Cubic critical, medium range

Percentage Accurate: 31.8% → 94.7%

Time: 3.9s

Alternatives: 9

Speedup: 2.9×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 31.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 94.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -500:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {t\_0}^{1.5}\right)}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 + b \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875, a \cdot c, -0.5625 \cdot \left(b \cdot b\right)\right)}{{b}^{6}} \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}\right), a, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -500.0)
     (/
      (/ (fma (* b b) (- b) (pow t_0 1.5)) (fma b b (+ (* t_1 t_1) (* b t_1))))
      (* 3.0 a))
     (/
      (fma
       (fma
        (*
         (/ (fma -1.0546875 (* a c) (* -0.5625 (* b b))) (pow b 6.0))
         (pow c 3.0))
        a
        (/ (* (* c c) -0.375) (* b b)))
       a
       (* -0.5 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -500.0) {
		tmp = (fma((b * b), -b, pow(t_0, 1.5)) / fma(b, b, ((t_1 * t_1) + (b * t_1)))) / (3.0 * a);
	} else {
		tmp = fma(fma(((fma(-1.0546875, (a * c), (-0.5625 * (b * b))) / pow(b, 6.0)) * pow(c, 3.0)), a, (((c * c) * -0.375) / (b * b))), a, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -500.0)
		tmp = Float64(Float64(fma(Float64(b * b), Float64(-b), (t_0 ^ 1.5)) / fma(b, b, Float64(Float64(t_1 * t_1) + Float64(b * t_1)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(fma(Float64(Float64(fma(-1.0546875, Float64(a * c), Float64(-0.5625 * Float64(b * b))) / (b ^ 6.0)) * (c ^ 3.0)), a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b))), a, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -500.0], N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-1.0546875 * N[(a * c), $MachinePrecision] + N[(-0.5625 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] * a + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -500

   1. Initial program 85.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 85.2%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. sqr-neg-rev N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sqrt-pow2 N/A

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

      17. metadata-eval N/A

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

      18. lower-pow.f64 N/A

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

   6. Applied rewrites 87.1%

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

   if -500 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 29.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 96.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{-1}{2} \cdot c + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right) + \frac{-1}{2} \cdot c}{b} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right) \cdot a + \frac{-1}{2} \cdot c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{4}}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

   8. Applied rewrites 96.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{{c}^{3}}{{b}^{4}}, -0.5625, \frac{-1.0546875 \cdot \left({c}^{4} \cdot a\right)}{{b}^{6}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}\right), a, -0.5 \cdot c\right)}{b} \]

   9. Taylor expanded in c around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left({c}^{3} \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot c}{{b}^{6}} - \frac{9}{16} \cdot \frac{1}{{b}^{4}}\right), a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-135}{128} \cdot \frac{a \cdot c}{{b}^{6}} - \frac{9}{16} \cdot \frac{1}{{b}^{4}}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-135}{128} \cdot \frac{a \cdot c}{{b}^{6}} - \frac{9}{16} \cdot \frac{1}{{b}^{4}}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-135}{128} \cdot \frac{a \cdot c}{{b}^{6}} + \left(\mathsf{neg}\left(\frac{9}{16}\right)\right) \cdot \frac{1}{{b}^{4}}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       4. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{a \cdot c}{{b}^{6}} \cdot \frac{-135}{128} + \left(\mathsf{neg}\left(\frac{9}{16}\right)\right) \cdot \frac{1}{{b}^{4}}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a \cdot c}{{b}^{6}}, \frac{-135}{128}, \left(\mathsf{neg}\left(\frac{9}{16}\right)\right) \cdot \frac{1}{{b}^{4}}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       6. associate-/l* N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{6}}, \frac{-135}{128}, \left(\mathsf{neg}\left(\frac{9}{16}\right)\right) \cdot \frac{1}{{b}^{4}}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{6}}, \frac{-135}{128}, \left(\mathsf{neg}\left(\frac{9}{16}\right)\right) \cdot \frac{1}{{b}^{4}}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{6}}, \frac{-135}{128}, \left(\mathsf{neg}\left(\frac{9}{16}\right)\right) \cdot \frac{1}{{b}^{4}}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       9. lift-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{6}}, \frac{-135}{128}, \left(\mathsf{neg}\left(\frac{9}{16}\right)\right) \cdot \frac{1}{{b}^{4}}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       10. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{6}}, \frac{-135}{128}, \frac{-9}{16} \cdot \frac{1}{{b}^{4}}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{6}}, \frac{-135}{128}, \frac{-9}{16} \cdot \frac{1}{{b}^{4}}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       12. pow-flip N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{6}}, \frac{-135}{128}, \frac{-9}{16} \cdot {b}^{\left(\mathsf{neg}\left(4\right)\right)}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       13. lower-pow.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{6}}, \frac{-135}{128}, \frac{-9}{16} \cdot {b}^{\left(\mathsf{neg}\left(4\right)\right)}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       14. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{6}}, \frac{-135}{128}, \frac{-9}{16} \cdot {b}^{-4}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       15. lift-pow.f64 96.6

           \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{6}}, -1.0546875, -0.5625 \cdot {b}^{-4}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}\right), a, -0.5 \cdot c\right)}{b} \]

   11. Applied rewrites 96.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{6}}, -1.0546875, -0.5625 \cdot {b}^{-4}\right) \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}\right), a, -0.5 \cdot c\right)}{b} \]

   12. Taylor expanded in b around 0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{-135}{128} \cdot \left(a \cdot c\right) + \frac{-9}{16} \cdot {b}^{2}}{{b}^{6}} \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{-135}{128} \cdot \left(a \cdot c\right) + \frac{-9}{16} \cdot {b}^{2}}{{b}^{6}} \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-135}{128}, a \cdot c, \frac{-9}{16} \cdot {b}^{2}\right)}{{b}^{6}} \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-135}{128}, a \cdot c, \frac{-9}{16} \cdot {b}^{2}\right)}{{b}^{6}} \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-135}{128}, a \cdot c, \frac{-9}{16} \cdot {b}^{2}\right)}{{b}^{6}} \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       5. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-135}{128}, a \cdot c, \frac{-9}{16} \cdot \left(b \cdot b\right)\right)}{{b}^{6}} \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-135}{128}, a \cdot c, \frac{-9}{16} \cdot \left(b \cdot b\right)\right)}{{b}^{6}} \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

       7. lift-pow.f64 96.6

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875, a \cdot c, -0.5625 \cdot \left(b \cdot b\right)\right)}{{b}^{6}} \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}\right), a, -0.5 \cdot c\right)}{b} \]

   14. Applied rewrites 96.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875, a \cdot c, -0.5625 \cdot \left(b \cdot b\right)\right)}{{b}^{6}} \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}\right), a, -0.5 \cdot c\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -500:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {\left(\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875, a \cdot c, -0.5625 \cdot \left(b \cdot b\right)\right)}{{b}^{6}} \cdot {c}^{3}, a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}\right), a, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -500:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {t\_0}^{1.5}\right)}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 + b \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, -0.5625, \left(c \cdot c\right) \cdot -0.375\right)}{b \cdot b}, a, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -500.0)
     (/
      (/ (fma (* b b) (- b) (pow t_0 1.5)) (fma b b (+ (* t_1 t_1) (* b t_1))))
      (* 3.0 a))
     (/
      (fma
       (/
        (fma (/ (* (pow c 3.0) a) (* b b)) -0.5625 (* (* c c) -0.375))
        (* b b))
       a
       (* -0.5 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -500.0) {
		tmp = (fma((b * b), -b, pow(t_0, 1.5)) / fma(b, b, ((t_1 * t_1) + (b * t_1)))) / (3.0 * a);
	} else {
		tmp = fma((fma(((pow(c, 3.0) * a) / (b * b)), -0.5625, ((c * c) * -0.375)) / (b * b)), a, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -500.0)
		tmp = Float64(Float64(fma(Float64(b * b), Float64(-b), (t_0 ^ 1.5)) / fma(b, b, Float64(Float64(t_1 * t_1) + Float64(b * t_1)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(fma(Float64(Float64((c ^ 3.0) * a) / Float64(b * b)), -0.5625, Float64(Float64(c * c) * -0.375)) / Float64(b * b)), a, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -500.0], N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * a + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -500

   1. Initial program 85.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 85.2%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. sqr-neg-rev N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sqrt-pow2 N/A

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

      17. metadata-eval N/A

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

      18. lower-pow.f64 N/A

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

   6. Applied rewrites 87.1%

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

   if -500 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 29.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 96.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{-1}{2} \cdot c + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right) + \frac{-1}{2} \cdot c}{b} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right) \cdot a + \frac{-1}{2} \cdot c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{4}}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

   8. Applied rewrites 96.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{{c}^{3}}{{b}^{4}}, -0.5625, \frac{-1.0546875 \cdot \left({c}^{4} \cdot a\right)}{{b}^{6}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}\right), a, -0.5 \cdot c\right)}{b} \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{a \cdot {c}^{3}}{{b}^{2}} \cdot \frac{-9}{16} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{a \cdot {c}^{3}}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{a \cdot {c}^{3}}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       6. lift-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       8. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       10. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, \frac{-9}{16}, {c}^{2} \cdot \frac{-3}{8}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       11. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, \frac{-9}{16}, \left(c \cdot c\right) \cdot \frac{-3}{8}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, \frac{-9}{16}, \left(c \cdot c\right) \cdot \frac{-3}{8}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, \frac{-9}{16}, \left(c \cdot c\right) \cdot \frac{-3}{8}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       14. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, \frac{-9}{16}, \left(c \cdot c\right) \cdot \frac{-3}{8}\right)}{b \cdot b}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       15. lift-*.f64 95.4

           \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, -0.5625, \left(c \cdot c\right) \cdot -0.375\right)}{b \cdot b}, a, -0.5 \cdot c\right)}{b} \]

   11. Applied rewrites 95.4%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, -0.5625, \left(c \cdot c\right) \cdot -0.375\right)}{b \cdot b}, a, -0.5 \cdot c\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -500:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {\left(\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, -0.5625, \left(c \cdot c\right) \cdot -0.375\right)}{b \cdot b}, a, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -500:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0 \cdot t\_0}{\left(-b\right) - t\_0}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, -0.5625, \left(c \cdot c\right) \cdot -0.375\right)}{b \cdot b}, a, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -3.0 a) c (* b b)))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -500.0)
     (/ (/ (- (* b b) (* t_0 t_0)) (- (- b) t_0)) (* 3.0 a))
     (/
      (fma
       (/
        (fma (/ (* (pow c 3.0) a) (* b b)) -0.5625 (* (* c c) -0.375))
        (* b b))
       a
       (* -0.5 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-3.0 * a), c, (b * b)));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -500.0) {
		tmp = (((b * b) - (t_0 * t_0)) / (-b - t_0)) / (3.0 * a);
	} else {
		tmp = fma((fma(((pow(c, 3.0) * a) / (b * b)), -0.5625, ((c * c) * -0.375)) / (b * b)), a, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-3.0 * a), c, Float64(b * b)))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -500.0)
		tmp = Float64(Float64(Float64(Float64(b * b) - Float64(t_0 * t_0)) / Float64(Float64(-b) - t_0)) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(fma(Float64(Float64((c ^ 3.0) * a) / Float64(b * b)), -0.5625, Float64(Float64(c * c) * -0.375)) / Float64(b * b)), a, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -500.0], N[(N[(N[(N[(b * b), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * a + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -500

   1. Initial program 85.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 86.2%

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

   if -500 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 29.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 96.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{-1}{2} \cdot c + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right)}{b} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right) + \frac{-1}{2} \cdot c}{b} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{4}}\right)\right) \cdot a + \frac{-1}{2} \cdot c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{2}} + a \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{4}}\right), a, \frac{-1}{2} \cdot c\right)}{b} \]

   8. Applied rewrites 96.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{{c}^{3}}{{b}^{4}}, -0.5625, \frac{-1.0546875 \cdot \left({c}^{4} \cdot a\right)}{{b}^{6}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}\right), a, -0.5 \cdot c\right)}{b} \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{a \cdot {c}^{3}}{{b}^{2}} \cdot \frac{-9}{16} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{a \cdot {c}^{3}}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{a \cdot {c}^{3}}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       6. lift-pow.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       8. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       10. *-commutative N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, \frac{-9}{16}, {c}^{2} \cdot \frac{-3}{8}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       11. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, \frac{-9}{16}, \left(c \cdot c\right) \cdot \frac{-3}{8}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, \frac{-9}{16}, \left(c \cdot c\right) \cdot \frac{-3}{8}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       13. lift-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, \frac{-9}{16}, \left(c \cdot c\right) \cdot \frac{-3}{8}\right)}{{b}^{2}}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       14. pow2 N/A

           \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, \frac{-9}{16}, \left(c \cdot c\right) \cdot \frac{-3}{8}\right)}{b \cdot b}, a, \frac{-1}{2} \cdot c\right)}{b} \]

       15. lift-*.f64 95.4

           \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, -0.5625, \left(c \cdot c\right) \cdot -0.375\right)}{b \cdot b}, a, -0.5 \cdot c\right)}{b} \]

   11. Applied rewrites 95.4%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{c}^{3} \cdot a}{b \cdot b}, -0.5625, \left(c \cdot c\right) \cdot -0.375\right)}{b \cdot b}, a, -0.5 \cdot c\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -500:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0 \cdot t\_0}{\left(-b\right) - t\_0}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, -0.5625, -0.375 \cdot a\right)}{b \cdot b} \cdot c - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -3.0 a) c (* b b)))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -500.0)
     (/ (/ (- (* b b) (* t_0 t_0)) (- (- b) t_0)) (* 3.0 a))
     (/
      (*
       (-
        (* (/ (fma (* (* a a) (/ c (* b b))) -0.5625 (* -0.375 a)) (* b b)) c)
        0.5)
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-3.0 * a), c, (b * b)));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -500.0) {
		tmp = (((b * b) - (t_0 * t_0)) / (-b - t_0)) / (3.0 * a);
	} else {
		tmp = ((((fma(((a * a) * (c / (b * b))), -0.5625, (-0.375 * a)) / (b * b)) * c) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-3.0 * a), c, Float64(b * b)))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -500.0)
		tmp = Float64(Float64(Float64(Float64(b * b) - Float64(t_0 * t_0)) / Float64(Float64(-b) - t_0)) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(Float64(Float64(a * a) * Float64(c / Float64(b * b))), -0.5625, Float64(-0.375 * a)) / Float64(b * b)) * c) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -500.0], N[(N[(N[(N[(b * b), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.375 * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -500

   1. Initial program 85.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 86.2%

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

   if -500 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 29.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 96.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) - \frac{1}{2}\right)}{b} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) - \frac{1}{2}\right) \cdot c}{b} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) - \frac{1}{2}\right) \cdot c}{b} \]

   8. Applied rewrites 95.3%

      \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{a}{b \cdot b}, -0.375, \frac{-0.5625 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{4}}\right) \cdot c - 0.5\right) \cdot c}{b} \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{\left(\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\left(\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       2. *-commutative N/A

          \[\leadsto \frac{\left(\frac{\frac{{a}^{2} \cdot c}{{b}^{2}} \cdot \frac{-9}{16} + \frac{-3}{8} \cdot a}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{{a}^{2} \cdot c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       4. associate-/l* N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left({a}^{2} \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left({a}^{2} \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       6. pow2 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       9. pow2 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       12. pow2 N/A

           \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{b \cdot b} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       13. lift-*.f64 95.3

           \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, -0.5625, -0.375 \cdot a\right)}{b \cdot b} \cdot c - 0.5\right) \cdot c}{b} \]

   11. Applied rewrites 95.3%

       \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, -0.5625, -0.375 \cdot a\right)}{b \cdot b} \cdot c - 0.5\right) \cdot c}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -500:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, -0.5625, -0.375 \cdot a\right)}{b \cdot b} \cdot c - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -500.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/
    (*
     (-
      (* (/ (fma (* (* a a) (/ c (* b b))) -0.5625 (* -0.375 a)) (* b b)) c)
      0.5)
     c)
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -500.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = ((((fma(((a * a) * (c / (b * b))), -0.5625, (-0.375 * a)) / (b * b)) * c) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -500.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(Float64(Float64(a * a) * Float64(c / Float64(b * b))), -0.5625, Float64(-0.375 * a)) / Float64(b * b)) * c) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -500.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.375 * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -500

   1. Initial program 85.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 86.0

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

   4. Applied rewrites 86.0%

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

   if -500 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 29.0%

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

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   5. Applied rewrites 96.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) - \frac{1}{2}\right)}{b} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) - \frac{1}{2}\right) \cdot c}{b} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + \frac{-3}{8} \cdot \frac{a}{{b}^{2}}\right) - \frac{1}{2}\right) \cdot c}{b} \]

   8. Applied rewrites 95.3%

      \[\leadsto \frac{\left(\mathsf{fma}\left(\frac{a}{b \cdot b}, -0.375, \frac{-0.5625 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{4}}\right) \cdot c - 0.5\right) \cdot c}{b} \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{\left(\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\left(\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       2. *-commutative N/A

          \[\leadsto \frac{\left(\frac{\frac{{a}^{2} \cdot c}{{b}^{2}} \cdot \frac{-9}{16} + \frac{-3}{8} \cdot a}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\frac{{a}^{2} \cdot c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       4. associate-/l* N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left({a}^{2} \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left({a}^{2} \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       6. pow2 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{2}}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       9. pow2 N/A

          \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{{b}^{2}} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       12. pow2 N/A

           \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, \frac{-9}{16}, \frac{-3}{8} \cdot a\right)}{b \cdot b} \cdot c - \frac{1}{2}\right) \cdot c}{b} \]

       13. lift-*.f64 95.3

           \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, -0.5625, -0.375 \cdot a\right)}{b \cdot b} \cdot c - 0.5\right) \cdot c}{b} \]

   11. Applied rewrites 95.3%

       \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{b \cdot b}, -0.5625, -0.375 \cdot a\right)}{b \cdot b} \cdot c - 0.5\right) \cdot c}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -40:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -40.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (fma (/ (* (* c c) a) (* b b)) -0.375 (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -40.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma((((c * c) * a) / (b * b)), -0.375, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -40.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(c * c) * a) / Float64(b * b)), -0.375, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -40.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -40

   1. Initial program 83.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 83.6

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

   4. Applied rewrites 83.6%

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

   if -40 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 28.0%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{\color{blue}{b}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}{b} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b}^{2}} \cdot \frac{-3}{8} + \frac{-1}{2} \cdot c}{b} \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

      12. lower-*.f64 93.1

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b} \]

   5. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -40:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(a \cdot \frac{c}{b \cdot b}\right) \cdot -0.375 - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -40.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (* (- (* (* a (/ c (* b b))) -0.375) 0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -40.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = ((((a * (c / (b * b))) * -0.375) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -40.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * Float64(c / Float64(b * b))) * -0.375) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -40.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -40

   1. Initial program 83.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 83.6

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

   4. Applied rewrites 83.6%

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

   if -40 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 28.0%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{\color{blue}{b}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}{b} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b}^{2}} \cdot \frac{-3}{8} + \frac{-1}{2} \cdot c}{b} \]

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

      12. lower-*.f64 93.1

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b} \]

   5. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}} \]

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\frac{a \cdot c}{{b}^{2}} \cdot \frac{-3}{8} - \frac{1}{2}\right) \cdot c}{b} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{a \cdot c}{{b}^{2}} \cdot \frac{-3}{8} - \frac{1}{2}\right) \cdot c}{b} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\left(\left(a \cdot \frac{c}{{b}^{2}}\right) \cdot \frac{-3}{8} - \frac{1}{2}\right) \cdot c}{b} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(a \cdot \frac{c}{{b}^{2}}\right) \cdot \frac{-3}{8} - \frac{1}{2}\right) \cdot c}{b} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(a \cdot \frac{c}{{b}^{2}}\right) \cdot \frac{-3}{8} - \frac{1}{2}\right) \cdot c}{b} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\left(a \cdot \frac{c}{b \cdot b}\right) \cdot \frac{-3}{8} - \frac{1}{2}\right) \cdot c}{b} \]

      10. lift-*.f64 93.0

          \[\leadsto \frac{\left(\left(a \cdot \frac{c}{b \cdot b}\right) \cdot -0.375 - 0.5\right) \cdot c}{b} \]

   8. Applied rewrites 93.0%

      \[\leadsto \frac{\left(\left(a \cdot \frac{c}{b \cdot b}\right) \cdot -0.375 - 0.5\right) \cdot c}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 90.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(a \cdot \frac{c}{b \cdot b}\right) \cdot -0.375 - 0.5\right) \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* (- (* (* a (/ c (* b b))) -0.375) 0.5) c) b))

C

double code(double a, double b, double c) {
	return ((((a * (c / (b * b))) * -0.375) - 0.5) * c) / b;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((((a * (c / (b * b))) * (-0.375d0)) - 0.5d0) * c) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((((a * (c / (b * b))) * -0.375) - 0.5) * c) / b;
}

Python

def code(a, b, c):
	return ((((a * (c / (b * b))) * -0.375) - 0.5) * c) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(Float64(a * Float64(c / Float64(b * b))) * -0.375) - 0.5) * c) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((((a * (c / (b * b))) * -0.375) - 0.5) * c) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{\color{blue}{b}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}{b} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b}^{2}} \cdot \frac{-3}{8} + \frac{-1}{2} \cdot c}{b} \]

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

   12. lower-*.f64 89.0

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b} \]

5. Applied rewrites 89.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}} \]

6. Taylor expanded in c around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

   3. lower--.f64 N/A

      \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\frac{a \cdot c}{{b}^{2}} \cdot \frac{-3}{8} - \frac{1}{2}\right) \cdot c}{b} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\left(\frac{a \cdot c}{{b}^{2}} \cdot \frac{-3}{8} - \frac{1}{2}\right) \cdot c}{b} \]

   6. associate-/l* N/A

      \[\leadsto \frac{\left(\left(a \cdot \frac{c}{{b}^{2}}\right) \cdot \frac{-3}{8} - \frac{1}{2}\right) \cdot c}{b} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\left(\left(a \cdot \frac{c}{{b}^{2}}\right) \cdot \frac{-3}{8} - \frac{1}{2}\right) \cdot c}{b} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\left(\left(a \cdot \frac{c}{{b}^{2}}\right) \cdot \frac{-3}{8} - \frac{1}{2}\right) \cdot c}{b} \]

   9. pow2 N/A

      \[\leadsto \frac{\left(\left(a \cdot \frac{c}{b \cdot b}\right) \cdot \frac{-3}{8} - \frac{1}{2}\right) \cdot c}{b} \]

   10. lift-*.f64 88.9

       \[\leadsto \frac{\left(\left(a \cdot \frac{c}{b \cdot b}\right) \cdot -0.375 - 0.5\right) \cdot c}{b} \]

8. Applied rewrites 88.9%

   \[\leadsto \frac{\left(\left(a \cdot \frac{c}{b \cdot b}\right) \cdot -0.375 - 0.5\right) \cdot c}{b} \]
9. Add Preprocessing

Alternative 9: 81.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot -0.5 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ c b) -0.5))

C

double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

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) * (-0.5d0)
end function

Java

public static double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

Python

def code(a, b, c):
	return (c / b) * -0.5

Julia

function code(a, b, c)
	return Float64(Float64(c / b) * -0.5)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c / b) * -0.5;
end

Wolfram

code[a_, b_, c_] := N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 33.9%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 79.3

      \[\leadsto \frac{c}{b} \cdot -0.5 \]

5. Applied rewrites 79.3%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025080 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))