Cubic critical, narrow range

Percentage Accurate: 55.9% → 91.2%

Time: 14.4s

Alternatives: 11

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: 55.9% 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: 91.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\\ t_1 := {\left(a \cdot c\right)}^{2}\\ t_2 := \mathsf{fma}\left(9, t\_1, 18 \cdot t\_1\right) - 0.25 \cdot {t\_0}^{2}\\ t_3 := -27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(t\_0 \cdot t\_2\right)\\ \frac{\frac{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, {t\_2}^{2}, 0.5 \cdot \left(t\_0 \cdot t\_3\right)\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, t\_0, \mathsf{fma}\left(0.5, \frac{t\_3}{{b}^{4}}, 0.5 \cdot \frac{t\_2}{b \cdot b}\right)\right)\right)}{\mathsf{fma}\left(2, b \cdot b, c \cdot \left(\mathsf{fma}\left(-3, a, c \cdot \left(-1.6875 \cdot \frac{{a}^{3} \cdot c}{{b}^{4}} - 1.125 \cdot \frac{a \cdot a}{b \cdot b}\right)\right) - 1.5 \cdot a\right)\right) + b \cdot b}}{3 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -6.0 (* a c) (* -3.0 (* a c))))
        (t_1 (pow (* a c) 2.0))
        (t_2 (- (fma 9.0 t_1 (* 18.0 t_1)) (* 0.25 (pow t_0 2.0))))
        (t_3 (- (* -27.0 (pow (* a c) 3.0)) (* 0.5 (* t_0 t_2)))))
   (/
    (/
     (*
      b
      (fma
       -0.5
       (/ (fma 0.25 (pow t_2 2.0) (* 0.5 (* t_0 t_3))) (pow b 6.0))
       (fma 0.5 t_0 (fma 0.5 (/ t_3 (pow b 4.0)) (* 0.5 (/ t_2 (* b b)))))))
     (+
      (fma
       2.0
       (* b b)
       (*
        c
        (-
         (fma
          -3.0
          a
          (*
           c
           (-
            (* -1.6875 (/ (* (pow a 3.0) c) (pow b 4.0)))
            (* 1.125 (/ (* a a) (* b b))))))
         (* 1.5 a))))
      (* b b)))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-6.0, (a * c), (-3.0 * (a * c)));
	double t_1 = pow((a * c), 2.0);
	double t_2 = fma(9.0, t_1, (18.0 * t_1)) - (0.25 * pow(t_0, 2.0));
	double t_3 = (-27.0 * pow((a * c), 3.0)) - (0.5 * (t_0 * t_2));
	return ((b * fma(-0.5, (fma(0.25, pow(t_2, 2.0), (0.5 * (t_0 * t_3))) / pow(b, 6.0)), fma(0.5, t_0, fma(0.5, (t_3 / pow(b, 4.0)), (0.5 * (t_2 / (b * b))))))) / (fma(2.0, (b * b), (c * (fma(-3.0, a, (c * ((-1.6875 * ((pow(a, 3.0) * c) / pow(b, 4.0))) - (1.125 * ((a * a) / (b * b)))))) - (1.5 * a)))) + (b * b))) / (3.0 * a);
}

Julia

function code(a, b, c)
	t_0 = fma(-6.0, Float64(a * c), Float64(-3.0 * Float64(a * c)))
	t_1 = Float64(a * c) ^ 2.0
	t_2 = Float64(fma(9.0, t_1, Float64(18.0 * t_1)) - Float64(0.25 * (t_0 ^ 2.0)))
	t_3 = Float64(Float64(-27.0 * (Float64(a * c) ^ 3.0)) - Float64(0.5 * Float64(t_0 * t_2)))
	return Float64(Float64(Float64(b * fma(-0.5, Float64(fma(0.25, (t_2 ^ 2.0), Float64(0.5 * Float64(t_0 * t_3))) / (b ^ 6.0)), fma(0.5, t_0, fma(0.5, Float64(t_3 / (b ^ 4.0)), Float64(0.5 * Float64(t_2 / Float64(b * b))))))) / Float64(fma(2.0, Float64(b * b), Float64(c * Float64(fma(-3.0, a, Float64(c * Float64(Float64(-1.6875 * Float64(Float64((a ^ 3.0) * c) / (b ^ 4.0))) - Float64(1.125 * Float64(Float64(a * a) / Float64(b * b)))))) - Float64(1.5 * a)))) + Float64(b * b))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-6.0 * N[(a * c), $MachinePrecision] + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(9.0 * t$95$1 + N[(18.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-27.0 * N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(b * N[(-0.5 * N[(N[(0.25 * N[Power[t$95$2, 2.0], $MachinePrecision] + N[(0.5 * N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$0 + N[(0.5 * N[(t$95$3 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$2 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * N[(b * b), $MachinePrecision] + N[(c * N[(N[(-3.0 * a + N[(c * N[(N[(-1.6875 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.125 * N[(N[(a * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.5 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 55.1%

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

   1. flip3-+ N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 55.1%

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

5. Taylor expanded in b around inf

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

7. Applied rewrites 91.7%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(0.5, \frac{-27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)}{{b}^{4}}, 0.5 \cdot \frac{\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

8. Taylor expanded in c around 0

   \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{1}{4}, {\left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)}^{2}, \frac{1}{2} \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(\frac{1}{2}, \frac{-27 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)}{{b}^{4}}, \frac{1}{2} \cdot \frac{\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{\color{blue}{\left(2 \cdot {b}^{2} + c \cdot \left(\left(-3 \cdot a + c \cdot \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{4}} - \frac{9}{8} \cdot \frac{{a}^{2}}{{b}^{2}}\right)\right) - \frac{3}{2} \cdot a\right)\right) - -1 \cdot {b}^{2}}}}{3 \cdot a} \]

10. Applied rewrites 91.9%

    \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(0.5, \frac{-27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)}{{b}^{4}}, 0.5 \cdot \frac{\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(2, b \cdot b, c \cdot \left(\mathsf{fma}\left(-3, a, c \cdot \left(-1.6875 \cdot \frac{{a}^{3} \cdot c}{{b}^{4}} - 1.125 \cdot \frac{a \cdot a}{b \cdot b}\right)\right) - 1.5 \cdot a\right)\right) - -1 \cdot \left(b \cdot b\right)}}}{3 \cdot a} \]

11. Final simplification 91.9%

    \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(0.5, \frac{-27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)}{{b}^{4}}, 0.5 \cdot \frac{\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{\mathsf{fma}\left(2, b \cdot b, c \cdot \left(\mathsf{fma}\left(-3, a, c \cdot \left(-1.6875 \cdot \frac{{a}^{3} \cdot c}{{b}^{4}} - 1.125 \cdot \frac{a \cdot a}{b \cdot b}\right)\right) - 1.5 \cdot a\right)\right) + b \cdot b}}{3 \cdot a} \]
12. Add Preprocessing

Alternative 2: 91.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, a, -3 \cdot a\right)\\ t_1 := \mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {t\_0}^{2}\\ t_2 := -27 \cdot {a}^{3} - 0.5 \cdot \left(t\_0 \cdot t\_1\right)\\ t_3 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ \frac{\frac{b \cdot \left(c \cdot \mathsf{fma}\left(0.5, t\_0, c \cdot \mathsf{fma}\left(0.5, \frac{t\_1}{b \cdot b}, c \cdot \mathsf{fma}\left(-0.5, \frac{c \cdot \mathsf{fma}\left(0.25, {t\_1}^{2}, 0.5 \cdot \left(t\_0 \cdot t\_2\right)\right)}{{b}^{6}}, 0.5 \cdot \frac{t\_2}{{b}^{4}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, t\_3 \cdot t\_3 + b \cdot t\_3\right)}}{3 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -6.0 a (* -3.0 a)))
        (t_1 (- (fma 9.0 (* a a) (* 18.0 (* a a))) (* 0.25 (pow t_0 2.0))))
        (t_2 (- (* -27.0 (pow a 3.0)) (* 0.5 (* t_0 t_1))))
        (t_3 (sqrt (fma (* -3.0 a) c (* b b)))))
   (/
    (/
     (*
      b
      (*
       c
       (fma
        0.5
        t_0
        (*
         c
         (fma
          0.5
          (/ t_1 (* b b))
          (*
           c
           (fma
            -0.5
            (/ (* c (fma 0.25 (pow t_1 2.0) (* 0.5 (* t_0 t_2)))) (pow b 6.0))
            (* 0.5 (/ t_2 (pow b 4.0))))))))))
     (fma b b (+ (* t_3 t_3) (* b t_3))))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-6.0, a, (-3.0 * a));
	double t_1 = fma(9.0, (a * a), (18.0 * (a * a))) - (0.25 * pow(t_0, 2.0));
	double t_2 = (-27.0 * pow(a, 3.0)) - (0.5 * (t_0 * t_1));
	double t_3 = sqrt(fma((-3.0 * a), c, (b * b)));
	return ((b * (c * fma(0.5, t_0, (c * fma(0.5, (t_1 / (b * b)), (c * fma(-0.5, ((c * fma(0.25, pow(t_1, 2.0), (0.5 * (t_0 * t_2)))) / pow(b, 6.0)), (0.5 * (t_2 / pow(b, 4.0)))))))))) / fma(b, b, ((t_3 * t_3) + (b * t_3)))) / (3.0 * a);
}

Julia

function code(a, b, c)
	t_0 = fma(-6.0, a, Float64(-3.0 * a))
	t_1 = Float64(fma(9.0, Float64(a * a), Float64(18.0 * Float64(a * a))) - Float64(0.25 * (t_0 ^ 2.0)))
	t_2 = Float64(Float64(-27.0 * (a ^ 3.0)) - Float64(0.5 * Float64(t_0 * t_1)))
	t_3 = sqrt(fma(Float64(-3.0 * a), c, Float64(b * b)))
	return Float64(Float64(Float64(b * Float64(c * fma(0.5, t_0, Float64(c * fma(0.5, Float64(t_1 / Float64(b * b)), Float64(c * fma(-0.5, Float64(Float64(c * fma(0.25, (t_1 ^ 2.0), Float64(0.5 * Float64(t_0 * t_2)))) / (b ^ 6.0)), Float64(0.5 * Float64(t_2 / (b ^ 4.0)))))))))) / fma(b, b, Float64(Float64(t_3 * t_3) + Float64(b * t_3)))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-6.0 * a + N[(-3.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(9.0 * N[(a * a), $MachinePrecision] + N[(18.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-27.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(b * N[(c * N[(0.5 * t$95$0 + N[(c * N[(0.5 * N[(t$95$1 / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * N[(-0.5 * N[(N[(c * N[(0.25 * N[Power[t$95$1, 2.0], $MachinePrecision] + N[(0.5 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$2 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(t$95$3 * t$95$3), $MachinePrecision] + N[(b * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 55.1%

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

   1. flip3-+ N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 55.1%

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

5. Taylor expanded in b around inf

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

7. Applied rewrites 91.7%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(0.5, \frac{-27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)}{{b}^{4}}, 0.5 \cdot \frac{\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

8. Taylor expanded in c around 0

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

10. Applied rewrites 91.8%

    \[\leadsto \frac{\frac{b \cdot \left(c \cdot \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(-6, a, -3 \cdot a\right), c \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}}{b \cdot b}, c \cdot \mathsf{fma}\left(-0.5, \frac{c \cdot \mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(-27 \cdot {a}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{6}}, 0.5 \cdot \frac{-27 \cdot {a}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)\right)}{{b}^{4}}\right)\right)\right)}\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

11. Final simplification 91.8%

    \[\leadsto \frac{\frac{b \cdot \left(c \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(-6, a, -3 \cdot a\right), c \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}}{b \cdot b}, c \cdot \mathsf{fma}\left(-0.5, \frac{c \cdot \mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(-27 \cdot {a}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{6}}, 0.5 \cdot \frac{-27 \cdot {a}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)\right)}{{b}^{4}}\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]
12. Add Preprocessing

Alternative 3: 91.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ t_1 := \mathsf{fma}\left(-6, a, -3 \cdot a\right)\\ t_2 := \mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {t\_1}^{2}\\ t_3 := -27 \cdot {a}^{3} - 0.5 \cdot \left(t\_1 \cdot t\_2\right)\\ \frac{\frac{c \cdot \mathsf{fma}\left(0.5, b \cdot t\_1, c \cdot \mathsf{fma}\left(0.5, \frac{t\_2}{b}, c \cdot \mathsf{fma}\left(-0.5, \frac{c \cdot \mathsf{fma}\left(0.25, {t\_2}^{2}, 0.5 \cdot \left(t\_1 \cdot t\_3\right)\right)}{{b}^{5}}, 0.5 \cdot \frac{t\_3}{{b}^{3}}\right)\right)\right)}{\mathsf{fma}\left(b, b, t\_0 \cdot t\_0 + b \cdot t\_0\right)}}{3 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -3.0 a) c (* b b))))
        (t_1 (fma -6.0 a (* -3.0 a)))
        (t_2 (- (fma 9.0 (* a a) (* 18.0 (* a a))) (* 0.25 (pow t_1 2.0))))
        (t_3 (- (* -27.0 (pow a 3.0)) (* 0.5 (* t_1 t_2)))))
   (/
    (/
     (*
      c
      (fma
       0.5
       (* b t_1)
       (*
        c
        (fma
         0.5
         (/ t_2 b)
         (*
          c
          (fma
           -0.5
           (/ (* c (fma 0.25 (pow t_2 2.0) (* 0.5 (* t_1 t_3)))) (pow b 5.0))
           (* 0.5 (/ t_3 (pow b 3.0)))))))))
     (fma b b (+ (* t_0 t_0) (* b t_0))))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-3.0 * a), c, (b * b)));
	double t_1 = fma(-6.0, a, (-3.0 * a));
	double t_2 = fma(9.0, (a * a), (18.0 * (a * a))) - (0.25 * pow(t_1, 2.0));
	double t_3 = (-27.0 * pow(a, 3.0)) - (0.5 * (t_1 * t_2));
	return ((c * fma(0.5, (b * t_1), (c * fma(0.5, (t_2 / b), (c * fma(-0.5, ((c * fma(0.25, pow(t_2, 2.0), (0.5 * (t_1 * t_3)))) / pow(b, 5.0)), (0.5 * (t_3 / pow(b, 3.0))))))))) / fma(b, b, ((t_0 * t_0) + (b * t_0)))) / (3.0 * a);
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-3.0 * a), c, Float64(b * b)))
	t_1 = fma(-6.0, a, Float64(-3.0 * a))
	t_2 = Float64(fma(9.0, Float64(a * a), Float64(18.0 * Float64(a * a))) - Float64(0.25 * (t_1 ^ 2.0)))
	t_3 = Float64(Float64(-27.0 * (a ^ 3.0)) - Float64(0.5 * Float64(t_1 * t_2)))
	return Float64(Float64(Float64(c * fma(0.5, Float64(b * t_1), Float64(c * fma(0.5, Float64(t_2 / b), Float64(c * fma(-0.5, Float64(Float64(c * fma(0.25, (t_2 ^ 2.0), Float64(0.5 * Float64(t_1 * t_3)))) / (b ^ 5.0)), Float64(0.5 * Float64(t_3 / (b ^ 3.0))))))))) / fma(b, b, Float64(Float64(t_0 * t_0) + Float64(b * t_0)))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-6.0 * a + N[(-3.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(9.0 * N[(a * a), $MachinePrecision] + N[(18.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-27.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(c * N[(0.5 * N[(b * t$95$1), $MachinePrecision] + N[(c * N[(0.5 * N[(t$95$2 / b), $MachinePrecision] + N[(c * N[(-0.5 * N[(N[(c * N[(0.25 * N[Power[t$95$2, 2.0], $MachinePrecision] + N[(0.5 * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$3 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 55.1%

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

   1. flip3-+ N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 55.1%

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

5. Taylor expanded in b around inf

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

7. Applied rewrites 91.7%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(0.5, \frac{-27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)}{{b}^{4}}, 0.5 \cdot \frac{\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

8. Taylor expanded in c around 0

   \[\leadsto \frac{\frac{c \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(b \cdot \left(-6 \cdot a + -3 \cdot a\right)\right) + c \cdot \left(\frac{1}{2} \cdot \frac{\left(9 \cdot {a}^{2} + 18 \cdot {a}^{2}\right) - \frac{1}{4} \cdot {\left(-6 \cdot a + -3 \cdot a\right)}^{2}}{b} + c \cdot \left(\frac{-1}{2} \cdot \frac{c \cdot \left(\frac{1}{4} \cdot {\left(\left(9 \cdot {a}^{2} + 18 \cdot {a}^{2}\right) - \frac{1}{4} \cdot {\left(-6 \cdot a + -3 \cdot a\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-6 \cdot a + -3 \cdot a\right) \cdot \left(-27 \cdot {a}^{3} - \frac{1}{2} \cdot \left(\left(-6 \cdot a + -3 \cdot a\right) \cdot \left(\left(9 \cdot {a}^{2} + 18 \cdot {a}^{2}\right) - \frac{1}{4} \cdot {\left(-6 \cdot a + -3 \cdot a\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{5}} + \frac{1}{2} \cdot \frac{-27 \cdot {a}^{3} - \frac{1}{2} \cdot \left(\left(-6 \cdot a + -3 \cdot a\right) \cdot \left(\left(9 \cdot {a}^{2} + 18 \cdot {a}^{2}\right) - \frac{1}{4} \cdot {\left(-6 \cdot a + -3 \cdot a\right)}^{2}\right)\right)}{{b}^{3}}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

10. Applied rewrites 91.7%

    \[\leadsto \frac{\frac{c \cdot \color{blue}{\mathsf{fma}\left(0.5, b \cdot \mathsf{fma}\left(-6, a, -3 \cdot a\right), c \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}}{b}, c \cdot \mathsf{fma}\left(-0.5, \frac{c \cdot \mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(-27 \cdot {a}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{5}}, 0.5 \cdot \frac{-27 \cdot {a}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)\right)}{{b}^{3}}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

11. Final simplification 91.7%

    \[\leadsto \frac{\frac{c \cdot \mathsf{fma}\left(0.5, b \cdot \mathsf{fma}\left(-6, a, -3 \cdot a\right), c \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}}{b}, c \cdot \mathsf{fma}\left(-0.5, \frac{c \cdot \mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(-27 \cdot {a}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{5}}, 0.5 \cdot \frac{-27 \cdot {a}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)\right)}{{b}^{3}}\right)\right)\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]
12. Add Preprocessing

Alternative 4: 91.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ t_1 := \mathsf{fma}\left(-6, c, -3 \cdot c\right)\\ t_2 := \mathsf{fma}\left(9, c \cdot c, 18 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {t\_1}^{2}\\ t_3 := -27 \cdot {c}^{3} - 0.5 \cdot \left(t\_1 \cdot t\_2\right)\\ \frac{\frac{a \cdot \mathsf{fma}\left(0.5, b \cdot t\_1, a \cdot \mathsf{fma}\left(0.5, \frac{t\_2}{b}, a \cdot \mathsf{fma}\left(-0.5, \frac{a \cdot \mathsf{fma}\left(0.25, {t\_2}^{2}, 0.5 \cdot \left(t\_1 \cdot t\_3\right)\right)}{{b}^{5}}, 0.5 \cdot \frac{t\_3}{{b}^{3}}\right)\right)\right)}{\mathsf{fma}\left(b, b, t\_0 \cdot t\_0 + b \cdot t\_0\right)}}{3 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -3.0 a) c (* b b))))
        (t_1 (fma -6.0 c (* -3.0 c)))
        (t_2 (- (fma 9.0 (* c c) (* 18.0 (* c c))) (* 0.25 (pow t_1 2.0))))
        (t_3 (- (* -27.0 (pow c 3.0)) (* 0.5 (* t_1 t_2)))))
   (/
    (/
     (*
      a
      (fma
       0.5
       (* b t_1)
       (*
        a
        (fma
         0.5
         (/ t_2 b)
         (*
          a
          (fma
           -0.5
           (/ (* a (fma 0.25 (pow t_2 2.0) (* 0.5 (* t_1 t_3)))) (pow b 5.0))
           (* 0.5 (/ t_3 (pow b 3.0)))))))))
     (fma b b (+ (* t_0 t_0) (* b t_0))))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-3.0 * a), c, (b * b)));
	double t_1 = fma(-6.0, c, (-3.0 * c));
	double t_2 = fma(9.0, (c * c), (18.0 * (c * c))) - (0.25 * pow(t_1, 2.0));
	double t_3 = (-27.0 * pow(c, 3.0)) - (0.5 * (t_1 * t_2));
	return ((a * fma(0.5, (b * t_1), (a * fma(0.5, (t_2 / b), (a * fma(-0.5, ((a * fma(0.25, pow(t_2, 2.0), (0.5 * (t_1 * t_3)))) / pow(b, 5.0)), (0.5 * (t_3 / pow(b, 3.0))))))))) / fma(b, b, ((t_0 * t_0) + (b * t_0)))) / (3.0 * a);
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-3.0 * a), c, Float64(b * b)))
	t_1 = fma(-6.0, c, Float64(-3.0 * c))
	t_2 = Float64(fma(9.0, Float64(c * c), Float64(18.0 * Float64(c * c))) - Float64(0.25 * (t_1 ^ 2.0)))
	t_3 = Float64(Float64(-27.0 * (c ^ 3.0)) - Float64(0.5 * Float64(t_1 * t_2)))
	return Float64(Float64(Float64(a * fma(0.5, Float64(b * t_1), Float64(a * fma(0.5, Float64(t_2 / b), Float64(a * fma(-0.5, Float64(Float64(a * fma(0.25, (t_2 ^ 2.0), Float64(0.5 * Float64(t_1 * t_3)))) / (b ^ 5.0)), Float64(0.5 * Float64(t_3 / (b ^ 3.0))))))))) / fma(b, b, Float64(Float64(t_0 * t_0) + Float64(b * t_0)))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-6.0 * c + N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(9.0 * N[(c * c), $MachinePrecision] + N[(18.0 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-27.0 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(a * N[(0.5 * N[(b * t$95$1), $MachinePrecision] + N[(a * N[(0.5 * N[(t$95$2 / b), $MachinePrecision] + N[(a * N[(-0.5 * N[(N[(a * N[(0.25 * N[Power[t$95$2, 2.0], $MachinePrecision] + N[(0.5 * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$3 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 55.1%

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

   1. flip3-+ N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 55.1%

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

5. Taylor expanded in b around inf

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

7. Applied rewrites 91.7%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(0.5, \frac{-27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)}{{b}^{4}}, 0.5 \cdot \frac{\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

8. Taylor expanded in a around 0

   \[\leadsto \frac{\frac{a \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(b \cdot \left(-6 \cdot c + -3 \cdot c\right)\right) + a \cdot \left(\frac{1}{2} \cdot \frac{\left(9 \cdot {c}^{2} + 18 \cdot {c}^{2}\right) - \frac{1}{4} \cdot {\left(-6 \cdot c + -3 \cdot c\right)}^{2}}{b} + a \cdot \left(\frac{-1}{2} \cdot \frac{a \cdot \left(\frac{1}{4} \cdot {\left(\left(9 \cdot {c}^{2} + 18 \cdot {c}^{2}\right) - \frac{1}{4} \cdot {\left(-6 \cdot c + -3 \cdot c\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-6 \cdot c + -3 \cdot c\right) \cdot \left(-27 \cdot {c}^{3} - \frac{1}{2} \cdot \left(\left(-6 \cdot c + -3 \cdot c\right) \cdot \left(\left(9 \cdot {c}^{2} + 18 \cdot {c}^{2}\right) - \frac{1}{4} \cdot {\left(-6 \cdot c + -3 \cdot c\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{5}} + \frac{1}{2} \cdot \frac{-27 \cdot {c}^{3} - \frac{1}{2} \cdot \left(\left(-6 \cdot c + -3 \cdot c\right) \cdot \left(\left(9 \cdot {c}^{2} + 18 \cdot {c}^{2}\right) - \frac{1}{4} \cdot {\left(-6 \cdot c + -3 \cdot c\right)}^{2}\right)\right)}{{b}^{3}}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

10. Applied rewrites 91.6%

    \[\leadsto \frac{\frac{a \cdot \color{blue}{\mathsf{fma}\left(0.5, b \cdot \mathsf{fma}\left(-6, c, -3 \cdot c\right), a \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(9, c \cdot c, 18 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, c, -3 \cdot c\right)\right)}^{2}}{b}, a \cdot \mathsf{fma}\left(-0.5, \frac{a \cdot \mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, c \cdot c, 18 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, c, -3 \cdot c\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, c, -3 \cdot c\right) \cdot \left(-27 \cdot {c}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, c, -3 \cdot c\right) \cdot \left(\mathsf{fma}\left(9, c \cdot c, 18 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, c, -3 \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{5}}, 0.5 \cdot \frac{-27 \cdot {c}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, c, -3 \cdot c\right) \cdot \left(\mathsf{fma}\left(9, c \cdot c, 18 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, c, -3 \cdot c\right)\right)}^{2}\right)\right)}{{b}^{3}}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

11. Final simplification 91.6%

    \[\leadsto \frac{\frac{a \cdot \mathsf{fma}\left(0.5, b \cdot \mathsf{fma}\left(-6, c, -3 \cdot c\right), a \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(9, c \cdot c, 18 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, c, -3 \cdot c\right)\right)}^{2}}{b}, a \cdot \mathsf{fma}\left(-0.5, \frac{a \cdot \mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, c \cdot c, 18 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, c, -3 \cdot c\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, c, -3 \cdot c\right) \cdot \left(-27 \cdot {c}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, c, -3 \cdot c\right) \cdot \left(\mathsf{fma}\left(9, c \cdot c, 18 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, c, -3 \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{5}}, 0.5 \cdot \frac{-27 \cdot {c}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, c, -3 \cdot c\right) \cdot \left(\mathsf{fma}\left(9, c \cdot c, 18 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, c, -3 \cdot c\right)\right)}^{2}\right)\right)}{{b}^{3}}\right)\right)\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]
12. Add Preprocessing

Alternative 5: 90.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 91.3%

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

6. Taylor expanded in c around 0

   \[\leadsto \mathsf{fma}\left({c}^{2} \cdot \left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right), a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

8. Applied rewrites 91.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{b}^{5}}, -0.5625, \frac{-1.0546875 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{7}}\right), c, -0.375 \cdot {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right) \]

9. Taylor expanded in b around 0

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

    1. lower-/.f64 N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. pow2 N/A

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

    5. lower-*.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. pow2 N/A

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

    9. lower-*.f64 N/A

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

    10. lower-pow.f64 91.3

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

11. Applied rewrites 91.3%

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

Alternative 6: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq 7.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {t\_0}^{1.5}\right)}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 + b \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5625 \cdot \frac{a \cdot c}{b \cdot b} - 0.375}{{b}^{3}} \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= b 7.5)
     (/
      (/ (fma (* b b) (- b) (pow t_0 1.5)) (fma b b (+ (* t_1 t_1) (* b t_1))))
      (* 3.0 a))
     (fma
      (* (/ (- (* -0.5625 (/ (* a c) (* b b))) 0.375) (pow b 3.0)) (* c c))
      a
      (* (/ c b) -0.5)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (b <= 7.5) {
		tmp = (fma((b * b), -b, pow(t_0, 1.5)) / fma(b, b, ((t_1 * t_1) + (b * t_1)))) / (3.0 * a);
	} else {
		tmp = fma(((((-0.5625 * ((a * c) / (b * b))) - 0.375) / pow(b, 3.0)) * (c * c)), a, ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (b <= 7.5)
		tmp = Float64(Float64(fma(Float64(b * b), Float64(-b), (t_0 ^ 1.5)) / fma(b, b, Float64(Float64(t_1 * t_1) + Float64(b * t_1)))) / Float64(3.0 * a));
	else
		tmp = fma(Float64(Float64(Float64(Float64(-0.5625 * Float64(Float64(a * c) / Float64(b * b))) - 0.375) / (b ^ 3.0)) * Float64(c * c)), a, Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[b, 7.5], N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.5625 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.375), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.5

   1. Initial program 81.1%

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

      1. flip3-+ N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 81.3%

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

      1. unpow3 N/A

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

      2. sqr-neg-rev N/A

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. sqrt-pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 82.2

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

   6. Applied rewrites 82.2%

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

   if 7.5 < b

   1. Initial program 47.4%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left({c}^{2} \cdot \left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right), a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

   8. Applied rewrites 95.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{b}^{5}}, -0.5625, \frac{-1.0546875 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{7}}\right), c, -0.375 \cdot {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right) \]

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-pow.f64 93.5

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

   11. Applied rewrites 93.5%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {\left(\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5625 \cdot \frac{a \cdot c}{b \cdot b} - 0.375}{{b}^{3}} \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5625 \cdot \frac{a \cdot c}{b \cdot b} - 0.375}{{b}^{3}} \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.8)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (fma
    (* (/ (- (* -0.5625 (/ (* a c) (* b b))) 0.375) (pow b 3.0)) (* c c))
    a
    (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.8) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma(((((-0.5625 * ((a * c) / (b * b))) - 0.375) / pow(b, 3.0)) * (c * c)), a, ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.8)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = fma(Float64(Float64(Float64(Float64(-0.5625 * Float64(Float64(a * c) / Float64(b * b))) - 0.375) / (b ^ 3.0)) * Float64(c * c)), a, Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.8], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.5625 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.375), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.79999999999999982

   1. Initial program 81.2%

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

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

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.4

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

   4. Applied rewrites 81.4%

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

   if 4.79999999999999982 < b

   1. Initial program 47.9%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left({c}^{2} \cdot \left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right), a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

   8. Applied rewrites 95.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{b}^{5}}, -0.5625, \frac{-1.0546875 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{7}}\right), c, -0.375 \cdot {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right) \]

   9. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-pow.f64 93.3

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

   11. Applied rewrites 93.3%

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

Alternative 8: 85.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, -0.375, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (fma (/ (* (* c c) a) (pow b 3.0)) -0.375 (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma((((c * c) * a) / pow(b, 3.0)), -0.375, ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = fma(Float64(Float64(Float64(c * c) * a) / (b ^ 3.0)), -0.375, Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375 + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8

   1. Initial program 81.0%

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

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

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.1

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

   4. Applied rewrites 81.1%

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

   if 8 < b

   1. Initial program 47.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 88.6

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

   5. Applied rewrites 88.6%

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

Alternative 9: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -6e-7)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 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)) <= -6e-7) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (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)) <= -6e-7)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / 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], -6e-7], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(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)) < -5.9999999999999997e-7

   1. Initial program 71.0%

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

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

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 71.1

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

   4. Applied rewrites 71.1%

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

   if -5.9999999999999997e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 29.8%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

Alternative 10: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (fma (/ (* (* c c) a) (* b b)) -0.375 (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma((((c * c) * a) / (b * b)), -0.375, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(c * c) * a) / Float64(b * b)), -0.375, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8

   1. Initial program 81.0%

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

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

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.1

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

   4. Applied rewrites 81.1%

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

   if 8 < b

   1. Initial program 47.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

Alternative 11: 64.0% 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 55.1%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.7

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

5. Applied rewrites 64.7%

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

Reproduce

herbie shell --seed 2025043 
(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)))