdot \left(2 \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c c) 12.0))
        (t_1 (fma -64.0 (pow c 3.0) (* 6.0 (* c t_0))))
        (t_2 (fma (* -4.0 a) c (* b b)))
        (t_3 (sqrt t_2)))
   (if (<= b 0.86)
     (/ (- (* b b) t_2) (* (- (- b) t_3) (* 2.0 a)))
     (/
      (*
       b
       (*
        a
        (fma
         a
         (fma
          a
          (fma
           -0.5
           (/ (* a (fma 0.25 (pow t_0 2.0) (* -6.0 (* c t_1)))) (pow b 6.0))
           (* 0.5 (/ t_1 (pow b 4.0))))
          (* 0.5 (/ t_0 (* b b))))
         (* -6.0 c))))
      (* (fma b b (+ t_2 (* b t_3))) (* 2.0 a))))))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) * 12.0;
	double t_1 = fma(-64.0, pow(c, 3.0), (6.0 * (c * t_0)));
	double t_2 = fma((-4.0 * a), c, (b * b));
	double t_3 = sqrt(t_2);
	double tmp;
	if (b <= 0.86) {
		tmp = ((b * b) - t_2) / ((-b - t_3) * (2.0 * a));
	} else {
		tmp = (b * (a * fma(a, fma(a, fma(-0.5, ((a * fma(0.25, pow(t_0, 2.0), (-6.0 * (c * t_1)))) / pow(b, 6.0)), (0.5 * (t_1 / pow(b, 4.0)))), (0.5 * (t_0 / (b * b)))), (-6.0 * c)))) / (fma(b, b, (t_2 + (b * t_3))) * (2.0 * a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) * 12.0)
	t_1 = fma(-64.0, (c ^ 3.0), Float64(6.0 * Float64(c * t_0)))
	t_2 = fma(Float64(-4.0 * a), c, Float64(b * b))
	t_3 = sqrt(t_2)
	tmp = 0.0
	if (b <= 0.86)
		tmp = Float64(Float64(Float64(b * b) - t_2) / Float64(Float64(Float64(-b) - t_3) * Float64(2.0 * a)));
	else
		tmp = Float64(Float64(b * Float64(a * fma(a, fma(a, fma(-0.5, Float64(Float64(a * fma(0.25, (t_0 ^ 2.0), Float64(-6.0 * Float64(c * t_1)))) / (b ^ 6.0)), Float64(0.5 * Float64(t_1 / (b ^ 4.0)))), Float64(0.5 * Float64(t_0 / Float64(b * b)))), Float64(-6.0 * c)))) / Float64(fma(b, b, Float64(t_2 + Float64(b * t_3))) * Float64(2.0 * a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] * 12.0), $MachinePrecision]}, Block[{t$95$1 = N[(-64.0 * N[Power[c, 3.0], $MachinePrecision] + N[(6.0 * N[(c * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, If[LessEqual[b, 0.86], N[(N[(N[(b * b), $MachinePrecision] - t$95$2), $MachinePrecision] / N[(N[((-b) - t$95$3), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(a * N[(a * N[(a * N[(-0.5 * N[(N[(a * N[(0.25 * N[Power[t$95$0, 2.0], $MachinePrecision] + N[(-6.0 * N[(c * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$1 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-6.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b + N[(t$95$2 + N[(b * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.859999999999999987

   1. Initial program 83.8%

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

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 85.9%

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

   if 0.859999999999999987 < b

   1. Initial program 53.2%

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

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 54.4%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5} + {\left(-b\right)}^{3}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{2}}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

   7. Applied rewrites 94.8%

      \[\leadsto \frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -12\right), \mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{b \cdot \left(a \cdot \color{blue}{\left(-6 \cdot c + a \cdot \left(\frac{1}{2} \cdot \left(-36 \cdot \frac{{c}^{2}}{{b}^{2}} + 48 \cdot \frac{{c}^{2}}{{b}^{2}}\right) + a \cdot \left(\frac{-1}{2} \cdot \frac{a \cdot \left(-6 \cdot \left(c \cdot \left(-64 \cdot {c}^{3} + 6 \cdot \left(c \cdot \left(-36 \cdot {c}^{2} + 48 \cdot {c}^{2}\right)\right)\right)\right) + \frac{1}{4} \cdot {\left(-36 \cdot {c}^{2} + 48 \cdot {c}^{2}\right)}^{2}\right)}{{b}^{6}} + \frac{1}{2} \cdot \left(-64 \cdot \frac{{c}^{3}}{{b}^{4}} + 6 \cdot \frac{c \cdot \left(-36 \cdot {c}^{2} + 48 \cdot {c}^{2}\right)}{{b}^{4}}\right)\right)\right)\right)}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

   10. Applied rewrites 94.8%

       \[\leadsto \frac{b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(-0.5, \frac{a \cdot \mathsf{fma}\left(0.25, {\left(\left(c \cdot c\right) \cdot 12\right)}^{2}, -6 \cdot \left(c \cdot \mathsf{fma}\left(-64, {c}^{3}, 6 \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot 12\right)\right)\right)\right)\right)}{{b}^{6}}, 0.5 \cdot \frac{\mathsf{fma}\left(-64, {c}^{3}, 6 \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot 12\right)\right)\right)}{{b}^{4}}\right), 0.5 \cdot \frac{\left(c \cdot c\right) \cdot 12}{b \cdot b}\right), -6 \cdot c\right)}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.86:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(\left(-b\right) - \sqrt{t\_0}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(-2 \cdot a, c, \left(-b\right) \cdot b\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 0.86)
     (/ (- (* b b) t_0) (* (- (- b) (sqrt t_0)) (* 2.0 a)))
     (fma
      (/
       (fma
        (* -5.0 (* a a))
        (pow c 4.0)
        (* (* b b) (* (fma (* -2.0 a) c (* (- b) b)) (* c c))))
       (pow b 7.0))
      a
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 0.86) {
		tmp = ((b * b) - t_0) / ((-b - sqrt(t_0)) * (2.0 * a));
	} else {
		tmp = fma((fma((-5.0 * (a * a)), pow(c, 4.0), ((b * b) * (fma((-2.0 * a), c, (-b * b)) * (c * c)))) / pow(b, 7.0)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 0.86)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(Float64(-b) - sqrt(t_0)) * Float64(2.0 * a)));
	else
		tmp = fma(Float64(fma(Float64(-5.0 * Float64(a * a)), (c ^ 4.0), Float64(Float64(b * b) * Float64(fma(Float64(-2.0 * a), c, Float64(Float64(-b) * b)) * Float64(c * c)))) / (b ^ 7.0)), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.86], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-5.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(N[(N[(-2.0 * a), $MachinePrecision] * c + N[((-b) * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.859999999999999987

   1. Initial program 83.8%

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

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 85.9%

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

   if 0.859999999999999987 < b

   1. Initial program 53.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + -1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + -1 \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, -1 \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \frac{\left(-c\right) \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\frac{-5 \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(-2 \cdot \left(a \cdot {c}^{3}\right) + -1 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 94.6%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left(b \cdot b\right) \cdot \mathsf{fma}\left(-b \cdot b, c \cdot c, -2 \cdot \left(a \cdot {c}^{3}\right)\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]

   9. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left(b \cdot b\right) \cdot \left({c}^{2} \cdot \left(-2 \cdot \left(a \cdot c\right) + -1 \cdot {b}^{2}\right)\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 94.6%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(-2 \cdot a, c, -b \cdot b\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.86:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(-2 \cdot a, c, \left(-b\right) \cdot b\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.7:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(\left(-b\right) - \sqrt{t\_0}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, \frac{a}{b} \cdot \frac{c \cdot c}{b}\right) + c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 1.7)
     (/ (- (* b b) t_0) (* (- (- b) (sqrt t_0)) (* 2.0 a)))
     (/
      (+
       (fma
        2.0
        (/ (* (* a a) (pow c 3.0)) (pow b 4.0))
        (* (/ a b) (/ (* c c) b)))
       c)
      (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 1.7) {
		tmp = ((b * b) - t_0) / ((-b - sqrt(t_0)) * (2.0 * a));
	} else {
		tmp = (fma(2.0, (((a * a) * pow(c, 3.0)) / pow(b, 4.0)), ((a / b) * ((c * c) / b))) + c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 1.7)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(Float64(-b) - sqrt(t_0)) * Float64(2.0 * a)));
	else
		tmp = Float64(Float64(fma(2.0, Float64(Float64(Float64(a * a) * (c ^ 3.0)) / (b ^ 4.0)), Float64(Float64(a / b) * Float64(Float64(c * c) / b))) + c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.7], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[(N[(N[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.69999999999999996

   1. Initial program 83.6%

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

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 85.8%

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

   if 1.69999999999999996 < b

   1. Initial program 52.4%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + -1 \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + -1 \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, -1 \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \frac{\left(-c\right) \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 91.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.7:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, \frac{a}{b} \cdot \frac{c \cdot c}{b}\right) + c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.7:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(\left(-b\right) - \sqrt{t\_0}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{{b}^{4}}, -2, \frac{\left(-c\right) \cdot c}{b \cdot b}\right), a, -c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 1.7)
     (/ (- (* b b) t_0) (* (- (- b) (sqrt t_0)) (* 2.0 a)))
     (/
      (fma
       (fma (/ (* (pow c 3.0) a) (pow b 4.0)) -2.0 (/ (* (- c) c) (* b b)))
       a
       (- c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 1.7) {
		tmp = ((b * b) - t_0) / ((-b - sqrt(t_0)) * (2.0 * a));
	} else {
		tmp = fma(fma(((pow(c, 3.0) * a) / pow(b, 4.0)), -2.0, ((-c * c) / (b * b))), a, -c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 1.7)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(Float64(-b) - sqrt(t_0)) * Float64(2.0 * a)));
	else
		tmp = Float64(fma(fma(Float64(Float64((c ^ 3.0) * a) / (b ^ 4.0)), -2.0, Float64(Float64(Float64(-c) * c) / Float64(b * b))), a, Float64(-c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.7], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[((-c) * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + (-c)), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.69999999999999996

   1. Initial program 83.6%

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

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 85.8%

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

   if 1.69999999999999996 < b

   1. Initial program 52.4%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. distribute-neg-frac N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 91.9%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 91.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.7:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{{b}^{4}}, -2, \frac{\left(-c\right) \cdot c}{b \cdot b}\right), a, -c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.7:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(\left(-b\right) - \sqrt{t\_0}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right), -2 \cdot \left(\frac{a \cdot a}{b \cdot b} \cdot \frac{{c}^{3}}{b \cdot b}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 1.7)
     (/ (- (* b b) t_0) (* (- (- b) (sqrt t_0)) (* 2.0 a)))
     (/
      (fma
       -1.0
       (fma a (/ (* c c) (* b b)) c)
       (* -2.0 (* (/ (* a a) (* b b)) (/ (pow c 3.0) (* b b)))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 1.7) {
		tmp = ((b * b) - t_0) / ((-b - sqrt(t_0)) * (2.0 * a));
	} else {
		tmp = fma(-1.0, fma(a, ((c * c) / (b * b)), c), (-2.0 * (((a * a) / (b * b)) * (pow(c, 3.0) / (b * b))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 1.7)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(Float64(-b) - sqrt(t_0)) * Float64(2.0 * a)));
	else
		tmp = Float64(fma(-1.0, fma(a, Float64(Float64(c * c) / Float64(b * b)), c), Float64(-2.0 * Float64(Float64(Float64(a * a) / Float64(b * b)) * Float64((c ^ 3.0) / Float64(b * b))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.7], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] + N[(-2.0 * N[(N[(N[(a * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.69999999999999996

   1. Initial program 83.6%

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

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 85.8%

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

   if 1.69999999999999996 < b

   1. Initial program 52.4%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. distribute-neg-frac N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 91.9%

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

   13. Applied rewrites 91.9%

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

Alternative 7: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 1.7:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(\left(-b\right) - \sqrt{t\_0}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{4}}, -2, \frac{a}{\left(-b\right) \cdot b}\right) \cdot c - 1\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 1.7)
     (/ (- (* b b) t_0) (* (- (- b) (sqrt t_0)) (* 2.0 a)))
     (/
      (*
       (- (* (fma (* (* a a) (/ c (pow b 4.0))) -2.0 (/ a (* (- b) b))) c) 1.0)
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 1.7) {
		tmp = ((b * b) - t_0) / ((-b - sqrt(t_0)) * (2.0 * a));
	} else {
		tmp = (((fma(((a * a) * (c / pow(b, 4.0))), -2.0, (a / (-b * b))) * c) - 1.0) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 1.7)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(Float64(-b) - sqrt(t_0)) * Float64(2.0 * a)));
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(Float64(a * a) * Float64(c / (b ^ 4.0))), -2.0, Float64(a / Float64(Float64(-b) * b))) * c) - 1.0) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.7], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(c / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(a / N[((-b) * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.69999999999999996

   1. Initial program 83.6%

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

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 85.8%

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

   if 1.69999999999999996 < b

   1. Initial program 52.4%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. distribute-neg-frac N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 91.9%

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

   12. Taylor expanded in c around 0

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

   14. Applied rewrites 91.7%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.7:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{4}}, -2, \frac{a}{\left(-b\right) \cdot b}\right) \cdot c - 1\right) \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq 2.82:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(\left(-b\right) - t\_1\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \mathsf{fma}\left(0.5, \frac{\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot 12\right)}{b \cdot b}, -6 \cdot \left(a \cdot c\right)\right)}{\mathsf{fma}\left(b, b, t\_0 + b \cdot t\_1\right) \cdot \left(2 \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= b 2.82)
     (/ (- (* b b) t_0) (* (- (- b) t_1) (* 2.0 a)))
     (/
      (* b (fma 0.5 (/ (* (* c c) (* (* a a) 12.0)) (* b b)) (* -6.0 (* a c))))
      (* (fma b b (+ t_0 (* b t_1))) (* 2.0 a))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (b <= 2.82) {
		tmp = ((b * b) - t_0) / ((-b - t_1) * (2.0 * a));
	} else {
		tmp = (b * fma(0.5, (((c * c) * ((a * a) * 12.0)) / (b * b)), (-6.0 * (a * c)))) / (fma(b, b, (t_0 + (b * t_1))) * (2.0 * a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (b <= 2.82)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(Float64(-b) - t_1) * Float64(2.0 * a)));
	else
		tmp = Float64(Float64(b * fma(0.5, Float64(Float64(Float64(c * c) * Float64(Float64(a * a) * 12.0)) / Float64(b * b)), Float64(-6.0 * Float64(a * c)))) / Float64(fma(b, b, Float64(t_0 + Float64(b * t_1))) * Float64(2.0 * a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[b, 2.82], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[((-b) - t$95$1), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(0.5 * N[(N[(N[(c * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 12.0), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-6.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b + N[(t$95$0 + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.81999999999999984

   1. Initial program 82.7%

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

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 85.1%

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

   if 2.81999999999999984 < b

   1. Initial program 51.8%

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

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 53.0%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5} + {\left(-b\right)}^{3}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{2}}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

   7. Applied rewrites 94.9%

      \[\leadsto \frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -12\right), \mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

   8. Taylor expanded in b around inf

      \[\leadsto \frac{b \cdot \left(-6 \cdot \left(a \cdot c\right) + \color{blue}{\frac{1}{2} \cdot \frac{-36 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 48 \cdot \left({a}^{2} \cdot {c}^{2}\right)}{{b}^{2}}}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 86.4%

       \[\leadsto \frac{b \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot 12\right)}{b \cdot b}}, -6 \cdot \left(a \cdot c\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 85.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq 2.82:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(\left(-b\right) - t\_1\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \left(a \cdot \mathsf{fma}\left(0.5 \cdot a, \frac{\left(c \cdot c\right) \cdot 12}{b \cdot b}, -6 \cdot c\right)\right)}{\mathsf{fma}\left(b, b, t\_0 + b \cdot t\_1\right) \cdot \left(2 \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= b 2.82)
     (/ (- (* b b) t_0) (* (- (- b) t_1) (* 2.0 a)))
     (/
      (* b (* a (fma (* 0.5 a) (/ (* (* c c) 12.0) (* b b)) (* -6.0 c))))
      (* (fma b b (+ t_0 (* b t_1))) (* 2.0 a))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (b <= 2.82) {
		tmp = ((b * b) - t_0) / ((-b - t_1) * (2.0 * a));
	} else {
		tmp = (b * (a * fma((0.5 * a), (((c * c) * 12.0) / (b * b)), (-6.0 * c)))) / (fma(b, b, (t_0 + (b * t_1))) * (2.0 * a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (b <= 2.82)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(Float64(-b) - t_1) * Float64(2.0 * a)));
	else
		tmp = Float64(Float64(b * Float64(a * fma(Float64(0.5 * a), Float64(Float64(Float64(c * c) * 12.0) / Float64(b * b)), Float64(-6.0 * c)))) / Float64(fma(b, b, Float64(t_0 + Float64(b * t_1))) * Float64(2.0 * a)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[b, 2.82], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[((-b) - t$95$1), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(a * N[(N[(0.5 * a), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * 12.0), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-6.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b + N[(t$95$0 + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.81999999999999984

   1. Initial program 82.7%

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

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 85.1%

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

   if 2.81999999999999984 < b

   1. Initial program 51.8%

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

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 53.0%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{1.5} + {\left(-b\right)}^{3}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)}} \]

   5. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{2}}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

   7. Applied rewrites 94.9%

      \[\leadsto \frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -12\right), \mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-64, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -12\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 48, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -12\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -12\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{b \cdot \left(a \cdot \color{blue}{\left(-6 \cdot c + \frac{1}{2} \cdot \left(a \cdot \left(-36 \cdot \frac{{c}^{2}}{{b}^{2}} + 48 \cdot \frac{{c}^{2}}{{b}^{2}}\right)\right)\right)}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 86.4%

       \[\leadsto \frac{b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot a, \frac{\left(c \cdot c\right) \cdot 12}{b \cdot b}, -6 \cdot c\right)}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 2.85:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(\left(-b\right) - \sqrt{t\_0}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 2.85)
     (/ (- (* b b) t_0) (* (- (- b) (sqrt t_0)) (* 2.0 a)))
     (/ (fma a (/ (* c c) (* b b)) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 2.85) {
		tmp = ((b * b) - t_0) / ((-b - sqrt(t_0)) * (2.0 * a));
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 2.85)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(Float64(-b) - sqrt(t_0)) * Float64(2.0 * a)));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.85], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.85000000000000009

   1. Initial program 82.6%

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

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 85.0%

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

   if 2.85000000000000009 < b

   1. Initial program 51.7%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. distribute-neg-frac N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.1%

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

   9. Taylor expanded in b around inf

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

       1. distribute-lft-out N/A

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

       2. associate-*r/ N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 85.9

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

   11. Applied rewrites 85.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.85:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.85:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{c \cdot \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.85)
   (/ (+ (- b) (sqrt (* c (fma -4.0 a (/ (* b b) c))))) (* 2.0 a))
   (/ (fma a (/ (* c c) (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.85) {
		tmp = (-b + sqrt((c * fma(-4.0, a, ((b * b) / c))))) / (2.0 * a);
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.85)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(c * fma(-4.0, a, Float64(Float64(b * b) / c))))) / Float64(2.0 * a));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.85], N[(N[((-b) + N[Sqrt[N[(c * N[(-4.0 * a + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.85000000000000009

   1. Initial program 82.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in c around inf

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

   8. Applied rewrites 82.6%

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

   if 2.85000000000000009 < b

   1. Initial program 51.7%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. distribute-neg-frac N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.1%

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

   9. Taylor expanded in b around inf

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

       1. distribute-lft-out N/A

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

       2. associate-*r/ N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 85.9

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

   11. Applied rewrites 85.9%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.85:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{c \cdot \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.85:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.85)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (/ (fma a (/ (* c c) (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.85) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.85)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.85], 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[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.85000000000000009

   1. Initial program 82.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing
   3. 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. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 82.6

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

   4. Applied rewrites 82.6%

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

   if 2.85000000000000009 < b

   1. Initial program 51.7%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. distribute-neg-frac N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.1%

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

   9. Taylor expanded in b around inf

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

       1. distribute-lft-out N/A

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

       2. associate-*r/ N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 85.9

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

   11. Applied rewrites 85.9%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.85:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.85:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.85)
   (/ (+ (- b) (sqrt (fma -4.0 (* c a) (* b b)))) (* 2.0 a))
   (/ (fma a (/ (* c c) (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.85) {
		tmp = (-b + sqrt(fma(-4.0, (c * a), (b * b)))) / (2.0 * a);
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.85)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.85], N[(N[((-b) + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.85000000000000009

   1. Initial program 82.6%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if 2.85000000000000009 < b

   1. Initial program 51.7%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*l/ N/A

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

      6. distribute-neg-frac N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.1%

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

   9. Taylor expanded in b around inf

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

       1. distribute-lft-out N/A

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

       2. associate-*r/ N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f64 85.9

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

   11. Applied rewrites 85.9%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.85:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

   5. associate-*l/ N/A

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

   6. distribute-neg-frac N/A

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

   7. mul-1-neg N/A

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

   8. lower-*.f64 N/A

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

5. Applied rewrites 80.4%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 62.5%

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

9. Taylor expanded in b around inf

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

    1. distribute-lft-out N/A

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

    2. associate-*r/ N/A

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

    3. mul-1-neg N/A

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

    4. lower-neg.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. +-commutative N/A

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

    7. associate-/l* N/A

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

    8. lower-fma.f64 N/A

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

    9. lower-/.f64 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f64 N/A

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

    12. unpow2 N/A

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

    13. lower-*.f64 80.5

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

11. Applied rewrites 80.5%

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

12. Final simplification 80.5%

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

Alternative 15: 64.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{-c}{b} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 62.5

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

5. Applied rewrites 62.5%

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

Reproduce

herbie shell --seed 2025009 
(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)))