Cubic critical, narrow range

Percentage Accurate: 55.0% → 91.3%

Time: 10.9s

Alternatives: 15

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: 55.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, {\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= b 11.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 3.0 a))
     (/
      (fma
       (* (* a a) -0.5625)
       (* (/ (* c c) (* b b)) (/ c (* b b)))
       (fma
        (/ -0.16666666666666666 (pow b 6.0))
        (* (pow (* a c) 4.0) (/ 6.328125 a))
        (fma (/ -0.375 b) (/ (* (* c c) a) b) (* -0.5 c))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (b <= 11.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (3.0 * a);
	} else {
		tmp = fma(((a * a) * -0.5625), (((c * c) / (b * b)) * (c / (b * b))), fma((-0.16666666666666666 / pow(b, 6.0)), (pow((a * c), 4.0) * (6.328125 / a)), fma((-0.375 / b), (((c * c) * a) / b), (-0.5 * c)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 11.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(a * a) * -0.5625), Float64(Float64(Float64(c * c) / Float64(b * b)) * Float64(c / Float64(b * b))), fma(Float64(-0.16666666666666666 / (b ^ 6.0)), Float64((Float64(a * c) ^ 4.0) * Float64(6.328125 / a)), fma(Float64(-0.375 / b), Float64(Float64(Float64(c * c) * a) / b), Float64(-0.5 * c)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 11.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] * N[(6.328125 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 / b), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 11

   1. Initial program 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 85.2%

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

   if 11 < b

   1. Initial program 44.4%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.5%

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

   7. Applied rewrites 94.5%

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

   9. Applied rewrites 94.5%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\frac{-0.16666666666666666}{{b}^{6}}, {\left(a \cdot c\right)}^{4} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(\frac{-0.375}{b}, \frac{\left(c \cdot c\right) \cdot a}{b}, -0.5 \cdot c\right)\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.375}{b}, \frac{c \cdot c}{b}, \frac{-1.0546875 \cdot \left(\left({c}^{4} \cdot a\right) \cdot a\right)}{{b}^{6}}\right), a, -0.5 \cdot c\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= b 11.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 3.0 a))
     (/
      (fma
       (* (* a a) -0.5625)
       (* (/ (* c c) (* b b)) (/ c (* b b)))
       (fma
        (fma
         (/ -0.375 b)
         (/ (* c c) b)
         (/ (* -1.0546875 (* (* (pow c 4.0) a) a)) (pow b 6.0)))
        a
        (* -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 <= 11.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (3.0 * a);
	} else {
		tmp = fma(((a * a) * -0.5625), (((c * c) / (b * b)) * (c / (b * b))), fma(fma((-0.375 / b), ((c * c) / b), ((-1.0546875 * ((pow(c, 4.0) * a) * a)) / pow(b, 6.0))), a, (-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 (b <= 11.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(a * a) * -0.5625), Float64(Float64(Float64(c * c) / Float64(b * b)) * Float64(c / Float64(b * b))), fma(fma(Float64(-0.375 / b), Float64(Float64(c * c) / b), Float64(Float64(-1.0546875 * Float64(Float64((c ^ 4.0) * a) * a)) / (b ^ 6.0))), a, Float64(-0.5 * c))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 11.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.375 / b), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] + N[(N[(-1.0546875 * N[(N[(N[Power[c, 4.0], $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 11

   1. Initial program 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 85.2%

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

   if 11 < b

   1. Initial program 44.4%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.5%

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

   7. Applied rewrites 94.5%

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

   9. Applied rewrites 94.5%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 94.5%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.375}{b}, \frac{c \cdot c}{b}, \frac{-1.0546875 \cdot \left(\left({c}^{4} \cdot a\right) \cdot a\right)}{{b}^{6}}\right), a, -0.5 \cdot c\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \left(\mathsf{fma}\left(\frac{-0.375}{b}, \frac{a}{b}, \left(-1.0546875 \cdot {a}^{3}\right) \cdot \left(c \cdot \frac{c}{{b}^{6}}\right)\right) \cdot c - 0.5\right) \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= b 11.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 3.0 a))
     (/
      (fma
       (* (* a a) -0.5625)
       (* (/ (* c c) (* b b)) (/ c (* b b)))
       (*
        (-
         (*
          (fma
           (/ -0.375 b)
           (/ a b)
           (* (* -1.0546875 (pow a 3.0)) (* c (/ c (pow b 6.0)))))
          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 <= 11.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (3.0 * a);
	} else {
		tmp = fma(((a * a) * -0.5625), (((c * c) / (b * b)) * (c / (b * b))), (((fma((-0.375 / b), (a / b), ((-1.0546875 * pow(a, 3.0)) * (c * (c / pow(b, 6.0))))) * 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 (b <= 11.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(a * a) * -0.5625), Float64(Float64(Float64(c * c) / Float64(b * b)) * Float64(c / Float64(b * b))), Float64(Float64(Float64(fma(Float64(-0.375 / b), Float64(a / b), Float64(Float64(-1.0546875 * (a ^ 3.0)) * Float64(c * Float64(c / (b ^ 6.0))))) * 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[b, 11.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(-0.375 / b), $MachinePrecision] * N[(a / b), $MachinePrecision] + N[(N[(-1.0546875 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] * N[(c * N[(c / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 11

   1. Initial program 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 85.2%

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

   if 11 < b

   1. Initial program 44.4%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.5%

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

   7. Applied rewrites 94.5%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 94.4%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{c \cdot c}{b \cdot b} \cdot \frac{c}{b \cdot b}, \left(\mathsf{fma}\left(\frac{-0.375}{b}, \frac{a}{b}, \left(-1.0546875 \cdot {a}^{3}\right) \cdot \left(c \cdot \frac{c}{{b}^{6}}\right)\right) \cdot c - 0.5\right) \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, -0.5 \cdot c\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= b 11.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 3.0 a))
     (/
      (fma
       (* (* a a) -0.5625)
       (/ (pow c 3.0) (pow b 4.0))
       (fma (/ (* -0.375 a) b) (/ (* c c) b) (* -0.5 c)))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (b <= 11.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (3.0 * a);
	} else {
		tmp = fma(((a * a) * -0.5625), (pow(c, 3.0) / pow(b, 4.0)), fma(((-0.375 * a) / b), ((c * c) / b), (-0.5 * c))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 11.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(a * a) * -0.5625), Float64((c ^ 3.0) / (b ^ 4.0)), fma(Float64(Float64(-0.375 * a) / b), Float64(Float64(c * c) / b), Float64(-0.5 * c))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 11.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * -0.5625), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.375 * a), $MachinePrecision] / b), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 11

   1. Initial program 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 85.2%

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

   if 11 < b

   1. Initial program 44.4%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 92.8%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -0.5625, \frac{{c}^{3}}{{b}^{4}}, \mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, -0.5 \cdot c\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(-0.375 \cdot \left(b \cdot b\right)\right) \cdot c, c, \left({c}^{3} \cdot a\right) \cdot -0.5625\right)}{{b}^{5}}, a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= b 11.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 3.0 a))
     (fma
      (/
       (fma (* (* -0.375 (* b b)) c) c (* (* (pow c 3.0) a) -0.5625))
       (pow b 5.0))
      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 <= 11.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (3.0 * a);
	} else {
		tmp = fma((fma(((-0.375 * (b * b)) * c), c, ((pow(c, 3.0) * a) * -0.5625)) / pow(b, 5.0)), 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 (b <= 11.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(3.0 * a));
	else
		tmp = fma(Float64(fma(Float64(Float64(-0.375 * Float64(b * b)) * c), c, Float64(Float64((c ^ 3.0) * a) * -0.5625)) / (b ^ 5.0)), 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[b, 11.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.375 * N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c + N[(N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 11

   1. Initial program 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 85.2%

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

   if 11 < b

   1. Initial program 44.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 92.8%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(-0.375 \cdot \left(b \cdot b\right)\right) \cdot c, c, \left({c}^{3} \cdot a\right) \cdot -0.5625\right)}{{b}^{5}}, a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{4}}, \frac{a}{b \cdot b} \cdot -0.375\right) \cdot c - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= b 11.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 3.0 a))
     (/
      (*
       (-
        (*
         (fma (* c -0.5625) (* a (/ a (pow b 4.0))) (* (/ 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 <= 11.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (3.0 * a);
	} else {
		tmp = (((fma((c * -0.5625), (a * (a / pow(b, 4.0))), ((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 (b <= 11.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(fma(Float64(c * -0.5625), Float64(a * Float64(a / (b ^ 4.0))), 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[b, 11.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(c * -0.5625), $MachinePrecision] * N[(a * N[(a / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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 b < 11

   1. Initial program 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 85.2%

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

   if 11 < b

   1. Initial program 44.4%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.5%

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

   7. Applied rewrites 94.5%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 92.7%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{4}}, \frac{a}{b \cdot b} \cdot -0.375\right) \cdot c - 0.5\right) \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 18.5:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= b 18.5)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 3.0 a))
     (/ (fma (/ (* -0.375 a) b) (/ (* c c) b) (* -0.5 c)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (b <= 18.5) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (3.0 * a);
	} else {
		tmp = fma(((-0.375 * a) / b), ((c * c) / b), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 18.5)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(-0.375 * a) / b), Float64(Float64(c * c) / b), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 18.5], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.375 * a), $MachinePrecision] / b), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 18.5

   1. Initial program 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 84.6%

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

   if 18.5 < b

   1. Initial program 43.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 94.9%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 89.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 18.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 18.5:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(\left(-b\right) - \sqrt{t\_0}\right) \cdot \left(3 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= b 18.5)
     (/ (- (* b b) t_0) (* (- (- b) (sqrt t_0)) (* 3.0 a)))
     (/ (fma (/ (* -0.375 a) b) (/ (* c c) b) (* -0.5 c)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (b <= 18.5) {
		tmp = ((b * b) - t_0) / ((-b - sqrt(t_0)) * (3.0 * a));
	} else {
		tmp = fma(((-0.375 * a) / b), ((c * c) / b), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 18.5)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(Float64(-b) - sqrt(t_0)) * Float64(3.0 * a)));
	else
		tmp = Float64(fma(Float64(Float64(-0.375 * a) / b), Float64(Float64(c * c) / b), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 18.5], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.375 * a), $MachinePrecision] / b), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 18.5

   1. Initial program 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 84.6%

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

   if 18.5 < b

   1. Initial program 43.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 94.9%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 89.5%

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

Alternative 9: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 18.5:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 18.5)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (/ (fma (/ (* -0.375 a) b) (/ (* c c) b) (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 18.5) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = fma(((-0.375 * a) / b), ((c * c) / b), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 18.5)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(-0.375 * a) / b), Float64(Float64(c * c) / b), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 18.5], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.375 * a), $MachinePrecision] / b), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 18.5

   1. Initial program 83.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-abs-rev N/A

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

      10. fabs-fabs N/A

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

      11. fabs-fabs N/A

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

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

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   4. Applied rewrites 83.5%

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

   if 18.5 < b

   1. Initial program 43.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 94.9%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 89.5%

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

Alternative 10: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 18.5:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(c \cdot \frac{a}{b \cdot b}\right) \cdot -0.375 - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 18.5)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (/ (* (- (* (* c (/ a (* b b))) -0.375) 0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 18.5) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = ((((c * (a / (b * b))) * -0.375) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 18.5)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(c * Float64(a / Float64(b * b))) * -0.375) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 18.5], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 18.5

   1. Initial program 83.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-abs-rev N/A

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

      10. fabs-fabs N/A

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

      11. fabs-fabs N/A

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

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

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   4. Applied rewrites 83.5%

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

   if 18.5 < b

   1. Initial program 43.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 94.9%

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 89.4%

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

Alternative 11: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 18.5:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(c \cdot \frac{a}{b \cdot b}\right) \cdot -0.375 - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 18.5)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (* (- (* (* c (/ a (* b b))) -0.375) 0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 18.5) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = ((((c * (a / (b * b))) * -0.375) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 18.5)
		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(c * Float64(a / Float64(b * b))) * -0.375) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 18.5], 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 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 18.5

   1. Initial program 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 83.5

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

   6. Applied rewrites 83.5%

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

   if 18.5 < b

   1. Initial program 43.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 94.9%

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in c around 0

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

   10. Applied rewrites 89.4%

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

Alternative 12: 81.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 88.8%

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

7. Applied rewrites 88.8%

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

8. Taylor expanded in c around 0

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

10. Applied rewrites 80.1%

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

Alternative 13: 81.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

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

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

   7. lower-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. metadata-eval N/A

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

   14. associate-*r/ N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 80.0

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

5. Applied rewrites 80.0%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 64.6%

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

9. Taylor expanded in b around -inf

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

11. Applied rewrites 80.0%

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

Alternative 14: 64.7% 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 54.7%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.7

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

5. Applied rewrites 64.7%

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

Alternative 15: 64.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

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

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

   7. lower-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. metadata-eval N/A

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

   14. associate-*r/ N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 80.0

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

5. Applied rewrites 80.0%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 64.6%

   \[\leadsto \frac{-0.5}{b} \cdot c \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025006 
(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)))