Cubic critical, narrow range

Percentage Accurate: 55.4% → 91.3%

Time: 8.1s

Alternatives: 16

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {t\_0}^{1.5}}{\mathsf{fma}\left(b, b, t\_0 + b \cdot \sqrt{t\_0}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \left(c \cdot c\right) \cdot \mathsf{fma}\left(-1.0546875, \frac{{a}^{3} \cdot \left(c \cdot c\right)}{{b}^{6}}, -0.375 \cdot \frac{a}{b \cdot b}\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
     (/
      (/ (+ (pow (- b) 3.0) (pow t_0 1.5)) (fma b b (+ t_0 (* b (sqrt t_0)))))
      (* 3.0 a))
     (/
      (fma
       (/ (* (* a a) (* (* c c) c)) (pow b 4.0))
       -0.5625
       (fma
        -0.5
        c
        (*
         (* c c)
         (fma
          -1.0546875
          (/ (* (pow a 3.0) (* c c)) (pow b 6.0))
          (* -0.375 (/ a (* b b)))))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		tmp = ((pow(-b, 3.0) + pow(t_0, 1.5)) / fma(b, b, (t_0 + (b * sqrt(t_0))))) / (3.0 * a);
	} else {
		tmp = fma((((a * a) * ((c * c) * c)) / pow(b, 4.0)), -0.5625, fma(-0.5, c, ((c * c) * fma(-1.0546875, ((pow(a, 3.0) * (c * c)) / pow(b, 6.0)), (-0.375 * (a / (b * b))))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.27)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + (t_0 ^ 1.5)) / fma(b, b, Float64(t_0 + Float64(b * sqrt(t_0))))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(a * a) * Float64(Float64(c * c) * c)) / (b ^ 4.0)), -0.5625, fma(-0.5, c, Float64(Float64(c * c) * fma(-1.0546875, Float64(Float64((a ^ 3.0) * Float64(c * c)) / (b ^ 6.0)), Float64(-0.375 * Float64(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]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.27], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(t$95$0 + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.5 * c + N[(N[(c * c), $MachinePrecision] * N[(-1.0546875 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.27000000000000002

   1. Initial program 84.3%

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

      1. lift-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 84.7%

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

   6. Applied rewrites 84.9%

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

   if -0.27000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 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

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

   5. Applied rewrites 95.8%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      2. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      6. lift-*.f64 95.8

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 95.8

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

   10. Applied rewrites 95.8%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \left(c \cdot c\right) \cdot \mathsf{fma}\left(-1.0546875, \frac{{a}^{3} \cdot \left(c \cdot c\right)}{{b}^{6}}, -0.375 \cdot \frac{a}{b \cdot b}\right)\right)\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {\left(\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\right)}^{1.5}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \left(c \cdot c\right) \cdot \mathsf{fma}\left(-1.0546875, \frac{{a}^{3} \cdot \left(c \cdot c\right)}{{b}^{6}}, -0.375 \cdot \frac{a}{b \cdot b}\right)\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot c\right)}^{3}\\ t_1 := \frac{a \cdot c}{b \cdot b}\\ t_2 := \mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\\ t_3 := {\left(a \cdot c\right)}^{2}\\ t_4 := \mathsf{fma}\left(9, t\_3, 18 \cdot t\_3\right) - 0.25 \cdot {t\_2}^{2}\\ t_5 := -27 \cdot t\_0 - 0.5 \cdot \left(t\_2 \cdot t\_4\right)\\ \frac{\frac{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, {t\_4}^{2}, 0.5 \cdot \left(t\_2 \cdot t\_5\right)\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, t\_2, \mathsf{fma}\left(0.5, \frac{t\_5}{{b}^{4}}, 0.5 \cdot \frac{t\_4}{b \cdot b}\right)\right)\right)}{\mathsf{fma}\left(b, b, \left(b \cdot b\right) \cdot \left(\left(2 + \mathsf{fma}\left(-3, t\_1, -1.6875 \cdot \frac{t\_0}{{b}^{6}}\right)\right) - \mathsf{fma}\left(1.125, \frac{t\_3}{{b}^{4}}, 1.5 \cdot t\_1\right)\right)\right)}}{3 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* a c) 3.0))
        (t_1 (/ (* a c) (* b b)))
        (t_2 (fma -6.0 (* a c) (* -3.0 (* a c))))
        (t_3 (pow (* a c) 2.0))
        (t_4 (- (fma 9.0 t_3 (* 18.0 t_3)) (* 0.25 (pow t_2 2.0))))
        (t_5 (- (* -27.0 t_0) (* 0.5 (* t_2 t_4)))))
   (/
    (/
     (*
      b
      (fma
       -0.5
       (/ (fma 0.25 (pow t_4 2.0) (* 0.5 (* t_2 t_5))) (pow b 6.0))
       (fma 0.5 t_2 (fma 0.5 (/ t_5 (pow b 4.0)) (* 0.5 (/ t_4 (* b b)))))))
     (fma
      b
      b
      (*
       (* b b)
       (-
        (+ 2.0 (fma -3.0 t_1 (* -1.6875 (/ t_0 (pow b 6.0)))))
        (fma 1.125 (/ t_3 (pow b 4.0)) (* 1.5 t_1))))))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = pow((a * c), 3.0);
	double t_1 = (a * c) / (b * b);
	double t_2 = fma(-6.0, (a * c), (-3.0 * (a * c)));
	double t_3 = pow((a * c), 2.0);
	double t_4 = fma(9.0, t_3, (18.0 * t_3)) - (0.25 * pow(t_2, 2.0));
	double t_5 = (-27.0 * t_0) - (0.5 * (t_2 * t_4));
	return ((b * fma(-0.5, (fma(0.25, pow(t_4, 2.0), (0.5 * (t_2 * t_5))) / pow(b, 6.0)), fma(0.5, t_2, fma(0.5, (t_5 / pow(b, 4.0)), (0.5 * (t_4 / (b * b))))))) / fma(b, b, ((b * b) * ((2.0 + fma(-3.0, t_1, (-1.6875 * (t_0 / pow(b, 6.0))))) - fma(1.125, (t_3 / pow(b, 4.0)), (1.5 * t_1)))))) / (3.0 * a);
}

Julia

function code(a, b, c)
	t_0 = Float64(a * c) ^ 3.0
	t_1 = Float64(Float64(a * c) / Float64(b * b))
	t_2 = fma(-6.0, Float64(a * c), Float64(-3.0 * Float64(a * c)))
	t_3 = Float64(a * c) ^ 2.0
	t_4 = Float64(fma(9.0, t_3, Float64(18.0 * t_3)) - Float64(0.25 * (t_2 ^ 2.0)))
	t_5 = Float64(Float64(-27.0 * t_0) - Float64(0.5 * Float64(t_2 * t_4)))
	return Float64(Float64(Float64(b * fma(-0.5, Float64(fma(0.25, (t_4 ^ 2.0), Float64(0.5 * Float64(t_2 * t_5))) / (b ^ 6.0)), fma(0.5, t_2, fma(0.5, Float64(t_5 / (b ^ 4.0)), Float64(0.5 * Float64(t_4 / Float64(b * b))))))) / fma(b, b, Float64(Float64(b * b) * Float64(Float64(2.0 + fma(-3.0, t_1, Float64(-1.6875 * Float64(t_0 / (b ^ 6.0))))) - fma(1.125, Float64(t_3 / (b ^ 4.0)), Float64(1.5 * t_1)))))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-6.0 * N[(a * c), $MachinePrecision] + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(9.0 * t$95$3 + N[(18.0 * t$95$3), $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(-27.0 * t$95$0), $MachinePrecision] - N[(0.5 * N[(t$95$2 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(b * N[(-0.5 * N[(N[(0.25 * N[Power[t$95$4, 2.0], $MachinePrecision] + N[(0.5 * N[(t$95$2 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$2 + N[(0.5 * N[(t$95$5 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$4 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(b * b), $MachinePrecision] * N[(N[(2.0 + N[(-3.0 * t$95$1 + N[(-1.6875 * N[(t$95$0 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.125 * N[(t$95$3 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 55.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 55.6%

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

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 92.7%

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

8. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lower--.f64 N/A

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

10. Applied rewrites 92.8%

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

Alternative 3: 91.1% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 55.6%

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

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 92.7%

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

8. Taylor expanded in c around 0

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

10. Applied rewrites 92.7%

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

11. Final simplification 92.7%

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

Alternative 4: 91.1% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 55.6%

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

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 92.7%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 92.6%

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

11. Final simplification 92.6%

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

Alternative 5: 91.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {t\_0}^{1.5}}{\mathsf{fma}\left(b, b, t\_0 + b \cdot \sqrt{t\_0}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a \cdot a}{{b}^{4}}, -0.5625, \frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{6}}\right), c, \frac{a}{b \cdot b} \cdot -0.375\right) \cdot c - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
     (/
      (/ (+ (pow (- b) 3.0) (pow t_0 1.5)) (fma b b (+ t_0 (* b (sqrt t_0)))))
      (* 3.0 a))
     (/
      (*
       (-
        (*
         (fma
          (fma
           (/ (* a a) (pow b 4.0))
           -0.5625
           (/ (* -1.0546875 (* (pow a 3.0) c)) (pow b 6.0)))
          c
          (* (/ a (* b b)) -0.375))
         c)
        0.5)
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		tmp = ((pow(-b, 3.0) + pow(t_0, 1.5)) / fma(b, b, (t_0 + (b * sqrt(t_0))))) / (3.0 * a);
	} else {
		tmp = (((fma(fma(((a * a) / pow(b, 4.0)), -0.5625, ((-1.0546875 * (pow(a, 3.0) * c)) / pow(b, 6.0))), c, ((a / (b * b)) * -0.375)) * c) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.27)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + (t_0 ^ 1.5)) / fma(b, b, Float64(t_0 + Float64(b * sqrt(t_0))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(fma(fma(Float64(Float64(a * a) / (b ^ 4.0)), -0.5625, Float64(Float64(-1.0546875 * Float64((a ^ 3.0) * c)) / (b ^ 6.0))), c, Float64(Float64(a / Float64(b * b)) * -0.375)) * c) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.27], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(t$95$0 + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(N[(-1.0546875 * N[(N[Power[a, 3.0], $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.27000000000000002

   1. Initial program 84.3%

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

      1. lift-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 84.7%

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

   6. Applied rewrites 84.9%

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

   if -0.27000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 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

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.8%

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a \cdot a}{{b}^{4}}, -0.5625, \frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{6}}\right), c, \frac{a}{b \cdot b} \cdot -0.375\right) \cdot c - 0.5\right) \cdot c}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {\left(\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\right)}^{1.5}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a \cdot a}{{b}^{4}}, -0.5625, \frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{6}}\right), c, \frac{a}{b \cdot b} \cdot -0.375\right) \cdot c - 0.5\right) \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {t\_0}^{1.5}}{\mathsf{fma}\left(b, b, t\_0 + b \cdot \sqrt{t\_0}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{5}}, -0.5625, -0.375 \cdot {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
     (/
      (/ (+ (pow (- b) 3.0) (pow t_0 1.5)) (fma b b (+ t_0 (* b (sqrt t_0)))))
      (* 3.0 a))
     (fma
      (* (fma (* a (/ c (pow b 5.0))) -0.5625 (* -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 tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		tmp = ((pow(-b, 3.0) + pow(t_0, 1.5)) / fma(b, b, (t_0 + (b * sqrt(t_0))))) / (3.0 * a);
	} else {
		tmp = fma((fma((a * (c / pow(b, 5.0))), -0.5625, (-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))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.27)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + (t_0 ^ 1.5)) / fma(b, b, Float64(t_0 + Float64(b * sqrt(t_0))))) / Float64(3.0 * a));
	else
		tmp = fma(Float64(fma(Float64(a * Float64(c / (b ^ 5.0))), -0.5625, 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]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.27], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(t$95$0 + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625 + 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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.27000000000000002

   1. Initial program 84.3%

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

      1. lift-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 84.7%

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

   6. Applied rewrites 84.9%

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

   if -0.27000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. pow-flip N/A

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

      13. metadata-eval N/A

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

      14. lower-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 93.7

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

   8. Applied rewrites 93.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {\left(\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\right)}^{1.5}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{5}}, -0.5625, -0.375 \cdot {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + t\_0 \cdot t\_1}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 + b \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{5}}, -0.5625, -0.375 \cdot {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
     (/
      (/ (+ (pow (- b) 3.0) (* t_0 t_1)) (fma b b (+ (* t_1 t_1) (* b t_1))))
      (* 3.0 a))
     (fma
      (* (fma (* a (/ c (pow b 5.0))) -0.5625 (* -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 + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		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((a * (c / pow(b, 5.0))), -0.5625, (-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 (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.27)
		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(a * Float64(c / (b ^ 5.0))), -0.5625, 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[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.27], 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[(a * N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625 + 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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.27000000000000002

   1. Initial program 84.3%

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

      1. lift-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 84.7%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{3} + {\left(\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}^{3}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}}{3 \cdot a} \]
   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 84.8

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

   6. Applied rewrites 84.8%

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

   if -0.27000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. pow-flip N/A

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

      13. metadata-eval N/A

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

      14. lower-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 93.7

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

   8. Applied rewrites 93.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{5}}, -0.5625, -0.375 \cdot {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {t\_0}^{1.5}\right)}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 + b \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{5}}, -0.5625, -0.375 \cdot {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
     (/
      (/ (fma (* b b) (- b) (pow t_0 1.5)) (fma b b (+ (* t_1 t_1) (* b t_1))))
      (* 3.0 a))
     (fma
      (* (fma (* a (/ c (pow b 5.0))) -0.5625 (* -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 + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		tmp = (fma((b * b), -b, pow(t_0, 1.5)) / fma(b, b, ((t_1 * t_1) + (b * t_1)))) / (3.0 * a);
	} else {
		tmp = fma((fma((a * (c / pow(b, 5.0))), -0.5625, (-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 (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.27)
		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(fma(Float64(a * Float64(c / (b ^ 5.0))), -0.5625, 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[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.27], 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[(a * N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5625 + 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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.27000000000000002

   1. Initial program 84.3%

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

      1. lift-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 84.7%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{3} + {\left(\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}^{3}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}}{3 \cdot a} \]
   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 84.7%

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

   if -0.27000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. pow-flip N/A

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

      13. metadata-eval N/A

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

      14. lower-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 93.7

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

   8. Applied rewrites 93.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {\left(\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{c}{{b}^{5}}, -0.5625, -0.375 \cdot {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {t\_0}^{1.5}\right)}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 + b \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
     (/
      (/ (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) (* (* c c) c)) (pow b 4.0))
       -0.5625
       (fma -0.5 c (* -0.375 (/ (* a (* c c)) (* 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 + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		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) * ((c * c) * c)) / pow(b, 4.0)), -0.5625, fma(-0.5, c, (-0.375 * ((a * (c * c)) / (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 (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.27)
		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) * Float64(Float64(c * c) * c)) / (b ^ 4.0)), -0.5625, fma(-0.5, c, Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / 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[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.27], 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[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.5 * c + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.27000000000000002

   1. Initial program 84.3%

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

      1. lift-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 84.7%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{3} + {\left(\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}^{3}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}}{3 \cdot a} \]
   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 84.7%

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

   if -0.27000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 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

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

   5. Applied rewrites 95.8%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      2. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      6. lift-*.f64 95.8

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 93.7

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

   10. Applied rewrites 93.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {\left(\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\right)}^{1.5}\right)}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/
    (fma
     (/ (* (* a a) (* (* c c) c)) (pow b 4.0))
     -0.5625
     (fma -0.5 c (* -0.375 (/ (* a (* c c)) (* b b)))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma((((a * a) * ((c * c) * c)) / pow(b, 4.0)), -0.5625, fma(-0.5, c, (-0.375 * ((a * (c * c)) / (b * b))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.27)
		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(a * a) * Float64(Float64(c * c) * c)) / (b ^ 4.0)), -0.5625, fma(-0.5, c, Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / Float64(b * b))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.27], 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 * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.5 * c + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.27000000000000002

   1. Initial program 84.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 84.5

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

   4. Applied rewrites 84.5%

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

   if -0.27000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 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

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

   5. Applied rewrites 95.8%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      2. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      6. lift-*.f64 95.8

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

   7. Applied rewrites 95.8%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 93.7

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

   10. Applied rewrites 93.7%

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

Alternative 11: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{a}{b \cdot b}, -0.375, \frac{-0.5625 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{4}}\right) \cdot c - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/
    (*
     (-
      (*
       (fma (/ a (* b b)) -0.375 (/ (* -0.5625 (* (* a a) c)) (pow b 4.0)))
       c)
      0.5)
     c)
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = (((fma((a / (b * b)), -0.375, ((-0.5625 * ((a * a) * c)) / pow(b, 4.0))) * c) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.27)
		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(fma(Float64(a / Float64(b * b)), -0.375, Float64(Float64(-0.5625 * Float64(Float64(a * a) * c)) / (b ^ 4.0))) * c) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.27], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(N[(-0.5625 * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.27000000000000002

   1. Initial program 84.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 84.5

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

   4. Applied rewrites 84.5%

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

   if -0.27000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 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

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 93.7%

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

Alternative 12: 84.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, -0.375, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (fma (/ (* (* c c) a) (pow b 3.0)) -0.375 (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma((((c * c) * a) / pow(b, 3.0)), -0.375, ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.27)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = fma(Float64(Float64(Float64(c * c) * a) / (b ^ 3.0)), -0.375, Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.27], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375 + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.27000000000000002

   1. Initial program 84.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 84.5

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

   4. Applied rewrites 84.5%

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

   if -0.27000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 88.5

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

   5. Applied rewrites 88.5%

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

Alternative 13: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (fma (/ (* (* c c) a) (* b b)) -0.375 (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma((((c * c) * a) / (b * b)), -0.375, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.27)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(c * c) * a) / Float64(b * b)), -0.375, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.27], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.27000000000000002

   1. Initial program 84.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 84.5

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

   4. Applied rewrites 84.5%

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

   if -0.27000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 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

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   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 88.5

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

   5. Applied rewrites 88.5%

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

Alternative 14: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (* (- (/ (* -0.375 (* c a)) (* b b)) 0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = ((((-0.375 * (c * a)) / (b * b)) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.27)
		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(-0.375 * Float64(c * a)) / Float64(b * b)) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.27], 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[(-0.375 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.27000000000000002

   1. Initial program 84.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 84.5

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

   4. Applied rewrites 84.5%

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

   if -0.27000000000000002 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 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

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
   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. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 88.5

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

   8. Applied rewrites 88.5%

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

Alternative 15: 81.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 92.4%

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

6. Taylor expanded in c around 0

   \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
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. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 83.5

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

8. Applied rewrites 83.5%

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

Alternative 16: 64.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 64.9

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

5. Applied rewrites 64.9%

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

Reproduce

herbie shell --seed 2025084 
(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)))