Cubic critical, narrow range

Percentage Accurate: 55.2% → 92.0%

Time: 5.8s

Alternatives: 9

Speedup: 3.3×

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: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.019:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\frac{-1.0546875 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.5625\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b}, a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}\right), -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= b 0.019)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (/
      (fma
       a
       (fma
        (/
         (* (- (/ (* -1.0546875 (* c a)) (* b b)) 0.5625) (* (* c c) c))
         (* (* (* b b) b) b))
        a
        (/ (* -0.375 (* c c)) (* b b)))
       (* -0.5 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (b <= 0.019) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(a, fma((((((-1.0546875 * (c * a)) / (b * b)) - 0.5625) * ((c * c) * c)) / (((b * b) * b) * b)), a, ((-0.375 * (c * c)) / (b * b))), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 0.019)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(a, fma(Float64(Float64(Float64(Float64(Float64(-1.0546875 * Float64(c * a)) / Float64(b * b)) - 0.5625) * Float64(Float64(c * c) * c)) / Float64(Float64(Float64(b * b) * b) * b)), a, Float64(Float64(-0.375 * Float64(c * c)) / Float64(b * b))), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.019], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(N[(N[(N[(-1.0546875 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - 0.5625), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a + N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0189999999999999995

   1. Initial program 55.2%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 56.7%

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

   if 0.0189999999999999995 < b

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 90.9%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 90.9%

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

   11. Taylor expanded in c around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. associate-*r/ N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow3 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 90.9

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

   13. Applied rewrites 90.9%

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

Alternative 2: 90.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -4

   1. Initial program 55.2%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 56.7%

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

   if -4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 90.9%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 87.8%

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

Alternative 3: 89.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -4

   1. Initial program 55.2%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 56.7%

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

   if -4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 87.6%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 87.6%

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

   11. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. associate-*r/ N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 87.6

           \[\leadsto \frac{\left(\frac{\left(\frac{-0.5625 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.375\right) \cdot a}{b \cdot b} \cdot c - 0.5\right) \cdot c}{b} \]

   13. Applied rewrites 87.6%

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

Alternative 4: 89.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -124:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(\frac{-0.5625 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.375\right) \cdot a}{b \cdot b} \cdot c - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -124.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/
    (*
     (- (* (/ (* (- (/ (* -0.5625 (* c a)) (* b b)) 0.375) a) (* b b)) c) 0.5)
     c)
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -124.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = ((((((((-0.5625 * (c * a)) / (b * b)) - 0.375) * a) / (b * b)) * c) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -124.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.5625 * Float64(c * a)) / Float64(b * b)) - 0.375) * a) / Float64(b * b)) * c) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -124.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(-0.5625 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - 0.375), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -124

   1. Initial program 55.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 55.3

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

   3. Applied rewrites 55.3%

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

   if -124 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 87.6%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 87.6%

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

   11. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. associate-*r/ N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 87.6

           \[\leadsto \frac{\left(\frac{\left(\frac{-0.5625 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.375\right) \cdot a}{b \cdot b} \cdot c - 0.5\right) \cdot c}{b} \]

   13. Applied rewrites 87.6%

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

Alternative 5: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.001:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} + \left(-b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.001)
   (/ (+ (sqrt (fma b b (* (* -3.0 a) c))) (- b)) (* a 3.0))
   (/ (fma (/ (* (* c c) a) (* b b)) -0.375 (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.001) {
		tmp = (sqrt(fma(b, b, ((-3.0 * a) * c))) + -b) / (a * 3.0);
	} else {
		tmp = fma((((c * c) * a) / (b * b)), -0.375, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.001)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c))) + Float64(-b)) / Float64(a * 3.0));
	else
		tmp = Float64(fma(Float64(Float64(Float64(c * c) * a) / Float64(b * b)), -0.375, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.001], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1e-3

   1. Initial program 55.2%

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

   3. Applied rewrites 55.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

   if -1e-3 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 81.6

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

   4. Applied rewrites 81.6%

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

Alternative 6: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.001:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} + \left(-b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.001)
   (/ (+ (sqrt (fma b b (* (* -3.0 a) c))) (- b)) (* a 3.0))
   (/ (* (- (/ (* -0.375 (* c a)) (* b b)) 0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.001) {
		tmp = (sqrt(fma(b, b, ((-3.0 * a) * c))) + -b) / (a * 3.0);
	} else {
		tmp = ((((-0.375 * (c * a)) / (b * b)) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.001)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c))) + Float64(-b)) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.375 * Float64(c * a)) / Float64(b * b)) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.001], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.375 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1e-3

   1. Initial program 55.2%

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

   3. Applied rewrites 55.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

   if -1e-3 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 81.5

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

   7. Applied rewrites 81.5%

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

Alternative 7: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.001:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.001)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (* (- (/ (* -0.375 (* c a)) (* b b)) 0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.001) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = ((((-0.375 * (c * a)) / (b * b)) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.001)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.375 * Float64(c * a)) / Float64(b * b)) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.001], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.375 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1e-3

   1. Initial program 55.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 55.3

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

   3. Applied rewrites 55.3%

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

   if -1e-3 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 81.5

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

   7. Applied rewrites 81.5%

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

Alternative 8: 81.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 90.9%

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

5. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 81.5

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

7. Applied rewrites 81.5%

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

Alternative 9: 64.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

2. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.5

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

4. Applied rewrites 64.5%

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

Reproduce

herbie shell --seed 2025140 
(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)))