Cubic critical, narrow range

Percentage Accurate: 55.5% → 92.1%

Time: 10.3s

Alternatives: 15

Speedup: 3.3×

Specification

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

Math

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

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: 55.5% 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: 92.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}\\ t_1 := \left(b \cdot b\right) \cdot b\\ t_2 := t\_1 \cdot t\_1\\ t_3 := \frac{-1}{\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b}\\ t_4 := \frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\\ t_5 := {\left(a \cdot c\right)}^{4}\\ t_6 := \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.58:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, t\_3, \left(3 \cdot \left(a \cdot c\right) - b \cdot b\right) \cdot t\_3\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot -3\right) \cdot c, b, \mathsf{fma}\left(t\_4, -1.6875, \mathsf{fma}\left(-1.5 \cdot a, c, \mathsf{fma}\left(t\_0, -1.125, \mathsf{fma}\left(\frac{t\_5 \cdot 6.328125}{t\_2}, -0.5, \mathsf{fma}\left(3.375, t\_4, \mathsf{fma}\left(4.5, t\_0, \frac{t\_5 \cdot 5.0625}{t\_2}\right)\right)\right)\right)\right)\right) \cdot b\right)}{\mathsf{fma}\left(b, b, t\_6 + b \cdot \sqrt{t\_6}\right) \cdot \left(a \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* (* c c) (* a a)) (* b b)))
        (t_1 (* (* b b) b))
        (t_2 (* t_1 t_1))
        (t_3 (/ -1.0 (+ (sqrt (fma (* a c) -3.0 (* b b))) b)))
        (t_4 (/ (* (* (* c c) c) (* (* a a) a)) (* (* b b) (* b b))))
        (t_5 (pow (* a c) 4.0))
        (t_6 (fma -3.0 (* c a) (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.58)
     (/ (fma (* b b) t_3 (* (- (* 3.0 (* a c)) (* b b)) t_3)) (* 3.0 a))
     (/
      (fma
       (* (* a -3.0) c)
       b
       (*
        (fma
         t_4
         -1.6875
         (fma
          (* -1.5 a)
          c
          (fma
           t_0
           -1.125
           (fma
            (/ (* t_5 6.328125) t_2)
            -0.5
            (fma 3.375 t_4 (fma 4.5 t_0 (/ (* t_5 5.0625) t_2)))))))
        b))
      (* (fma b b (+ t_6 (* b (sqrt t_6)))) (* a 3.0))))))

C

double code(double a, double b, double c) {
	double t_0 = ((c * c) * (a * a)) / (b * b);
	double t_1 = (b * b) * b;
	double t_2 = t_1 * t_1;
	double t_3 = -1.0 / (sqrt(fma((a * c), -3.0, (b * b))) + b);
	double t_4 = (((c * c) * c) * ((a * a) * a)) / ((b * b) * (b * b));
	double t_5 = pow((a * c), 4.0);
	double t_6 = fma(-3.0, (c * a), (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.58) {
		tmp = fma((b * b), t_3, (((3.0 * (a * c)) - (b * b)) * t_3)) / (3.0 * a);
	} else {
		tmp = fma(((a * -3.0) * c), b, (fma(t_4, -1.6875, fma((-1.5 * a), c, fma(t_0, -1.125, fma(((t_5 * 6.328125) / t_2), -0.5, fma(3.375, t_4, fma(4.5, t_0, ((t_5 * 5.0625) / t_2))))))) * b)) / (fma(b, b, (t_6 + (b * sqrt(t_6)))) * (a * 3.0));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(c * c) * Float64(a * a)) / Float64(b * b))
	t_1 = Float64(Float64(b * b) * b)
	t_2 = Float64(t_1 * t_1)
	t_3 = Float64(-1.0 / Float64(sqrt(fma(Float64(a * c), -3.0, Float64(b * b))) + b))
	t_4 = Float64(Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) * a)) / Float64(Float64(b * b) * Float64(b * b)))
	t_5 = Float64(a * c) ^ 4.0
	t_6 = fma(-3.0, Float64(c * a), 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)) <= -0.58)
		tmp = Float64(fma(Float64(b * b), t_3, Float64(Float64(Float64(3.0 * Float64(a * c)) - Float64(b * b)) * t_3)) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(a * -3.0) * c), b, Float64(fma(t_4, -1.6875, fma(Float64(-1.5 * a), c, fma(t_0, -1.125, fma(Float64(Float64(t_5 * 6.328125) / t_2), -0.5, fma(3.375, t_4, fma(4.5, t_0, Float64(Float64(t_5 * 5.0625) / t_2))))))) * b)) / Float64(fma(b, b, Float64(t_6 + Float64(b * sqrt(t_6)))) * Float64(a * 3.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$6 = N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $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], -0.58], N[(N[(N[(b * b), $MachinePrecision] * t$95$3 + N[(N[(N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision] * b + N[(N[(t$95$4 * -1.6875 + N[(N[(-1.5 * a), $MachinePrecision] * c + N[(t$95$0 * -1.125 + N[(N[(N[(t$95$5 * 6.328125), $MachinePrecision] / t$95$2), $MachinePrecision] * -0.5 + N[(3.375 * t$95$4 + N[(4.5 * t$95$0 + N[(N[(t$95$5 * 5.0625), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b + N[(t$95$6 + N[(b * N[Sqrt[t$95$6], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.57999999999999996

   1. Initial program 55.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 54.7%

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 57.3%

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

   if -0.57999999999999996 < (/.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 55.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 57.3%

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

   4. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{b \cdot \left(-3 \cdot \left(a \cdot c\right) + \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-3}{2} \cdot \left(a \cdot c\right) + \left(\frac{-9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(\frac{-1}{2} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(\frac{27}{8} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{9}{2} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{81}{16} \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   6. Applied rewrites 91.4%

      \[\leadsto \frac{\color{blue}{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(4.5, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, 5.0625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   8. Applied rewrites 91.5%

      \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot -3\right) \cdot c, \color{blue}{b}, \mathsf{fma}\left(\frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -1.6875, \mathsf{fma}\left(-1.5 \cdot a, c, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}, -1.125, \mathsf{fma}\left(\frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}, -0.5, \mathsf{fma}\left(3.375, \frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \mathsf{fma}\left(4.5, \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}, \frac{{\left(a \cdot c\right)}^{4} \cdot 5.0625}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}\right)\right)\right)\right)\right)\right) \cdot b\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 92.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}\\ t_1 := {\left(a \cdot c\right)}^{4}\\ t_2 := \mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)\\ t_3 := \sqrt{t\_2}\\ t_4 := \left(b \cdot b\right) \cdot b\\ t_5 := t\_4 \cdot t\_4\\ t_6 := \frac{-1}{t\_3 + b}\\ t_7 := \frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.58:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, t\_6, \left(3 \cdot \left(a \cdot c\right) - b \cdot b\right) \cdot t\_6\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3 \cdot c, a, \mathsf{fma}\left(t\_7, -1.6875, \mathsf{fma}\left(-1.5 \cdot a, c, \mathsf{fma}\left(t\_0, -1.125, \mathsf{fma}\left(\frac{t\_1 \cdot 6.328125}{t\_5}, -0.5, \mathsf{fma}\left(3.375, t\_7, \mathsf{fma}\left(4.5, t\_0, \frac{t\_1 \cdot 5.0625}{t\_5}\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(t\_3, b, t\_2\right)\right)} \cdot \frac{0.3333333333333333}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* (* c c) (* a a)) (* b b)))
        (t_1 (pow (* a c) 4.0))
        (t_2 (fma (* a c) -3.0 (* b b)))
        (t_3 (sqrt t_2))
        (t_4 (* (* b b) b))
        (t_5 (* t_4 t_4))
        (t_6 (/ -1.0 (+ t_3 b)))
        (t_7 (/ (* (* (* c c) c) (* (* a a) a)) (* (* b b) (* b b)))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.58)
     (/ (fma (* b b) t_6 (* (- (* 3.0 (* a c)) (* b b)) t_6)) (* 3.0 a))
     (*
      (/
       (*
        (fma
         (* -3.0 c)
         a
         (fma
          t_7
          -1.6875
          (fma
           (* -1.5 a)
           c
           (fma
            t_0
            -1.125
            (fma
             (/ (* t_1 6.328125) t_5)
             -0.5
             (fma 3.375 t_7 (fma 4.5 t_0 (/ (* t_1 5.0625) t_5))))))))
        b)
       (fma b b (fma t_3 b t_2)))
      (/ 0.3333333333333333 a)))))

C

double code(double a, double b, double c) {
	double t_0 = ((c * c) * (a * a)) / (b * b);
	double t_1 = pow((a * c), 4.0);
	double t_2 = fma((a * c), -3.0, (b * b));
	double t_3 = sqrt(t_2);
	double t_4 = (b * b) * b;
	double t_5 = t_4 * t_4;
	double t_6 = -1.0 / (t_3 + b);
	double t_7 = (((c * c) * c) * ((a * a) * a)) / ((b * b) * (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.58) {
		tmp = fma((b * b), t_6, (((3.0 * (a * c)) - (b * b)) * t_6)) / (3.0 * a);
	} else {
		tmp = ((fma((-3.0 * c), a, fma(t_7, -1.6875, fma((-1.5 * a), c, fma(t_0, -1.125, fma(((t_1 * 6.328125) / t_5), -0.5, fma(3.375, t_7, fma(4.5, t_0, ((t_1 * 5.0625) / t_5)))))))) * b) / fma(b, b, fma(t_3, b, t_2))) * (0.3333333333333333 / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(c * c) * Float64(a * a)) / Float64(b * b))
	t_1 = Float64(a * c) ^ 4.0
	t_2 = fma(Float64(a * c), -3.0, Float64(b * b))
	t_3 = sqrt(t_2)
	t_4 = Float64(Float64(b * b) * b)
	t_5 = Float64(t_4 * t_4)
	t_6 = Float64(-1.0 / Float64(t_3 + b))
	t_7 = Float64(Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) * a)) / Float64(Float64(b * b) * 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)) <= -0.58)
		tmp = Float64(fma(Float64(b * b), t_6, Float64(Float64(Float64(3.0 * Float64(a * c)) - Float64(b * b)) * t_6)) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(fma(Float64(-3.0 * c), a, fma(t_7, -1.6875, fma(Float64(-1.5 * a), c, fma(t_0, -1.125, fma(Float64(Float64(t_1 * 6.328125) / t_5), -0.5, fma(3.375, t_7, fma(4.5, t_0, Float64(Float64(t_1 * 5.0625) / t_5)))))))) * b) / fma(b, b, fma(t_3, b, t_2))) * Float64(0.3333333333333333 / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * c), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(-1.0 / N[(t$95$3 + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * 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], -0.58], N[(N[(N[(b * b), $MachinePrecision] * t$95$6 + N[(N[(N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-3.0 * c), $MachinePrecision] * a + N[(t$95$7 * -1.6875 + N[(N[(-1.5 * a), $MachinePrecision] * c + N[(t$95$0 * -1.125 + N[(N[(N[(t$95$1 * 6.328125), $MachinePrecision] / t$95$5), $MachinePrecision] * -0.5 + N[(3.375 * t$95$7 + N[(4.5 * t$95$0 + N[(N[(t$95$1 * 5.0625), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(b * b + N[(t$95$3 * b + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.57999999999999996

   1. Initial program 55.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 54.7%

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 57.3%

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

   if -0.57999999999999996 < (/.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 55.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 57.3%

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

   4. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{b \cdot \left(-3 \cdot \left(a \cdot c\right) + \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-3}{2} \cdot \left(a \cdot c\right) + \left(\frac{-9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(\frac{-1}{2} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(\frac{27}{8} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{9}{2} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{81}{16} \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   6. Applied rewrites 91.4%

      \[\leadsto \frac{\color{blue}{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(4.5, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, 5.0625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   8. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-3 \cdot c, a, \mathsf{fma}\left(\frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -1.6875, \mathsf{fma}\left(-1.5 \cdot a, c, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}, -1.125, \mathsf{fma}\left(\frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}, -0.5, \mathsf{fma}\left(3.375, \frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \mathsf{fma}\left(4.5, \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}, \frac{{\left(a \cdot c\right)}^{4} \cdot 5.0625}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)}, b, \mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)\right)\right)} \cdot \frac{0.3333333333333333}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\\ t_1 := {\left(a \cdot c\right)}^{4}\\ t_2 := \left(b \cdot b\right) \cdot b\\ t_3 := \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}\\ t_4 := \mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)\\ t_5 := \sqrt{t\_4}\\ t_6 := \frac{-1}{t\_5 + b}\\ t_7 := t\_2 \cdot t\_2\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.58:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, t\_6, \left(3 \cdot \left(a \cdot c\right) - b \cdot b\right) \cdot t\_6\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-3 \cdot c, a, \mathsf{fma}\left(t\_0, -1.6875, \mathsf{fma}\left(-1.5 \cdot a, c, \mathsf{fma}\left(t\_3, -1.125, \mathsf{fma}\left(\frac{t\_1 \cdot 6.328125}{t\_7}, -0.5, \mathsf{fma}\left(3.375, t\_0, \mathsf{fma}\left(4.5, t\_3, \frac{t\_1 \cdot 5.0625}{t\_7}\right)\right)\right)\right)\right)\right)\right) \cdot b}{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(t\_5, b, t\_4\right)\right) \cdot a\right) \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* (* (* c c) c) (* (* a a) a)) (* (* b b) (* b b))))
        (t_1 (pow (* a c) 4.0))
        (t_2 (* (* b b) b))
        (t_3 (/ (* (* c c) (* a a)) (* b b)))
        (t_4 (fma (* a c) -3.0 (* b b)))
        (t_5 (sqrt t_4))
        (t_6 (/ -1.0 (+ t_5 b)))
        (t_7 (* t_2 t_2)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.58)
     (/ (fma (* b b) t_6 (* (- (* 3.0 (* a c)) (* b b)) t_6)) (* 3.0 a))
     (/
      (*
       (fma
        (* -3.0 c)
        a
        (fma
         t_0
         -1.6875
         (fma
          (* -1.5 a)
          c
          (fma
           t_3
           -1.125
           (fma
            (/ (* t_1 6.328125) t_7)
            -0.5
            (fma 3.375 t_0 (fma 4.5 t_3 (/ (* t_1 5.0625) t_7))))))))
       b)
      (* (* (fma b b (fma t_5 b t_4)) a) 3.0)))))

C

double code(double a, double b, double c) {
	double t_0 = (((c * c) * c) * ((a * a) * a)) / ((b * b) * (b * b));
	double t_1 = pow((a * c), 4.0);
	double t_2 = (b * b) * b;
	double t_3 = ((c * c) * (a * a)) / (b * b);
	double t_4 = fma((a * c), -3.0, (b * b));
	double t_5 = sqrt(t_4);
	double t_6 = -1.0 / (t_5 + b);
	double t_7 = t_2 * t_2;
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.58) {
		tmp = fma((b * b), t_6, (((3.0 * (a * c)) - (b * b)) * t_6)) / (3.0 * a);
	} else {
		tmp = (fma((-3.0 * c), a, fma(t_0, -1.6875, fma((-1.5 * a), c, fma(t_3, -1.125, fma(((t_1 * 6.328125) / t_7), -0.5, fma(3.375, t_0, fma(4.5, t_3, ((t_1 * 5.0625) / t_7)))))))) * b) / ((fma(b, b, fma(t_5, b, t_4)) * a) * 3.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) * a)) / Float64(Float64(b * b) * Float64(b * b)))
	t_1 = Float64(a * c) ^ 4.0
	t_2 = Float64(Float64(b * b) * b)
	t_3 = Float64(Float64(Float64(c * c) * Float64(a * a)) / Float64(b * b))
	t_4 = fma(Float64(a * c), -3.0, Float64(b * b))
	t_5 = sqrt(t_4)
	t_6 = Float64(-1.0 / Float64(t_5 + b))
	t_7 = Float64(t_2 * t_2)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.58)
		tmp = Float64(fma(Float64(b * b), t_6, Float64(Float64(Float64(3.0 * Float64(a * c)) - Float64(b * b)) * t_6)) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(-3.0 * c), a, fma(t_0, -1.6875, fma(Float64(-1.5 * a), c, fma(t_3, -1.125, fma(Float64(Float64(t_1 * 6.328125) / t_7), -0.5, fma(3.375, t_0, fma(4.5, t_3, Float64(Float64(t_1 * 5.0625) / t_7)))))))) * b) / Float64(Float64(fma(b, b, fma(t_5, b, t_4)) * a) * 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(a * c), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[(-1.0 / N[(t$95$5 + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$2 * t$95$2), $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], -0.58], N[(N[(N[(b * b), $MachinePrecision] * t$95$6 + N[(N[(N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-3.0 * c), $MachinePrecision] * a + N[(t$95$0 * -1.6875 + N[(N[(-1.5 * a), $MachinePrecision] * c + N[(t$95$3 * -1.125 + N[(N[(N[(t$95$1 * 6.328125), $MachinePrecision] / t$95$7), $MachinePrecision] * -0.5 + N[(3.375 * t$95$0 + N[(4.5 * t$95$3 + N[(N[(t$95$1 * 5.0625), $MachinePrecision] / t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(N[(N[(b * b + N[(t$95$5 * b + t$95$4), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.57999999999999996

   1. Initial program 55.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 54.7%

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 57.3%

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

   if -0.57999999999999996 < (/.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 55.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 57.3%

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

   4. Taylor expanded in b around inf

      \[\leadsto \frac{\color{blue}{b \cdot \left(-3 \cdot \left(a \cdot c\right) + \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-3}{2} \cdot \left(a \cdot c\right) + \left(\frac{-9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(\frac{-1}{2} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(\frac{27}{8} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{9}{2} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{81}{16} \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   6. Applied rewrites 91.4%

      \[\leadsto \frac{\color{blue}{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(4.5, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, 5.0625 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   8. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-3 \cdot c, a, \mathsf{fma}\left(\frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -1.6875, \mathsf{fma}\left(-1.5 \cdot a, c, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}, -1.125, \mathsf{fma}\left(\frac{{\left(a \cdot c\right)}^{4} \cdot 6.328125}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}, -0.5, \mathsf{fma}\left(3.375, \frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \mathsf{fma}\left(4.5, \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}, \frac{{\left(a \cdot c\right)}^{4} \cdot 5.0625}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}\right)\right)\right)\right)\right)\right)\right) \cdot b}{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)}, b, \mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)\right)\right) \cdot a\right) \cdot 3}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b}\\ t_1 := \left(b \cdot b\right) \cdot b\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.58:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, t\_0, \left(3 \cdot \left(a \cdot c\right) - b \cdot b\right) \cdot t\_0\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \frac{a \cdot a}{t\_1 \cdot b}, -0.5625, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{\left(a \cdot t\_1\right) \cdot t\_1}, -0.16666666666666666, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, -0.5 \cdot c\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ -1.0 (+ (sqrt (fma (* a c) -3.0 (* b b))) b)))
        (t_1 (* (* b b) b)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.58)
     (/ (fma (* b b) t_0 (* (- (* 3.0 (* a c)) (* b b)) t_0)) (* 3.0 a))
     (/
      (fma
       (* (* (* c c) c) (/ (* a a) (* t_1 b)))
       -0.5625
       (fma
        (* (pow (* c a) 4.0) (/ 6.328125 (* (* a t_1) t_1)))
        -0.16666666666666666
        (fma (* -0.375 a) (* c (/ c (* b b))) (* -0.5 c))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = -1.0 / (sqrt(fma((a * c), -3.0, (b * b))) + b);
	double t_1 = (b * b) * b;
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.58) {
		tmp = fma((b * b), t_0, (((3.0 * (a * c)) - (b * b)) * t_0)) / (3.0 * a);
	} else {
		tmp = fma((((c * c) * c) * ((a * a) / (t_1 * b))), -0.5625, fma((pow((c * a), 4.0) * (6.328125 / ((a * t_1) * t_1))), -0.16666666666666666, fma((-0.375 * a), (c * (c / (b * b))), (-0.5 * c)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(-1.0 / Float64(sqrt(fma(Float64(a * c), -3.0, Float64(b * b))) + b))
	t_1 = Float64(Float64(b * 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)) <= -0.58)
		tmp = Float64(fma(Float64(b * b), t_0, Float64(Float64(Float64(3.0 * Float64(a * c)) - Float64(b * b)) * t_0)) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) / Float64(t_1 * b))), -0.5625, fma(Float64((Float64(c * a) ^ 4.0) * Float64(6.328125 / Float64(Float64(a * t_1) * t_1))), -0.16666666666666666, fma(Float64(-0.375 * a), Float64(c * Float64(c / Float64(b * b))), Float64(-0.5 * c)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-1.0 / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * b), $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], -0.58], N[(N[(N[(b * b), $MachinePrecision] * t$95$0 + N[(N[(N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] / N[(t$95$1 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(6.328125 / N[(N[(a * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(N[(-0.375 * a), $MachinePrecision] * N[(c * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.57999999999999996

   1. Initial program 55.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 54.7%

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 57.3%

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

   if -0.57999999999999996 < (/.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 55.5%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   6. Applied rewrites 90.9%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \frac{a \cdot a}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b}, -0.5625, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{\left(a \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}, -0.16666666666666666, \mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, -0.5 \cdot c\right)\right)\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.58:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, t\_0, \left(3 \cdot \left(a \cdot c\right) - b \cdot b\right) \cdot t\_0\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.5625, \frac{a \cdot {c}^{3}}{{b}^{5}}, -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ -1.0 (+ (sqrt (fma (* a c) -3.0 (* b b))) b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.58)
     (/ (fma (* b b) t_0 (* (- (* 3.0 (* a c)) (* b b)) t_0)) (* 3.0 a))
     (fma
      -0.5
      (/ c b)
      (*
       a
       (fma
        -0.5625
        (/ (* a (pow c 3.0)) (pow b 5.0))
        (* -0.375 (/ (pow c 2.0) (pow b 3.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = -1.0 / (sqrt(fma((a * c), -3.0, (b * b))) + b);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.58) {
		tmp = fma((b * b), t_0, (((3.0 * (a * c)) - (b * b)) * t_0)) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), (a * fma(-0.5625, ((a * pow(c, 3.0)) / pow(b, 5.0)), (-0.375 * (pow(c, 2.0) / pow(b, 3.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(-1.0 / Float64(sqrt(fma(Float64(a * c), -3.0, Float64(b * 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)) <= -0.58)
		tmp = Float64(fma(Float64(b * b), t_0, Float64(Float64(Float64(3.0 * Float64(a * c)) - Float64(b * b)) * t_0)) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(a * fma(-0.5625, Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0)), Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-1.0 / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $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], -0.58], N[(N[(N[(b * b), $MachinePrecision] * t$95$0 + N[(N[(N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(a * N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.57999999999999996

   1. Initial program 55.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 54.7%

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 57.3%

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

   if -0.57999999999999996 < (/.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 55.5%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{-1}{2} \cdot \frac{c}{b} + \color{blue}{a \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{\color{blue}{b}}, a \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{c}{b}, a \cdot \mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{5}}, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   7. Applied rewrites 87.8%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{c}{b}}, a \cdot \mathsf{fma}\left(-0.5625, \frac{a \cdot {c}^{3}}{{b}^{5}}, -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sqrt{\mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)} + b}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.58:\\ \;\;\;\;\frac{\mathsf{fma}\left(b \cdot b, t\_0, \left(3 \cdot \left(a \cdot c\right) - b \cdot b\right) \cdot t\_0\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot c}{{b}^{4}}, -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ -1.0 (+ (sqrt (fma (* a c) -3.0 (* b b))) b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.58)
     (/ (fma (* b b) t_0 (* (- (* 3.0 (* a c)) (* b b)) t_0)) (* 3.0 a))
     (/
      (*
       c
       (-
        (*
         c
         (fma
          -0.5625
          (/ (* (pow a 2.0) c) (pow b 4.0))
          (* -0.375 (/ a (pow b 2.0)))))
        0.5))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = -1.0 / (sqrt(fma((a * c), -3.0, (b * b))) + b);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.58) {
		tmp = fma((b * b), t_0, (((3.0 * (a * c)) - (b * b)) * t_0)) / (3.0 * a);
	} else {
		tmp = (c * ((c * fma(-0.5625, ((pow(a, 2.0) * c) / pow(b, 4.0)), (-0.375 * (a / pow(b, 2.0))))) - 0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(-1.0 / Float64(sqrt(fma(Float64(a * c), -3.0, Float64(b * 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)) <= -0.58)
		tmp = Float64(fma(Float64(b * b), t_0, Float64(Float64(Float64(3.0 * Float64(a * c)) - Float64(b * b)) * t_0)) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * fma(-0.5625, Float64(Float64((a ^ 2.0) * c) / (b ^ 4.0)), Float64(-0.375 * Float64(a / (b ^ 2.0))))) - 0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-1.0 / N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $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], -0.58], N[(N[(N[(b * b), $MachinePrecision] * t$95$0 + N[(N[(N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $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)) < -0.57999999999999996

   1. Initial program 55.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 54.7%

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 57.3%

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

   if -0.57999999999999996 < (/.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 55.5%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   7. Applied rewrites 87.7%

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

Alternative 7: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.015:\\ \;\;\;\;\frac{\left(b \cdot b - t\_0\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* c a) (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.015)
     (/ (* (- (* b b) t_0) (/ 0.3333333333333333 a)) (- (- b) (sqrt t_0)))
     (fma -0.5 (/ c b) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (c * a), (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.015) {
		tmp = (((b * b) - t_0) * (0.3333333333333333 / a)) / (-b - sqrt(t_0));
	} else {
		tmp = fma(-0.5, (c / b), (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(c * a), 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)) <= -0.015)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) * Float64(0.3333333333333333 / a)) / Float64(Float64(-b) - sqrt(t_0)));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $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], -0.015], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.014999999999999999

   1. Initial program 55.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\frac{\left(b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}}} \]

   if -0.014999999999999999 < (/.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 55.5%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.5

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   7. Applied rewrites 81.5%

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

Alternative 8: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.015:\\ \;\;\;\;\frac{\left(b \cdot b - t\_0\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -3.0 (* c a) (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.015)
     (/ (* (- (* b b) t_0) (/ 0.3333333333333333 a)) (- (- b) (sqrt t_0)))
     (/ (* c (- (* -0.375 (/ (* a c) (pow b 2.0))) 0.5)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, (c * a), (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.015) {
		tmp = (((b * b) - t_0) * (0.3333333333333333 / a)) / (-b - sqrt(t_0));
	} else {
		tmp = (c * ((-0.375 * ((a * c) / pow(b, 2.0))) - 0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-3.0, Float64(c * a), 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)) <= -0.015)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) * Float64(0.3333333333333333 / a)) / Float64(Float64(-b) - sqrt(t_0)));
	else
		tmp = Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 2.0))) - 0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $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], -0.015], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $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)) < -0.014999999999999999

   1. Initial program 55.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 57.0%

      \[\leadsto \color{blue}{\frac{\left(b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)\right) \cdot \frac{0.3333333333333333}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}}} \]

   if -0.014999999999999999 < (/.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 55.5%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 81.4

         \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b} \]

   7. Applied rewrites 81.4%

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

Alternative 9: 85.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.015:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a c) -3.0 (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.015)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 3.0 a))
     (/ (* c (- (* -0.375 (/ (* a c) (pow b 2.0))) 0.5)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * c), -3.0, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.015) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (3.0 * a);
	} else {
		tmp = (c * ((-0.375 * ((a * c) / pow(b, 2.0))) - 0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(a * c), -3.0, 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)) <= -0.015)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 2.0))) - 0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -3.0 + N[(b * b), $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], -0.015], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $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)) < -0.014999999999999999

   1. Initial program 55.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 54.7%

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

   5. Applied rewrites 56.3%

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

   7. Applied rewrites 57.0%

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

   if -0.014999999999999999 < (/.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 55.5%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 81.4

         \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b} \]

   7. Applied rewrites 81.4%

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

Alternative 10: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.015:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{1 - \left(a \cdot 3\right) \cdot \frac{c}{b \cdot b}}, \left|b\right|, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.015)
   (/
    (fma (sqrt (- 1.0 (* (* a 3.0) (/ c (* b b))))) (fabs b) (- b))
    (* 3.0 a))
   (/ (* c (- (* -0.375 (/ (* a c) (pow b 2.0))) 0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.015) {
		tmp = fma(sqrt((1.0 - ((a * 3.0) * (c / (b * b))))), fabs(b), -b) / (3.0 * a);
	} else {
		tmp = (c * ((-0.375 * ((a * c) / pow(b, 2.0))) - 0.5)) / 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)) <= -0.015)
		tmp = Float64(fma(sqrt(Float64(1.0 - Float64(Float64(a * 3.0) * Float64(c / Float64(b * b))))), abs(b), Float64(-b)) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 2.0))) - 0.5)) / 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], -0.015], N[(N[(N[Sqrt[N[(1.0 - N[(N[(a * 3.0), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[b], $MachinePrecision] + (-b)), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $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)) < -0.014999999999999999

   1. Initial program 55.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-to-mult N/A

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

      6. sqrt-prod N/A

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

      7. lift-*.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 56.1%

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

   if -0.014999999999999999 < (/.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 55.5%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 81.4

         \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b} \]

   7. Applied rewrites 81.4%

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

Alternative 11: 76.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{1 - \left(a \cdot 3\right) \cdot \frac{c}{b \cdot b}}, \left|b\right|, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -2.4e-7)
   (/
    (fma (sqrt (- 1.0 (* (* a 3.0) (/ c (* b b))))) (fabs b) (- b))
    (* 3.0 a))
   (* -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)) <= -2.4e-7) {
		tmp = fma(sqrt((1.0 - ((a * 3.0) * (c / (b * b))))), fabs(b), -b) / (3.0 * a);
	} else {
		tmp = -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)) <= -2.4e-7)
		tmp = Float64(fma(sqrt(Float64(1.0 - Float64(Float64(a * 3.0) * Float64(c / Float64(b * b))))), abs(b), Float64(-b)) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -2.4e-7], N[(N[(N[Sqrt[N[(1.0 - N[(N[(a * 3.0), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[b], $MachinePrecision] + (-b)), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2.39999999999999979e-7

   1. Initial program 55.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-to-mult N/A

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

      6. sqrt-prod N/A

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

      7. lift-*.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 56.1%

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

   if -2.39999999999999979e-7 < (/.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 55.5%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.3

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

   4. Applied rewrites 64.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -2.4e-7)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (* -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)) <= -2.4e-7) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = -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)) <= -2.4e-7)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -2.4e-7], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2.39999999999999979e-7

   1. Initial program 55.5%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 55.5

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

   3. Applied rewrites 55.5%

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

   if -2.39999999999999979e-7 < (/.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 55.5%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.3

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

   4. Applied rewrites 64.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -2.4e-7)
   (/ (* (- (sqrt (fma -3.0 (* c a) (* b b))) b) 0.3333333333333333) a)
   (* -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)) <= -2.4e-7) {
		tmp = ((sqrt(fma(-3.0, (c * a), (b * b))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = -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)) <= -2.4e-7)
		tmp = Float64(Float64(Float64(sqrt(fma(-3.0, Float64(c * a), Float64(b * b))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -2.4e-7], N[(N[(N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2.39999999999999979e-7

   1. Initial program 55.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   3. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}} \]

   if -2.39999999999999979e-7 < (/.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 55.5%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.3

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

   4. Applied rewrites 64.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -2.4e-7)
   (* (- (sqrt (fma -3.0 (* c a) (* b b))) b) (/ 0.3333333333333333 a))
   (* -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)) <= -2.4e-7) {
		tmp = (sqrt(fma(-3.0, (c * a), (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -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)) <= -2.4e-7)
		tmp = Float64(Float64(sqrt(fma(-3.0, Float64(c * a), Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -2.4e-7], N[(N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2.39999999999999979e-7

   1. Initial program 55.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

   3. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]

   if -2.39999999999999979e-7 < (/.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 55.5%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.3

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

   4. Applied rewrites 64.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 64.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return -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 = (-0.5d0) * (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.5%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 64.3

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

4. Applied rewrites 64.3%

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

Reproduce

herbie shell --seed 2025149 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))