Cubic critical, narrow range

Percentage Accurate: 55.6% → 91.1%

Time: 13.4s

Alternatives: 13

Speedup: 2.9×

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: 55.6% 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: 91.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot c\right) \cdot 6.75\\ t_1 := \mathsf{fma}\left(4.5, c \cdot t\_0, -27 \cdot {c}^{3}\right)\\ \frac{\frac{b \cdot \left(a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, \frac{t\_1}{{b}^{4}}, \frac{-0.5 \cdot \left(a \cdot \mathsf{fma}\left(-4.5, c \cdot t\_1, 0.25 \cdot {t\_0}^{2}\right)\right)}{{b}^{6}}\right), 0.5 \cdot \frac{t\_0}{b \cdot b}\right), -4.5 \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(\left(\mathsf{fma}\left(-3, a \cdot \frac{c}{b \cdot b}, 3\right) + \frac{-1.6875 \cdot {\left(a \cdot c\right)}^{3}}{{b}^{6}}\right) - \mathsf{fma}\left(1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}}, \frac{1.5 \cdot \left(a \cdot c\right)}{b \cdot b}\right)\right)}}{3 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c c) 6.75)) (t_1 (fma 4.5 (* c t_0) (* -27.0 (pow c 3.0)))))
   (/
    (/
     (*
      b
      (*
       a
       (fma
        a
        (fma
         a
         (fma
          0.5
          (/ t_1 (pow b 4.0))
          (/
           (* -0.5 (* a (fma -4.5 (* c t_1) (* 0.25 (pow t_0 2.0)))))
           (pow b 6.0)))
         (* 0.5 (/ t_0 (* b b))))
        (* -4.5 c))))
     (*
      (* b b)
      (-
       (+
        (fma -3.0 (* a (/ c (* b b))) 3.0)
        (/ (* -1.6875 (pow (* a c) 3.0)) (pow b 6.0)))
       (fma
        1.125
        (/ (* (* a a) (* c c)) (pow b 4.0))
        (/ (* 1.5 (* a c)) (* b b))))))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) * 6.75;
	double t_1 = fma(4.5, (c * t_0), (-27.0 * pow(c, 3.0)));
	return ((b * (a * fma(a, fma(a, fma(0.5, (t_1 / pow(b, 4.0)), ((-0.5 * (a * fma(-4.5, (c * t_1), (0.25 * pow(t_0, 2.0))))) / pow(b, 6.0))), (0.5 * (t_0 / (b * b)))), (-4.5 * c)))) / ((b * b) * ((fma(-3.0, (a * (c / (b * b))), 3.0) + ((-1.6875 * pow((a * c), 3.0)) / pow(b, 6.0))) - fma(1.125, (((a * a) * (c * c)) / pow(b, 4.0)), ((1.5 * (a * c)) / (b * b)))))) / (3.0 * a);
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) * 6.75)
	t_1 = fma(4.5, Float64(c * t_0), Float64(-27.0 * (c ^ 3.0)))
	return Float64(Float64(Float64(b * Float64(a * fma(a, fma(a, fma(0.5, Float64(t_1 / (b ^ 4.0)), Float64(Float64(-0.5 * Float64(a * fma(-4.5, Float64(c * t_1), Float64(0.25 * (t_0 ^ 2.0))))) / (b ^ 6.0))), Float64(0.5 * Float64(t_0 / Float64(b * b)))), Float64(-4.5 * c)))) / Float64(Float64(b * b) * Float64(Float64(fma(-3.0, Float64(a * Float64(c / Float64(b * b))), 3.0) + Float64(Float64(-1.6875 * (Float64(a * c) ^ 3.0)) / (b ^ 6.0))) - fma(1.125, Float64(Float64(Float64(a * a) * Float64(c * c)) / (b ^ 4.0)), Float64(Float64(1.5 * Float64(a * c)) / Float64(b * b)))))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] * 6.75), $MachinePrecision]}, Block[{t$95$1 = N[(4.5 * N[(c * t$95$0), $MachinePrecision] + N[(-27.0 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(b * N[(a * N[(a * N[(a * N[(0.5 * N[(t$95$1 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(a * N[(-4.5 * N[(c * t$95$1), $MachinePrecision] + N[(0.25 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(N[(N[(-3.0 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] + N[(N[(-1.6875 * N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.125 * N[(N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.5 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 55.8%

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

   1. lift-+.f64 N/A

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

   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. flip3-+ N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 56.9%

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

5. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-27 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{2}}\right)\right)\right)}}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \left(-b\right)\right)}}{3 \cdot a} \]

7. Applied rewrites 93.0%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-27, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \left(-b\right)\right)}}{3 \cdot a} \]

8. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

10. Applied rewrites 93.2%

    \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-27, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}{\color{blue}{\left(b \cdot b\right) \cdot \left(\left(\mathsf{fma}\left(-3, a \cdot \frac{c}{b \cdot b}, 3\right) + \frac{-1.6875 \cdot {\left(a \cdot c\right)}^{3}}{{b}^{6}}\right) - \mathsf{fma}\left(1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}}, \frac{1.5 \cdot \left(a \cdot c\right)}{b \cdot b}\right)\right)}}}{3 \cdot a} \]

11. Taylor expanded in a around 0

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

13. Applied rewrites 93.2%

    \[\leadsto \frac{\frac{b \cdot \left(a \cdot \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(4.5, c \cdot \left(\left(c \cdot c\right) \cdot 6.75\right), -27 \cdot {c}^{3}\right)}{{b}^{4}}, \frac{-0.5 \cdot \left(a \cdot \mathsf{fma}\left(-4.5, c \cdot \mathsf{fma}\left(4.5, c \cdot \left(\left(c \cdot c\right) \cdot 6.75\right), -27 \cdot {c}^{3}\right), 0.25 \cdot {\left(\left(c \cdot c\right) \cdot 6.75\right)}^{2}\right)\right)}{{b}^{6}}\right), 0.5 \cdot \frac{\left(c \cdot c\right) \cdot 6.75}{b \cdot b}\right), -4.5 \cdot c\right)}\right)}{\left(b \cdot b\right) \cdot \left(\left(\mathsf{fma}\left(-3, a \cdot \frac{c}{b \cdot b}, 3\right) + \frac{-1.6875 \cdot {\left(a \cdot c\right)}^{3}}{{b}^{6}}\right) - \mathsf{fma}\left(1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}}, \frac{1.5 \cdot \left(a \cdot c\right)}{b \cdot b}\right)\right)}}{3 \cdot a} \]
14. Add Preprocessing

Alternative 2: 91.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot 6.75\\ t_1 := \mathsf{fma}\left(-27, {a}^{3}, 4.5 \cdot \left(a \cdot t\_0\right)\right)\\ t_2 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \frac{\frac{b \cdot \left(c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-0.5, \frac{c \cdot \mathsf{fma}\left(0.25, {t\_0}^{2}, -4.5 \cdot \left(a \cdot t\_1\right)\right)}{{b}^{6}}, 0.5 \cdot \frac{t\_1}{{b}^{4}}\right), 0.5 \cdot \frac{t\_0}{b \cdot b}\right), -4.5 \cdot a\right)\right)}{t\_2 + \left(b \cdot b + \sqrt{t\_2} \cdot b\right)}}{3 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a a) 6.75))
        (t_1 (fma -27.0 (pow a 3.0) (* 4.5 (* a t_0))))
        (t_2 (fma (* -3.0 a) c (* b b))))
   (/
    (/
     (*
      b
      (*
       c
       (fma
        c
        (fma
         c
         (fma
          -0.5
          (/ (* c (fma 0.25 (pow t_0 2.0) (* -4.5 (* a t_1)))) (pow b 6.0))
          (* 0.5 (/ t_1 (pow b 4.0))))
         (* 0.5 (/ t_0 (* b b))))
        (* -4.5 a))))
     (+ t_2 (+ (* b b) (* (sqrt t_2) b))))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = (a * a) * 6.75;
	double t_1 = fma(-27.0, pow(a, 3.0), (4.5 * (a * t_0)));
	double t_2 = fma((-3.0 * a), c, (b * b));
	return ((b * (c * fma(c, fma(c, fma(-0.5, ((c * fma(0.25, pow(t_0, 2.0), (-4.5 * (a * t_1)))) / pow(b, 6.0)), (0.5 * (t_1 / pow(b, 4.0)))), (0.5 * (t_0 / (b * b)))), (-4.5 * a)))) / (t_2 + ((b * b) + (sqrt(t_2) * b)))) / (3.0 * a);
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * a) * 6.75)
	t_1 = fma(-27.0, (a ^ 3.0), Float64(4.5 * Float64(a * t_0)))
	t_2 = fma(Float64(-3.0 * a), c, Float64(b * b))
	return Float64(Float64(Float64(b * Float64(c * fma(c, fma(c, fma(-0.5, Float64(Float64(c * fma(0.25, (t_0 ^ 2.0), Float64(-4.5 * Float64(a * t_1)))) / (b ^ 6.0)), Float64(0.5 * Float64(t_1 / (b ^ 4.0)))), Float64(0.5 * Float64(t_0 / Float64(b * b)))), Float64(-4.5 * a)))) / Float64(t_2 + Float64(Float64(b * b) + Float64(sqrt(t_2) * b)))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * 6.75), $MachinePrecision]}, Block[{t$95$1 = N[(-27.0 * N[Power[a, 3.0], $MachinePrecision] + N[(4.5 * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(b * N[(c * N[(c * N[(c * N[(-0.5 * N[(N[(c * N[(0.25 * N[Power[t$95$0, 2.0], $MachinePrecision] + N[(-4.5 * N[(a * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$1 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-4.5 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 + N[(N[(b * b), $MachinePrecision] + N[(N[Sqrt[t$95$2], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 55.8%

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

   1. lift-+.f64 N/A

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

   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. flip3-+ N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 56.9%

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

5. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-27 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{2}}\right)\right)\right)}}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \left(-b\right)\right)}}{3 \cdot a} \]

7. Applied rewrites 93.0%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-27, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \left(-b\right)\right)}}{3 \cdot a} \]

8. Taylor expanded in c around 0

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

10. Applied rewrites 93.2%

    \[\leadsto \frac{\frac{b \cdot \left(c \cdot \color{blue}{\mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-0.5, \frac{c \cdot \mathsf{fma}\left(0.25, {\left(\left(a \cdot a\right) \cdot 6.75\right)}^{2}, -4.5 \cdot \left(a \cdot \mathsf{fma}\left(-27, {a}^{3}, 4.5 \cdot \left(a \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\right)\right)\right)\right)}{{b}^{6}}, 0.5 \cdot \frac{\mathsf{fma}\left(-27, {a}^{3}, 4.5 \cdot \left(a \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\right)\right)}{{b}^{4}}\right), 0.5 \cdot \frac{\left(a \cdot a\right) \cdot 6.75}{b \cdot b}\right), -4.5 \cdot a\right)}\right)}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \left(-b\right)\right)}}{3 \cdot a} \]

11. Final simplification 93.2%

    \[\leadsto \frac{\frac{b \cdot \left(c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(c, \mathsf{fma}\left(-0.5, \frac{c \cdot \mathsf{fma}\left(0.25, {\left(\left(a \cdot a\right) \cdot 6.75\right)}^{2}, -4.5 \cdot \left(a \cdot \mathsf{fma}\left(-27, {a}^{3}, 4.5 \cdot \left(a \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\right)\right)\right)\right)}{{b}^{6}}, 0.5 \cdot \frac{\mathsf{fma}\left(-27, {a}^{3}, 4.5 \cdot \left(a \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\right)\right)}{{b}^{4}}\right), 0.5 \cdot \frac{\left(a \cdot a\right) \cdot 6.75}{b \cdot b}\right), -4.5 \cdot a\right)\right)}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + \left(b \cdot b + \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot b\right)}}{3 \cdot a} \]
12. Add Preprocessing

Alternative 3: 90.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\\ t_1 := \mathsf{fma}\left(4.5, \left(a \cdot c\right) \cdot t\_0, -27 \cdot {\left(a \cdot c\right)}^{3}\right)\\ t_2 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \frac{\frac{b \cdot \frac{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(-4.5 \cdot a, c \cdot t\_1, 0.25 \cdot {t\_0}^{2}\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, t\_1, \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, t\_0, -4.5 \cdot \left(a \cdot \left(\left(b \cdot b\right) \cdot c\right)\right)\right)\right)\right)}{{b}^{6}}}{t\_2 + \left(b \cdot b + \sqrt{t\_2} \cdot b\right)}}{3 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c c) (* (* a a) 6.75)))
        (t_1 (fma 4.5 (* (* a c) t_0) (* -27.0 (pow (* a c) 3.0))))
        (t_2 (fma (* -3.0 a) c (* b b))))
   (/
    (/
     (*
      b
      (/
       (fma
        -0.5
        (fma (* -4.5 a) (* c t_1) (* 0.25 (pow t_0 2.0)))
        (*
         (* b b)
         (fma 0.5 t_1 (* (* b b) (fma 0.5 t_0 (* -4.5 (* a (* (* b b) c))))))))
       (pow b 6.0)))
     (+ t_2 (+ (* b b) (* (sqrt t_2) b))))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) * ((a * a) * 6.75);
	double t_1 = fma(4.5, ((a * c) * t_0), (-27.0 * pow((a * c), 3.0)));
	double t_2 = fma((-3.0 * a), c, (b * b));
	return ((b * (fma(-0.5, fma((-4.5 * a), (c * t_1), (0.25 * pow(t_0, 2.0))), ((b * b) * fma(0.5, t_1, ((b * b) * fma(0.5, t_0, (-4.5 * (a * ((b * b) * c)))))))) / pow(b, 6.0))) / (t_2 + ((b * b) + (sqrt(t_2) * b)))) / (3.0 * a);
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) * Float64(Float64(a * a) * 6.75))
	t_1 = fma(4.5, Float64(Float64(a * c) * t_0), Float64(-27.0 * (Float64(a * c) ^ 3.0)))
	t_2 = fma(Float64(-3.0 * a), c, Float64(b * b))
	return Float64(Float64(Float64(b * Float64(fma(-0.5, fma(Float64(-4.5 * a), Float64(c * t_1), Float64(0.25 * (t_0 ^ 2.0))), Float64(Float64(b * b) * fma(0.5, t_1, Float64(Float64(b * b) * fma(0.5, t_0, Float64(-4.5 * Float64(a * Float64(Float64(b * b) * c)))))))) / (b ^ 6.0))) / Float64(t_2 + Float64(Float64(b * b) + Float64(sqrt(t_2) * b)))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 6.75), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(4.5 * N[(N[(a * c), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(-27.0 * N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(b * N[(N[(-0.5 * N[(N[(-4.5 * a), $MachinePrecision] * N[(c * t$95$1), $MachinePrecision] + N[(0.25 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 * t$95$1 + N[(N[(b * b), $MachinePrecision] * N[(0.5 * t$95$0 + N[(-4.5 * N[(a * N[(N[(b * b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 + N[(N[(b * b), $MachinePrecision] + N[(N[Sqrt[t$95$2], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 55.8%

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

   1. lift-+.f64 N/A

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

   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. flip3-+ N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 56.9%

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

5. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-27 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{2}}\right)\right)\right)}}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \left(-b\right)\right)}}{3 \cdot a} \]

7. Applied rewrites 93.0%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-27, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \left(-b\right)\right)}}{3 \cdot a} \]

8. Taylor expanded in b around 0

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

10. Applied rewrites 92.9%

    \[\leadsto \frac{\frac{b \cdot \frac{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(-4.5 \cdot a, c \cdot \mathsf{fma}\left(4.5, \left(a \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\right), -27 \cdot {\left(a \cdot c\right)}^{3}\right), 0.25 \cdot {\left(\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\right)}^{2}\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(4.5, \left(a \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\right), -27 \cdot {\left(a \cdot c\right)}^{3}\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot 6.75\right), -4.5 \cdot \left(a \cdot \left(\left(b \cdot b\right) \cdot c\right)\right)\right)\right)\right)}{\color{blue}{{b}^{6}}}}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \left(-b\right)\right)}}{3 \cdot a} \]

11. Final simplification 92.9%

    \[\leadsto \frac{\frac{b \cdot \frac{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(-4.5 \cdot a, c \cdot \mathsf{fma}\left(4.5, \left(a \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\right), -27 \cdot {\left(a \cdot c\right)}^{3}\right), 0.25 \cdot {\left(\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\right)}^{2}\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(4.5, \left(a \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot 6.75\right)\right), -27 \cdot {\left(a \cdot c\right)}^{3}\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot 6.75\right), -4.5 \cdot \left(a \cdot \left(\left(b \cdot b\right) \cdot c\right)\right)\right)\right)\right)}{{b}^{6}}}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + \left(b \cdot b + \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot b\right)}}{3 \cdot a} \]
12. Add Preprocessing

Alternative 4: 90.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (fma
   (fma
    (* -0.16666666666666666 a)
    (* (/ (pow c 4.0) (pow b 6.0)) (/ 6.328125 b))
    (/ (* (pow c 3.0) -0.5625) (pow b 5.0)))
   a
   (/ (* (* c c) -0.375) (pow b 3.0)))
  a
  (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	return fma(fma(fma((-0.16666666666666666 * a), ((pow(c, 4.0) / pow(b, 6.0)) * (6.328125 / b)), ((pow(c, 3.0) * -0.5625) / pow(b, 5.0))), a, (((c * c) * -0.375) / pow(b, 3.0))), a, ((c / b) * -0.5));
}

Julia

function code(a, b, c)
	return fma(fma(fma(Float64(-0.16666666666666666 * a), Float64(Float64((c ^ 4.0) / (b ^ 6.0)) * Float64(6.328125 / b)), Float64(Float64((c ^ 3.0) * -0.5625) / (b ^ 5.0))), a, Float64(Float64(Float64(c * c) * -0.375) / (b ^ 3.0))), a, Float64(Float64(c / b) * -0.5))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-0.16666666666666666 * a), $MachinePrecision] * N[(N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * N[(6.328125 / b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[c, 3.0], $MachinePrecision] * -0.5625), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.8%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \frac{-1}{2} \cdot \frac{c}{b}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + \frac{-1}{2} \cdot \frac{c}{b} \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

5. Applied rewrites 92.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]
6. Add Preprocessing

Alternative 5: 90.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \left(b \cdot b\right) \cdot \mathsf{fma}\left(-0.5625 \cdot a, {c}^{3}, \left(-0.375 \cdot \left(b \cdot b\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (/
   (fma
    (* -1.0546875 (* a a))
    (pow c 4.0)
    (* (* b b) (fma (* -0.5625 a) (pow c 3.0) (* (* -0.375 (* b b)) (* c c)))))
   (pow b 7.0))
  a
  (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	return fma((fma((-1.0546875 * (a * a)), pow(c, 4.0), ((b * b) * fma((-0.5625 * a), pow(c, 3.0), ((-0.375 * (b * b)) * (c * c))))) / pow(b, 7.0)), a, ((c / b) * -0.5));
}

Julia

function code(a, b, c)
	return fma(Float64(fma(Float64(-1.0546875 * Float64(a * a)), (c ^ 4.0), Float64(Float64(b * b) * fma(Float64(-0.5625 * a), (c ^ 3.0), Float64(Float64(-0.375 * Float64(b * b)) * Float64(c * c))))) / (b ^ 7.0)), a, Float64(Float64(c / b) * -0.5))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-1.0546875 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(N[(-0.5625 * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision] + N[(N[(-0.375 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.8%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \frac{-1}{2} \cdot \frac{c}{b}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + \frac{-1}{2} \cdot \frac{c}{b} \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

5. Applied rewrites 92.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

6. Taylor expanded in b around 0

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

8. Applied rewrites 92.8%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot \left(a \cdot a\right), {c}^{4}, \left(b \cdot b\right) \cdot \mathsf{fma}\left(-0.5625 \cdot a, {c}^{3}, \left(-0.375 \cdot \left(b \cdot b\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]
9. Add Preprocessing

Alternative 6: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.005:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.005)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 3.0 a))
     (/ (fma -0.375 (/ (* a (* c c)) (* b b)) (* -0.5 c)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.005) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (3.0 * a);
	} else {
		tmp = fma(-0.375, ((a * (c * c)) / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.005)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(3.0 * a));
	else
		tmp = Float64(fma(-0.375, Float64(Float64(a * Float64(c * c)) / Float64(b * b)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, 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], -0.005], 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[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0050000000000000001

   1. Initial program 77.9%

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

      1. lift-+.f64 N/A

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

      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. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 79.5%

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

   if -0.0050000000000000001 < (/.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 45.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 73.5

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

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.4%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 90.0

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

   10. Applied rewrites 90.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.005:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.005:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(\left(-b\right) - \sqrt{t\_0}\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.005)
     (/ (- (* b b) t_0) (* (- (- b) (sqrt t_0)) (* a 3.0)))
     (/ (fma -0.375 (/ (* a (* c c)) (* b b)) (* -0.5 c)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.005) {
		tmp = ((b * b) - t_0) / ((-b - sqrt(t_0)) * (a * 3.0));
	} else {
		tmp = fma(-0.375, ((a * (c * c)) / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), 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)) <= -0.005)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(Float64(-b) - sqrt(t_0)) * Float64(a * 3.0)));
	else
		tmp = Float64(fma(-0.375, Float64(Float64(a * Float64(c * c)) / Float64(b * b)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + 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], -0.005], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0050000000000000001

   1. Initial program 77.9%

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

      1. lift-/.f64 N/A

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

      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. div-add N/A

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

      4. lower-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 75.9

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

   4. Applied rewrites 75.9%

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

   6. Applied rewrites 79.4%

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

   if -0.0050000000000000001 < (/.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 45.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 73.5

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

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.4%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 90.0

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

   10. Applied rewrites 90.0%

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

Alternative 8: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.005:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(\left(-b\right) - \sqrt{t\_0}\right) \cdot \left(a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.005)
     (/ (- (* b b) t_0) (* (- (- b) (sqrt t_0)) (* a 3.0)))
     (/ (fma -0.375 (/ (* a (* c c)) (* b b)) (* -0.5 c)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.005) {
		tmp = ((b * b) - t_0) / ((-b - sqrt(t_0)) * (a * 3.0));
	} else {
		tmp = fma(-0.375, ((a * (c * c)) / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.005)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(Float64(-b) - sqrt(t_0)) * Float64(a * 3.0)));
	else
		tmp = Float64(fma(-0.375, Float64(Float64(a * Float64(c * c)) / Float64(b * b)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, 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], -0.005], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0050000000000000001

   1. Initial program 77.9%

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

      1. lift-/.f64 N/A

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

      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. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 79.4%

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

   if -0.0050000000000000001 < (/.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 45.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 73.5

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

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.4%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 90.0

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

   10. Applied rewrites 90.0%

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

Alternative 9: 88.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{b \cdot \left(c \cdot \mathsf{fma}\left(0.5 \cdot c, \frac{\left(a \cdot a\right) \cdot 6.75}{b \cdot b}, -4.5 \cdot a\right)\right)}{\mathsf{fma}\left(2, b \cdot b, c \cdot \left(a \cdot -4.5\right)\right) + b \cdot b}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (* b (* c (fma (* 0.5 c) (/ (* (* a a) 6.75) (* b b)) (* -4.5 a))))
   (+ (fma 2.0 (* b b) (* c (* a -4.5))) (* b b)))
  (* 3.0 a)))

C

double code(double a, double b, double c) {
	return ((b * (c * fma((0.5 * c), (((a * a) * 6.75) / (b * b)), (-4.5 * a)))) / (fma(2.0, (b * b), (c * (a * -4.5))) + (b * b))) / (3.0 * a);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(b * Float64(c * fma(Float64(0.5 * c), Float64(Float64(Float64(a * a) * 6.75) / Float64(b * b)), Float64(-4.5 * a)))) / Float64(fma(2.0, Float64(b * b), Float64(c * Float64(a * -4.5))) + Float64(b * b))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(b * N[(c * N[(N[(0.5 * c), $MachinePrecision] * N[(N[(N[(a * a), $MachinePrecision] * 6.75), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-4.5 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.8%

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

   1. lift-+.f64 N/A

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

   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. flip3-+ N/A

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

   4. +-commutative N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 56.9%

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

5. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-27 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{2}}\right)\right)\right)}}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \left(-b\right)\right)}}{3 \cdot a} \]

7. Applied rewrites 93.0%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right), \mathsf{fma}\left(-27, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right), 0.25 \cdot {\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)}^{2}\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-27, {\left(a \cdot c\right)}^{3}, -0.5 \cdot \left(\left(\left(a \cdot c\right) \cdot -9\right) \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \left(c \cdot c\right), 27, -0.25 \cdot {\left(\left(a \cdot c\right) \cdot -9\right)}^{2}\right)}{b \cdot b}, 0.5 \cdot \left(\left(a \cdot c\right) \cdot -9\right)\right)\right)}}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \left(-b\right)\right)}}{3 \cdot a} \]

8. Taylor expanded in c around 0

   \[\leadsto \frac{\frac{b \cdot \left(c \cdot \color{blue}{\left(\frac{-9}{2} \cdot a + \frac{1}{2} \cdot \left(c \cdot \left(\frac{-81}{4} \cdot \frac{{a}^{2}}{{b}^{2}} + 27 \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right)\right)}\right)}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \left(-b\right)\right)}}{3 \cdot a} \]
9. Step-by-step derivation

10. Applied rewrites 83.0%

    \[\leadsto \frac{\frac{b \cdot \left(c \cdot \color{blue}{\mathsf{fma}\left(0.5 \cdot c, \frac{\left(a \cdot a\right) \cdot 6.75}{b \cdot b}, -4.5 \cdot a\right)}\right)}{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + \left(b \cdot b - \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \left(-b\right)\right)}}{3 \cdot a} \]

11. Taylor expanded in c around 0

    \[\leadsto \frac{\frac{b \cdot \left(c \cdot \mathsf{fma}\left(\frac{1}{2} \cdot c, \frac{\left(a \cdot a\right) \cdot \frac{27}{4}}{b \cdot b}, \frac{-9}{2} \cdot a\right)\right)}{\color{blue}{\left(2 \cdot {b}^{2} + c \cdot \left(-3 \cdot a - \frac{3}{2} \cdot a\right)\right) - -1 \cdot {b}^{2}}}}{3 \cdot a} \]
12. Step-by-step derivation

    1. lower--.f64 N/A

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

    2. lower-fma.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. distribute-rgt-out-- N/A

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

    7. metadata-eval N/A

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

    8. lower-*.f64 N/A

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

    9. mul-1-neg N/A

       \[\leadsto \frac{\frac{b \cdot \left(c \cdot \mathsf{fma}\left(\frac{1}{2} \cdot c, \frac{\left(a \cdot a\right) \cdot \frac{27}{4}}{b \cdot b}, \frac{-9}{2} \cdot a\right)\right)}{\mathsf{fma}\left(2, b \cdot b, c \cdot \left(a \cdot \frac{-9}{2}\right)\right) - \color{blue}{\left(\mathsf{neg}\left({b}^{2}\right)\right)}}}{3 \cdot a} \]

    10. lower-neg.f64 N/A

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

    11. unpow2 N/A

        \[\leadsto \frac{\frac{b \cdot \left(c \cdot \mathsf{fma}\left(\frac{1}{2} \cdot c, \frac{\left(a \cdot a\right) \cdot \frac{27}{4}}{b \cdot b}, \frac{-9}{2} \cdot a\right)\right)}{\mathsf{fma}\left(2, b \cdot b, c \cdot \left(a \cdot \frac{-9}{2}\right)\right) - \left(-\color{blue}{b \cdot b}\right)}}{3 \cdot a} \]

    12. lower-*.f64 89.9

        \[\leadsto \frac{\frac{b \cdot \left(c \cdot \mathsf{fma}\left(0.5 \cdot c, \frac{\left(a \cdot a\right) \cdot 6.75}{b \cdot b}, -4.5 \cdot a\right)\right)}{\mathsf{fma}\left(2, b \cdot b, c \cdot \left(a \cdot -4.5\right)\right) - \left(-\color{blue}{b \cdot b}\right)}}{3 \cdot a} \]

13. Applied rewrites 89.9%

    \[\leadsto \frac{\frac{b \cdot \left(c \cdot \mathsf{fma}\left(0.5 \cdot c, \frac{\left(a \cdot a\right) \cdot 6.75}{b \cdot b}, -4.5 \cdot a\right)\right)}{\color{blue}{\mathsf{fma}\left(2, b \cdot b, c \cdot \left(a \cdot -4.5\right)\right) - \left(-b \cdot b\right)}}}{3 \cdot a} \]

14. Final simplification 89.9%

    \[\leadsto \frac{\frac{b \cdot \left(c \cdot \mathsf{fma}\left(0.5 \cdot c, \frac{\left(a \cdot a\right) \cdot 6.75}{b \cdot b}, -4.5 \cdot a\right)\right)}{\mathsf{fma}\left(2, b \cdot b, c \cdot \left(a \cdot -4.5\right)\right) + b \cdot b}}{3 \cdot a} \]
15. Add Preprocessing

Alternative 10: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.005:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.375, \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.005)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (/ (fma -0.375 (/ (* a (* c c)) (* b b)) (* -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)) <= -0.005) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = fma(-0.375, ((a * (c * c)) / (b * b)), (-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)) <= -0.005)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(-0.375, Float64(Float64(a * Float64(c * c)) / Float64(b * b)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.005], 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[(N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0050000000000000001

   1. Initial program 77.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 78.1

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

   4. Applied rewrites 78.1%

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

   if -0.0050000000000000001 < (/.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 45.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 73.5

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

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.4%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 90.0

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

   10. Applied rewrites 90.0%

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

Alternative 11: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.4e-6)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.4e-6) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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)) <= -1.4e-6)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	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], -1.4e-6], 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[(N[(c / b), $MachinePrecision] * -0.5), $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)) < -1.39999999999999994e-6

   1. Initial program 73.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 73.5

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

   4. Applied rewrites 73.5%

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

   if -1.39999999999999994e-6 < (/.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 32.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 83.5

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

   5. Applied rewrites 83.5%

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

Alternative 12: 64.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.2

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

5. Applied rewrites 64.2%

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

Alternative 13: 64.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.2

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

5. Applied rewrites 64.2%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
6. Step-by-step derivation

7. Applied rewrites 64.2%

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

Reproduce

herbie shell --seed 2025009 
(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)))