Cubic critical, medium range

Percentage Accurate: 31.7% → 95.3%

Time: 6.8s

Alternatives: 14

Speedup: 3.3×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot b\\ \frac{\mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, -0.5 \cdot c\right) + \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{\left(t\_0 \cdot t\_0\right) \cdot a}, -0.16666666666666666, \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \frac{a \cdot a}{t\_0 \cdot b}\right) \cdot -0.5625\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* b b) b)))
   (/
    (+
     (fma (* -0.375 a) (* c (/ c (* b b))) (* -0.5 c))
     (fma
      (* (pow (* c a) 4.0) (/ 6.328125 (* (* t_0 t_0) a)))
      -0.16666666666666666
      (* (* (* (* c c) c) (/ (* a a) (* t_0 b))) -0.5625)))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) * b;
	return (fma((-0.375 * a), (c * (c / (b * b))), (-0.5 * c)) + fma((pow((c * a), 4.0) * (6.328125 / ((t_0 * t_0) * a))), -0.16666666666666666, ((((c * c) * c) * ((a * a) / (t_0 * b))) * -0.5625))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) * b)
	return Float64(Float64(fma(Float64(-0.375 * a), Float64(c * Float64(c / Float64(b * b))), Float64(-0.5 * c)) + fma(Float64((Float64(c * a) ^ 4.0) * Float64(6.328125 / Float64(Float64(t_0 * t_0) * a))), -0.16666666666666666, Float64(Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) / Float64(t_0 * b))) * -0.5625))) / b)
end

Wolfram

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

Derivation

1. Initial program 31.7%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 95.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

6. Applied rewrites 95.3%

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

Alternative 2: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ \frac{\mathsf{fma}\left(c, -0.5, \left(\left(\frac{c}{b \cdot b} \cdot c\right) \cdot -0.375\right) \cdot a - \mathsf{fma}\left(\frac{1.0546875}{\left(a \cdot \left(t\_0 \cdot b\right)\right) \cdot b}, {\left(c \cdot a\right)}^{4}, \left(\left(\frac{a}{t\_0} \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)\right) \cdot 0.5625\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* (* b b) b) b)))
   (/
    (fma
     c
     -0.5
     (-
      (* (* (* (/ c (* b b)) c) -0.375) a)
      (fma
       (/ 1.0546875 (* (* a (* t_0 b)) b))
       (pow (* c a) 4.0)
       (* (* (* (/ a t_0) a) (* (* c c) c)) 0.5625))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = ((b * b) * b) * b;
	return fma(c, -0.5, (((((c / (b * b)) * c) * -0.375) * a) - fma((1.0546875 / ((a * (t_0 * b)) * b)), pow((c * a), 4.0), ((((a / t_0) * a) * ((c * c) * c)) * 0.5625)))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(b * b) * b) * b)
	return Float64(fma(c, -0.5, Float64(Float64(Float64(Float64(Float64(c / Float64(b * b)) * c) * -0.375) * a) - fma(Float64(1.0546875 / Float64(Float64(a * Float64(t_0 * b)) * b)), (Float64(c * a) ^ 4.0), Float64(Float64(Float64(Float64(a / t_0) * a) * Float64(Float64(c * c) * c)) * 0.5625)))) / b)
end

Wolfram

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

Derivation

1. Initial program 31.7%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 95.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

6. Applied rewrites 95.3%

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

8. Applied rewrites 95.3%

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

Alternative 3: 93.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, -0.5 \cdot c\right) + -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (fma (* -0.375 a) (* c (/ c (* b b))) (* -0.5 c))
   (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0))))
  b))

C

double code(double a, double b, double c) {
	return (fma((-0.375 * a), (c * (c / (b * b))), (-0.5 * c)) + (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0)))) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(-0.375 * a), Float64(c * Float64(c / Float64(b * b))), Float64(-0.5 * c)) + Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-0.375 * a), $MachinePrecision] * N[(c * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.7%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 95.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

6. Applied rewrites 95.3%

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

7. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-pow.f64 93.8

      \[\leadsto \frac{\mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{b \cdot b}, -0.5 \cdot c\right) + -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}}{b} \]

9. Applied rewrites 93.8%

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

Alternative 4: 93.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot c}{{b}^{4}}, -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (-
    (*
     c
     (fma
      -0.5625
      (/ (* (pow a 2.0) c) (pow b 4.0))
      (* -0.375 (/ a (pow b 2.0)))))
    0.5))
  b))

C

double code(double a, double b, double c) {
	return (c * ((c * fma(-0.5625, ((pow(a, 2.0) * c) / pow(b, 4.0)), (-0.375 * (a / pow(b, 2.0))))) - 0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(c * fma(-0.5625, Float64(Float64((a ^ 2.0) * c) / (b ^ 4.0)), Float64(-0.375 * Float64(a / (b ^ 2.0))))) - 0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(c * N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.7%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 95.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

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. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

7. Applied rewrites 93.7%

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

Alternative 5: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -60:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(1 \cdot \left(-b\right), a, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot a\right)}{a \cdot a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -60.0)
   (/
    (/ (fma (* 1.0 (- b)) a (* (sqrt (fma (* -3.0 a) c (* b b))) a)) (* a a))
    3.0)
   (/ (fma -0.5 c (* -0.375 (/ (* a (pow c 2.0)) (pow b 2.0)))) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -60.0) {
		tmp = (fma((1.0 * -b), a, (sqrt(fma((-3.0 * a), c, (b * b))) * a)) / (a * a)) / 3.0;
	} else {
		tmp = fma(-0.5, c, (-0.375 * ((a * pow(c, 2.0)) / pow(b, 2.0)))) / 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)) <= -60.0)
		tmp = Float64(Float64(fma(Float64(1.0 * Float64(-b)), a, Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) * a)) / Float64(a * a)) / 3.0);
	else
		tmp = Float64(fma(-0.5, c, Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0)))) / 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], -60.0], N[(N[(N[(N[(1.0 * (-b)), $MachinePrecision] * a + N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(-0.5 * c + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -60

   1. Initial program 31.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-out N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 31.7

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

   3. Applied rewrites 31.7%

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

   5. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{a} \cdot \left(-b\right)\right) \cdot 3 - \frac{-\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}}{a}}{3}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 31.4

          \[\leadsto \frac{\mathsf{fma}\left(\frac{0.3333333333333333}{a}, \left(-b\right) \cdot 3, \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}}{a}}\right)}{3} \]

   7. Applied rewrites 31.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{0.3333333333333333}{a}, \left(-b\right) \cdot 3, \frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}{a}\right)}}{3} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. common-denominator N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

   9. Applied rewrites 33.0%

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

   if -60 < (/.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 31.7%

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

   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. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 90.7

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b} \]

   4. Applied rewrites 90.7%

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

Alternative 6: 90.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -60:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(1 \cdot \left(-b\right), a, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} \cdot a\right)}{a \cdot a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -60.0)
   (/
    (/ (fma (* 1.0 (- b)) a (* (sqrt (fma (* -3.0 a) c (* b b))) a)) (* a a))
    3.0)
   (/ (* c (- (* -0.375 (/ (* a c) (pow b 2.0))) 0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -60.0) {
		tmp = (fma((1.0 * -b), a, (sqrt(fma((-3.0 * a), c, (b * b))) * a)) / (a * a)) / 3.0;
	} else {
		tmp = (c * ((-0.375 * ((a * c) / pow(b, 2.0))) - 0.5)) / 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)) <= -60.0)
		tmp = Float64(Float64(fma(Float64(1.0 * Float64(-b)), a, Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) * a)) / Float64(a * a)) / 3.0);
	else
		tmp = Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 2.0))) - 0.5)) / 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], -60.0], N[(N[(N[(N[(1.0 * (-b)), $MachinePrecision] * a + N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $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)) < -60

   1. Initial program 31.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-out N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 31.7

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

   3. Applied rewrites 31.7%

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

   5. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{a} \cdot \left(-b\right)\right) \cdot 3 - \frac{-\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}}{a}}{3}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 31.4

          \[\leadsto \frac{\mathsf{fma}\left(\frac{0.3333333333333333}{a}, \left(-b\right) \cdot 3, \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}}{a}}\right)}{3} \]

   7. Applied rewrites 31.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{0.3333333333333333}{a}, \left(-b\right) \cdot 3, \frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}}{a}\right)}}{3} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. common-denominator N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

   9. Applied rewrites 33.0%

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

   if -60 < (/.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 31.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 90.6

         \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b} \]

   7. Applied rewrites 90.6%

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

Alternative 7: 90.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1 \cdot \left(-b\right)}{a}, a, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.1)
   (/
    (*
     (fma (/ (* 1.0 (- b)) a) a (sqrt (fma (* -3.0 a) c (* b b))))
     0.3333333333333333)
    a)
   (/ (* c (- (* -0.375 (/ (* a c) (pow b 2.0))) 0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.1) {
		tmp = (fma(((1.0 * -b) / a), a, sqrt(fma((-3.0 * a), c, (b * b)))) * 0.3333333333333333) / a;
	} else {
		tmp = (c * ((-0.375 * ((a * c) / pow(b, 2.0))) - 0.5)) / 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)) <= -0.1)
		tmp = Float64(Float64(fma(Float64(Float64(1.0 * Float64(-b)) / a), a, sqrt(fma(Float64(-3.0 * a), c, Float64(b * b)))) * 0.3333333333333333) / a);
	else
		tmp = Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / (b ^ 2.0))) - 0.5)) / 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], -0.1], N[(N[(N[(N[(N[(1.0 * (-b)), $MachinePrecision] / a), $MachinePrecision] * a + N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.10000000000000001

   1. Initial program 31.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-out N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 31.7

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

   3. Applied rewrites 31.7%

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

   5. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(-b\right) - \frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}}{-3 \cdot a}} \]

   7. Applied rewrites 33.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1 \cdot \left(-b\right)}{a}, a, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right) \cdot 0.3333333333333333}{a}} \]

   if -0.10000000000000001 < (/.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 31.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.265625, {a}^{4} \cdot {c}^{4}, 5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 90.6

         \[\leadsto \frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b} \]

   7. Applied rewrites 90.6%

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

Alternative 8: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1 \cdot \left(-b\right)}{a}, a, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot c, -0.5, \left(\left(c \cdot a\right) \cdot c\right) \cdot -0.375\right)}{b \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.1)
   (/
    (*
     (fma (/ (* 1.0 (- b)) a) a (sqrt (fma (* -3.0 a) c (* b b))))
     0.3333333333333333)
    a)
   (/ (/ (fma (* (* b b) c) -0.5 (* (* (* c a) c) -0.375)) (* b b)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.1) {
		tmp = (fma(((1.0 * -b) / a), a, sqrt(fma((-3.0 * a), c, (b * b)))) * 0.3333333333333333) / a;
	} else {
		tmp = (fma(((b * b) * c), -0.5, (((c * a) * c) * -0.375)) / (b * b)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.1)
		tmp = Float64(Float64(fma(Float64(Float64(1.0 * Float64(-b)) / a), a, sqrt(fma(Float64(-3.0 * a), c, Float64(b * b)))) * 0.3333333333333333) / a);
	else
		tmp = Float64(Float64(fma(Float64(Float64(b * b) * c), -0.5, Float64(Float64(Float64(c * a) * c) * -0.375)) / Float64(b * b)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[(N[(N[(N[(1.0 * (-b)), $MachinePrecision] / a), $MachinePrecision] * a + N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * c), $MachinePrecision] * -0.5 + N[(N[(N[(c * a), $MachinePrecision] * c), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.10000000000000001

   1. Initial program 31.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-out N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 31.7

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

   3. Applied rewrites 31.7%

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

   5. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(-b\right) - \frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)}}{-3 \cdot a}} \]

   7. Applied rewrites 33.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1 \cdot \left(-b\right)}{a}, a, \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right) \cdot 0.3333333333333333}{a}} \]

   if -0.10000000000000001 < (/.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 31.7%

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

   2. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\sqrt{{b}^{2}} - b}{a}, c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}} - 0.5 \cdot \frac{1}{\sqrt{{b}^{2}}}\right)\right)} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 90.3

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, {b}^{2} \cdot c, -0.375 \cdot \left(a \cdot {c}^{2}\right)\right)}{{b}^{3}} \]

   7. Applied rewrites 90.3%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, {b}^{2} \cdot c, -0.375 \cdot \left(a \cdot {c}^{2}\right)\right)}{\color{blue}{{b}^{3}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow3 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 90.5%

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

Alternative 9: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}, \frac{0.3333333333333333}{a}, \left(-b\right) \cdot \frac{0.3333333333333333}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot c, -0.5, \left(\left(c \cdot a\right) \cdot c\right) \cdot -0.375\right)}{b \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.1)
   (fma
    (sqrt (fma -3.0 (* c a) (* b b)))
    (/ 0.3333333333333333 a)
    (* (- b) (/ 0.3333333333333333 a)))
   (/ (/ (fma (* (* b b) c) -0.5 (* (* (* c a) c) -0.375)) (* b b)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.1) {
		tmp = fma(sqrt(fma(-3.0, (c * a), (b * b))), (0.3333333333333333 / a), (-b * (0.3333333333333333 / a)));
	} else {
		tmp = (fma(((b * b) * c), -0.5, (((c * a) * c) * -0.375)) / (b * b)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.1)
		tmp = fma(sqrt(fma(-3.0, Float64(c * a), Float64(b * b))), Float64(0.3333333333333333 / a), Float64(Float64(-b) * Float64(0.3333333333333333 / a)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(b * b) * c), -0.5, Float64(Float64(Float64(c * a) * c) * -0.375)) / Float64(b * b)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision] + N[((-b) * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * c), $MachinePrecision] * -0.5 + N[(N[(N[(c * a), $MachinePrecision] * c), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.10000000000000001

   1. Initial program 31.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. mult-flip N/A

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

      6. lower-fma.f64 N/A

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

   3. Applied rewrites 33.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}, \frac{0.3333333333333333}{a}, \left(-b\right) \cdot \frac{0.3333333333333333}{a}\right)} \]

   if -0.10000000000000001 < (/.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 31.7%

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

   2. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\sqrt{{b}^{2}} - b}{a}, c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}} - 0.5 \cdot \frac{1}{\sqrt{{b}^{2}}}\right)\right)} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 90.3

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, {b}^{2} \cdot c, -0.375 \cdot \left(a \cdot {c}^{2}\right)\right)}{{b}^{3}} \]

   7. Applied rewrites 90.3%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, {b}^{2} \cdot c, -0.375 \cdot \left(a \cdot {c}^{2}\right)\right)}{\color{blue}{{b}^{3}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow3 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

   9. Applied rewrites 90.5%

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

Alternative 10: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.1:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}, \frac{0.3333333333333333}{a}, \left(-b\right) \cdot \frac{0.3333333333333333}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(b \cdot b\right) \cdot -0.5, c, \left(\left(c \cdot c\right) \cdot a\right) \cdot -0.375\right)}{\left(b \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.1)
   (fma
    (sqrt (fma -3.0 (* c a) (* b b)))
    (/ 0.3333333333333333 a)
    (* (- b) (/ 0.3333333333333333 a)))
   (/ (fma (* (* b b) -0.5) c (* (* (* c c) a) -0.375)) (* (* b b) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.1) {
		tmp = fma(sqrt(fma(-3.0, (c * a), (b * b))), (0.3333333333333333 / a), (-b * (0.3333333333333333 / a)));
	} else {
		tmp = fma(((b * b) * -0.5), c, (((c * c) * a) * -0.375)) / ((b * b) * b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.1)
		tmp = fma(sqrt(fma(-3.0, Float64(c * a), Float64(b * b))), Float64(0.3333333333333333 / a), Float64(Float64(-b) * Float64(0.3333333333333333 / a)));
	else
		tmp = Float64(fma(Float64(Float64(b * b) * -0.5), c, Float64(Float64(Float64(c * c) * a) * -0.375)) / Float64(Float64(b * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.1], N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision] + N[((-b) * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] * -0.5), $MachinePrecision] * c + N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.10000000000000001

   1. Initial program 31.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. mult-flip N/A

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

      6. lower-fma.f64 N/A

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

   3. Applied rewrites 33.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}, \frac{0.3333333333333333}{a}, \left(-b\right) \cdot \frac{0.3333333333333333}{a}\right)} \]

   if -0.10000000000000001 < (/.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 31.7%

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

   2. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{\sqrt{{b}^{2}} - b}{a}, c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{\left(\sqrt{{b}^{2}}\right)}^{3}} - 0.5 \cdot \frac{1}{\sqrt{{b}^{2}}}\right)\right)} \]

   5. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 90.3

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, {b}^{2} \cdot c, -0.375 \cdot \left(a \cdot {c}^{2}\right)\right)}{{b}^{3}} \]

   7. Applied rewrites 90.3%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, {b}^{2} \cdot c, -0.375 \cdot \left(a \cdot {c}^{2}\right)\right)}{\color{blue}{{b}^{3}}} \]

   9. Applied rewrites 90.4%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \left(b \cdot b\right), \frac{c}{\color{blue}{\left(b \cdot b\right) \cdot b}}, -0.375 \cdot \frac{\left(c \cdot a\right) \cdot c}{\left(b \cdot b\right) \cdot b}\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lift-*.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-*r/ N/A

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

   11. Applied rewrites 90.4%

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

Alternative 11: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0022:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0022)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (* -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)) <= -0.0022) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = -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)) <= -0.0022)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(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], -0.0022], 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[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.00220000000000000013

   1. Initial program 31.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. sub-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-out N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 31.7

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

   3. Applied rewrites 31.7%

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

   if -0.00220000000000000013 < (/.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 31.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 12: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0022:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0022)
   (/ (* (- (sqrt (fma -3.0 (* c a) (* b b))) b) 0.3333333333333333) a)
   (* -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)) <= -0.0022) {
		tmp = ((sqrt(fma(-3.0, (c * a), (b * b))) - b) * 0.3333333333333333) / a;
	} else {
		tmp = -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)) <= -0.0022)
		tmp = Float64(Float64(Float64(sqrt(fma(-3.0, Float64(c * a), Float64(b * b))) - b) * 0.3333333333333333) / a);
	else
		tmp = Float64(-0.5 * Float64(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], -0.0022], N[(N[(N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.00220000000000000013

   1. Initial program 31.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

   3. Applied rewrites 31.7%

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b\right) \cdot 0.3333333333333333}{a}} \]

   if -0.00220000000000000013 < (/.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 31.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 13: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0022:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0022)
   (* (- (sqrt (fma -3.0 (* c a) (* b b))) b) (/ 0.3333333333333333 a))
   (* -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)) <= -0.0022) {
		tmp = (sqrt(fma(-3.0, (c * a), (b * b))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -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)) <= -0.0022)
		tmp = Float64(Float64(sqrt(fma(-3.0, Float64(c * a), Float64(b * b))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(-0.5 * Float64(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], -0.0022], N[(N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.00220000000000000013

   1. Initial program 31.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

   3. Applied rewrites 31.7%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]

   if -0.00220000000000000013 < (/.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 31.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 14: 81.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return -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 = (-0.5d0) * (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 81.1

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

4. Applied rewrites 81.1%

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

Reproduce

herbie shell --seed 2025142 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))