Cubic critical, narrow range

Percentage Accurate: 54.7% → 92.2%

Time: 8.8s

Alternatives: 13

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, a, -3 \cdot a\right)\\ t_1 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_2 := \sqrt{t\_1}\\ t_3 := \mathsf{fma}\left(b, b, t\_2 \cdot t\_2 + b \cdot t\_2\right)\\ t_4 := \mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {t\_0}^{2}\\ t_5 := -27 \cdot {a}^{3} - 0.5 \cdot \left(t\_0 \cdot t\_4\right)\\ \mathbf{if}\;b \leq 0.22:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + t\_1 \cdot t\_2}{t\_3}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \mathsf{fma}\left(0.5, b \cdot t\_0, c \cdot \mathsf{fma}\left(0.5, \frac{t\_4}{b}, c \cdot \mathsf{fma}\left(-0.5, \frac{c \cdot \mathsf{fma}\left(0.25, {t\_4}^{2}, 0.5 \cdot \left(t\_0 \cdot t\_5\right)\right)}{{b}^{5}}, 0.5 \cdot \frac{t\_5}{{b}^{3}}\right)\right)\right)}{t\_3}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -6.0 a (* -3.0 a)))
        (t_1 (fma (* -3.0 a) c (* b b)))
        (t_2 (sqrt t_1))
        (t_3 (fma b b (+ (* t_2 t_2) (* b t_2))))
        (t_4 (- (fma 9.0 (* a a) (* 18.0 (* a a))) (* 0.25 (pow t_0 2.0))))
        (t_5 (- (* -27.0 (pow a 3.0)) (* 0.5 (* t_0 t_4)))))
   (if (<= b 0.22)
     (/ (/ (+ (pow (- b) 3.0) (* t_1 t_2)) t_3) (* 3.0 a))
     (/
      (/
       (*
        c
        (fma
         0.5
         (* b t_0)
         (*
          c
          (fma
           0.5
           (/ t_4 b)
           (*
            c
            (fma
             -0.5
             (/ (* c (fma 0.25 (pow t_4 2.0) (* 0.5 (* t_0 t_5)))) (pow b 5.0))
             (* 0.5 (/ t_5 (pow b 3.0)))))))))
       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((-3.0 * a), c, (b * b));
	double t_2 = sqrt(t_1);
	double t_3 = fma(b, b, ((t_2 * t_2) + (b * t_2)));
	double t_4 = fma(9.0, (a * a), (18.0 * (a * a))) - (0.25 * pow(t_0, 2.0));
	double t_5 = (-27.0 * pow(a, 3.0)) - (0.5 * (t_0 * t_4));
	double tmp;
	if (b <= 0.22) {
		tmp = ((pow(-b, 3.0) + (t_1 * t_2)) / t_3) / (3.0 * a);
	} else {
		tmp = ((c * fma(0.5, (b * t_0), (c * fma(0.5, (t_4 / b), (c * fma(-0.5, ((c * fma(0.25, pow(t_4, 2.0), (0.5 * (t_0 * t_5)))) / pow(b, 5.0)), (0.5 * (t_5 / pow(b, 3.0))))))))) / t_3) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-6.0, a, Float64(-3.0 * a))
	t_1 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_2 = sqrt(t_1)
	t_3 = fma(b, b, Float64(Float64(t_2 * t_2) + Float64(b * t_2)))
	t_4 = Float64(fma(9.0, Float64(a * a), Float64(18.0 * Float64(a * a))) - Float64(0.25 * (t_0 ^ 2.0)))
	t_5 = Float64(Float64(-27.0 * (a ^ 3.0)) - Float64(0.5 * Float64(t_0 * t_4)))
	tmp = 0.0
	if (b <= 0.22)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + Float64(t_1 * t_2)) / t_3) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(c * fma(0.5, Float64(b * t_0), Float64(c * fma(0.5, Float64(t_4 / b), Float64(c * fma(-0.5, Float64(Float64(c * fma(0.25, (t_4 ^ 2.0), Float64(0.5 * Float64(t_0 * t_5)))) / (b ^ 5.0)), Float64(0.5 * Float64(t_5 / (b ^ 3.0))))))))) / t_3) / Float64(3.0 * a));
	end
	return tmp
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[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(b * b + N[(N[(t$95$2 * t$95$2), $MachinePrecision] + N[(b * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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$5 = N[(N[(-27.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(t$95$0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.22], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(0.5 * N[(b * t$95$0), $MachinePrecision] + N[(c * N[(0.5 * N[(t$95$4 / b), $MachinePrecision] + N[(c * N[(-0.5 * N[(N[(c * N[(0.25 * N[Power[t$95$4, 2.0], $MachinePrecision] + N[(0.5 * N[(t$95$0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$5 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.220000000000000001

   1. Initial program 86.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 85.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. unpow3 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sqrt.f64 86.8

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

   6. Applied rewrites 86.8%

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

   if 0.220000000000000001 < b

   1. Initial program 51.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 51.1%

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

   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 93.0%

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

   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 93.1%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.22:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot \mathsf{fma}\left(0.5, b \cdot \mathsf{fma}\left(-6, a, -3 \cdot a\right), c \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}}{b}, c \cdot \mathsf{fma}\left(-0.5, \frac{c \cdot \mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(-27 \cdot {a}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{5}}, 0.5 \cdot \frac{-27 \cdot {a}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a, -3 \cdot a\right) \cdot \left(\mathsf{fma}\left(9, a \cdot a, 18 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a, -3 \cdot a\right)\right)}^{2}\right)\right)}{{b}^{3}}\right)\right)\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, c, -3 \cdot c\right)\\ t_1 := \mathsf{fma}\left(9, c \cdot c, 18 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {t\_0}^{2}\\ t_2 := -27 \cdot {c}^{3} - 0.5 \cdot \left(t\_0 \cdot t\_1\right)\\ t_3 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_4 := \sqrt{t\_3}\\ t_5 := \mathsf{fma}\left(b, b, t\_4 \cdot t\_4 + b \cdot t\_4\right)\\ \mathbf{if}\;b \leq 0.22:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + t\_3 \cdot t\_4}{t\_5}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot \mathsf{fma}\left(0.5, b \cdot t\_0, a \cdot \mathsf{fma}\left(0.5, \frac{t\_1}{b}, a \cdot \mathsf{fma}\left(-0.5, \frac{a \cdot \mathsf{fma}\left(0.25, {t\_1}^{2}, 0.5 \cdot \left(t\_0 \cdot t\_2\right)\right)}{{b}^{5}}, 0.5 \cdot \frac{t\_2}{{b}^{3}}\right)\right)\right)}{t\_5}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -6.0 c (* -3.0 c)))
        (t_1 (- (fma 9.0 (* c c) (* 18.0 (* c c))) (* 0.25 (pow t_0 2.0))))
        (t_2 (- (* -27.0 (pow c 3.0)) (* 0.5 (* t_0 t_1))))
        (t_3 (fma (* -3.0 a) c (* b b)))
        (t_4 (sqrt t_3))
        (t_5 (fma b b (+ (* t_4 t_4) (* b t_4)))))
   (if (<= b 0.22)
     (/ (/ (+ (pow (- b) 3.0) (* t_3 t_4)) t_5) (* 3.0 a))
     (/
      (/
       (*
        a
        (fma
         0.5
         (* b t_0)
         (*
          a
          (fma
           0.5
           (/ t_1 b)
           (*
            a
            (fma
             -0.5
             (/ (* a (fma 0.25 (pow t_1 2.0) (* 0.5 (* t_0 t_2)))) (pow b 5.0))
             (* 0.5 (/ t_2 (pow b 3.0)))))))))
       t_5)
      (* 3.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-6.0, c, (-3.0 * c));
	double t_1 = fma(9.0, (c * c), (18.0 * (c * c))) - (0.25 * pow(t_0, 2.0));
	double t_2 = (-27.0 * pow(c, 3.0)) - (0.5 * (t_0 * t_1));
	double t_3 = fma((-3.0 * a), c, (b * b));
	double t_4 = sqrt(t_3);
	double t_5 = fma(b, b, ((t_4 * t_4) + (b * t_4)));
	double tmp;
	if (b <= 0.22) {
		tmp = ((pow(-b, 3.0) + (t_3 * t_4)) / t_5) / (3.0 * a);
	} else {
		tmp = ((a * fma(0.5, (b * t_0), (a * fma(0.5, (t_1 / b), (a * fma(-0.5, ((a * fma(0.25, pow(t_1, 2.0), (0.5 * (t_0 * t_2)))) / pow(b, 5.0)), (0.5 * (t_2 / pow(b, 3.0))))))))) / t_5) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-6.0, c, Float64(-3.0 * c))
	t_1 = Float64(fma(9.0, Float64(c * c), Float64(18.0 * Float64(c * c))) - Float64(0.25 * (t_0 ^ 2.0)))
	t_2 = Float64(Float64(-27.0 * (c ^ 3.0)) - Float64(0.5 * Float64(t_0 * t_1)))
	t_3 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_4 = sqrt(t_3)
	t_5 = fma(b, b, Float64(Float64(t_4 * t_4) + Float64(b * t_4)))
	tmp = 0.0
	if (b <= 0.22)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + Float64(t_3 * t_4)) / t_5) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(a * fma(0.5, Float64(b * t_0), Float64(a * fma(0.5, Float64(t_1 / b), Float64(a * fma(-0.5, Float64(Float64(a * fma(0.25, (t_1 ^ 2.0), Float64(0.5 * Float64(t_0 * t_2)))) / (b ^ 5.0)), Float64(0.5 * Float64(t_2 / (b ^ 3.0))))))))) / t_5) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-6.0 * c + N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(9.0 * N[(c * c), $MachinePrecision] + N[(18.0 * N[(c * c), $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[c, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[(b * b + N[(N[(t$95$4 * t$95$4), $MachinePrecision] + N[(b * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.22], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(0.5 * N[(b * t$95$0), $MachinePrecision] + N[(a * N[(0.5 * N[(t$95$1 / b), $MachinePrecision] + N[(a * N[(-0.5 * N[(N[(a * 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, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$2 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.220000000000000001

   1. Initial program 86.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 85.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. unpow3 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sqrt.f64 86.8

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

   6. Applied rewrites 86.8%

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

   if 0.220000000000000001 < b

   1. Initial program 51.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 51.1%

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

   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 93.0%

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

   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 93.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.22:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot \mathsf{fma}\left(0.5, b \cdot \mathsf{fma}\left(-6, c, -3 \cdot c\right), a \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(9, c \cdot c, 18 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, c, -3 \cdot c\right)\right)}^{2}}{b}, a \cdot \mathsf{fma}\left(-0.5, \frac{a \cdot \mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, c \cdot c, 18 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, c, -3 \cdot c\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, c, -3 \cdot c\right) \cdot \left(-27 \cdot {c}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, c, -3 \cdot c\right) \cdot \left(\mathsf{fma}\left(9, c \cdot c, 18 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, c, -3 \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{5}}, 0.5 \cdot \frac{-27 \cdot {c}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, c, -3 \cdot c\right) \cdot \left(\mathsf{fma}\left(9, c \cdot c, 18 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, c, -3 \cdot c\right)\right)}^{2}\right)\right)}{{b}^{3}}\right)\right)\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq 0.66:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + t\_0 \cdot t\_1}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 + b \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= b 0.66)
     (/
      (/ (+ (pow (- b) 3.0) (* t_0 t_1)) (fma b b (+ (* t_1 t_1) (* b t_1))))
      (* 3.0 a))
     (/
      (fma
       (/ (* (* a a) (pow c 3.0)) (pow b 4.0))
       -0.5625
       (fma
        -0.5
        c
        (fma
         (/ (/ (* (pow (* c a) 4.0) 6.328125) a) (pow b 6.0))
         -0.16666666666666666
         (/ (* -0.375 (* (* c c) a)) (* b b)))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (b <= 0.66) {
		tmp = ((pow(-b, 3.0) + (t_0 * t_1)) / fma(b, b, ((t_1 * t_1) + (b * t_1)))) / (3.0 * a);
	} else {
		tmp = fma((((a * a) * pow(c, 3.0)) / pow(b, 4.0)), -0.5625, fma(-0.5, c, fma((((pow((c * a), 4.0) * 6.328125) / a) / pow(b, 6.0)), -0.16666666666666666, ((-0.375 * ((c * c) * a)) / (b * b))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (b <= 0.66)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + Float64(t_0 * t_1)) / fma(b, b, Float64(Float64(t_1 * t_1) + Float64(b * t_1)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(a * a) * (c ^ 3.0)) / (b ^ 4.0)), -0.5625, fma(-0.5, c, fma(Float64(Float64(Float64((Float64(c * a) ^ 4.0) * 6.328125) / a) / (b ^ 6.0)), -0.16666666666666666, Float64(Float64(-0.375 * Float64(Float64(c * c) * a)) / Float64(b * b))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[b, 0.66], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[(t$95$0 * t$95$1), $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[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.5 * c + N[(N[(N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision] / a), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(N[(-0.375 * N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.660000000000000031

   1. Initial program 85.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 85.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. unpow3 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sqrt.f64 86.6

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

   6. Applied rewrites 86.6%

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

   if 0.660000000000000031 < b

   1. Initial program 50.4%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.66:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq 0.66:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + t\_0 \cdot t\_1}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 + b \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875, \left(a \cdot a\right) \cdot c, -0.5625 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}{{b}^{7}}, c, -0.375 \cdot {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= b 0.66)
     (/
      (/ (+ (pow (- b) 3.0) (* t_0 t_1)) (fma b b (+ (* t_1 t_1) (* b t_1))))
      (* 3.0 a))
     (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) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (b <= 0.66) {
		tmp = ((pow(-b, 3.0) + (t_0 * t_1)) / fma(b, b, ((t_1 * t_1) + (b * t_1)))) / (3.0 * a);
	} else {
		tmp = 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));
	}
	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 <= 0.66)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + Float64(t_0 * t_1)) / fma(b, b, Float64(Float64(t_1 * t_1) + Float64(b * t_1)))) / Float64(3.0 * a));
	else
		tmp = 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
	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, 0.66], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[(t$95$0 * t$95$1), $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[(-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. Split input into 2 regimes

2. if b < 0.660000000000000031

   1. Initial program 85.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 85.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. unpow3 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sqrt.f64 86.6

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

   6. Applied rewrites 86.6%

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

   if 0.660000000000000031 < b

   1. Initial program 50.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 93.1%

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

   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 93.1%

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

   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. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.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. lift-*.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. lift-pow.f64 93.1

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

   11. Applied rewrites 93.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875, \left(a \cdot a\right) \cdot c, -0.5625 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}{{b}^{7}}, c, -0.375 \cdot {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.66:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875, \left(a \cdot a\right) \cdot c, -0.5625 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}{{b}^{7}}, c, -0.375 \cdot {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq 0.7:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + t\_0 \cdot t\_1}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 + b \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= b 0.7)
     (/
      (/ (+ (pow (- b) 3.0) (* t_0 t_1)) (fma b b (+ (* t_1 t_1) (* b t_1))))
      (* 3.0 a))
     (/
      (fma
       (/ (* (* a a) (pow c 3.0)) (pow b 4.0))
       -0.5625
       (fma (/ (* (* c c) a) (* b b)) -0.375 (* -0.5 c)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (b <= 0.7) {
		tmp = ((pow(-b, 3.0) + (t_0 * t_1)) / fma(b, b, ((t_1 * t_1) + (b * t_1)))) / (3.0 * a);
	} else {
		tmp = fma((((a * a) * pow(c, 3.0)) / pow(b, 4.0)), -0.5625, fma((((c * c) * a) / (b * b)), -0.375, (-0.5 * c))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (b <= 0.7)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + Float64(t_0 * t_1)) / fma(b, b, Float64(Float64(t_1 * t_1) + Float64(b * t_1)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(a * a) * (c ^ 3.0)) / (b ^ 4.0)), -0.5625, 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_] := 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, 0.7], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[(t$95$0 * t$95$1), $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[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.69999999999999996

   1. Initial program 85.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 85.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. unpow3 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-sqrt.f64 86.6

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

   6. Applied rewrites 86.6%

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

   if 0.69999999999999996 < b

   1. Initial program 50.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 90.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.7:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {t\_0}^{1.5}\right)}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 + b \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= b 1.4)
     (/
      (/ (fma (* b b) (- b) (pow t_0 1.5)) (fma b b (+ (* t_1 t_1) (* b t_1))))
      (* 3.0 a))
     (/
      (fma
       (/ (* (* a a) (pow c 3.0)) (pow b 4.0))
       -0.5625
       (fma (/ (* (* c c) a) (* b b)) -0.375 (* -0.5 c)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (b <= 1.4) {
		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((((a * a) * pow(c, 3.0)) / pow(b, 4.0)), -0.5625, fma((((c * c) * a) / (b * b)), -0.375, (-0.5 * c))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (b <= 1.4)
		tmp = Float64(Float64(fma(Float64(b * b), Float64(-b), (t_0 ^ 1.5)) / fma(b, b, Float64(Float64(t_1 * t_1) + Float64(b * t_1)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(a * a) * (c ^ 3.0)) / (b ^ 4.0)), -0.5625, 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_] := 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, 1.4], 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[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3999999999999999

   1. Initial program 85.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 84.8%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. sqr-neg-rev N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sqrt-pow2 N/A

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

      17. metadata-eval N/A

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

      18. lower-pow.f64 N/A

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

   6. Applied rewrites 86.2%

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

   if 1.3999999999999999 < b

   1. Initial program 50.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 90.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {\left(\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {t\_0}^{1.5}\right)}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 + b \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5625 \cdot \frac{a \cdot c}{b \cdot b} - 0.375}{{b}^{3}} \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= b 1.4)
     (/
      (/ (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 <= 1.4) {
		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 <= 1.4)
		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, 1.4], 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 < 1.3999999999999999

   1. Initial program 85.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 84.8%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. sqr-neg-rev N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sqrt-pow2 N/A

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

      17. metadata-eval N/A

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

      18. lower-pow.f64 N/A

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

   6. Applied rewrites 86.2%

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

   if 1.3999999999999999 < b

   1. Initial program 50.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 93.2%

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

   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 93.2%

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

   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. lift-*.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 90.8

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

   11. Applied rewrites 90.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {\left(\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5625 \cdot \frac{a \cdot c}{b \cdot b} - 0.375}{{b}^{3}} \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {t\_0}^{1.5}\right)}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 + b \cdot \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5625 \cdot \frac{a \cdot c}{b \cdot b} - 0.375}{{b}^{3}} \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= b 1.4)
     (/
      (/
       (fma (* b b) (- b) (pow t_0 1.5))
       (fma b b (+ (* t_1 t_1) (* b (sqrt (fma -3.0 (* a c) (* b b)))))))
      (* 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 <= 1.4) {
		tmp = (fma((b * b), -b, pow(t_0, 1.5)) / fma(b, b, ((t_1 * t_1) + (b * sqrt(fma(-3.0, (a * c), (b * b))))))) / (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 <= 1.4)
		tmp = Float64(Float64(fma(Float64(b * b), Float64(-b), (t_0 ^ 1.5)) / fma(b, b, Float64(Float64(t_1 * t_1) + Float64(b * sqrt(fma(-3.0, Float64(a * c), Float64(b * b))))))) / 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, 1.4], 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 * N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $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 < 1.3999999999999999

   1. Initial program 85.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 84.8%

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

   5. Taylor expanded in a around 0

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

      1. rem-square-sqrt N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 84.7

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

   7. Applied rewrites 84.7%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. sqr-neg-rev N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sqrt-pow2 N/A

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

      17. rem-square-sqrt N/A

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

      18. metadata-eval N/A

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

      19. lower-pow.f64 N/A

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

   9. Applied rewrites 86.2%

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

   if 1.3999999999999999 < b

   1. Initial program 50.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 93.2%

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

   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 93.2%

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

   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. lift-*.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 90.8

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

   11. Applied rewrites 90.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {\left(\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5625 \cdot \frac{a \cdot c}{b \cdot b} - 0.375}{{b}^{3}} \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.7:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5625 \cdot \frac{a \cdot c}{b \cdot b} - 0.375}{{b}^{3}} \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.7)
   (/ (+ (- 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 <= 0.7) {
		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 <= 0.7)
		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, 0.7], 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[(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 < 0.69999999999999996

   1. Initial program 85.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 85.8

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

   4. Applied rewrites 85.8%

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

   if 0.69999999999999996 < b

   1. Initial program 50.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 93.1%

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

   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 93.1%

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

   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. lift-*.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 90.7

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

   11. Applied rewrites 90.7%

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

Alternative 10: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 16:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 16.0)
   (/ (+ (- 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 <= 16.0) {
		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 <= 16.0)
		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, 16.0], 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 < 16

   1. Initial program 81.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.0

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

   4. Applied rewrites 82.0%

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

   if 16 < 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. lift-*.f64 N/A

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

      12. lower-*.f64 87.5

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

   5. Applied rewrites 87.5%

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

Alternative 11: 85.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 16:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(a \cdot \frac{c}{b \cdot b}\right) \cdot -0.375 - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 16.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (* (- (* (* a (/ c (* b b))) -0.375) 0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 16.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = ((((a * (c / (b * b))) * -0.375) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 16.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * Float64(c / Float64(b * b))) * -0.375) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 16.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 16

   1. Initial program 81.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.0

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

   4. Applied rewrites 82.0%

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

   if 16 < 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. lift-*.f64 N/A

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

      12. lower-*.f64 87.5

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 87.4

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

   8. Applied rewrites 87.4%

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

Alternative 12: 81.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-*.f64 79.6

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

5. Applied rewrites 79.6%

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

6. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 79.6

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

8. Applied rewrites 79.6%

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

Alternative 13: 65.0% 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 56.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 63.6

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

5. Applied rewrites 63.6%

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

Reproduce

herbie shell --seed 2025037 
(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)))