Cubic critical, narrow range

Percentage Accurate: 55.2% → 92.1%

Time: 8.4s

Alternatives: 13

Speedup: 3.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -70:\\ \;\;\;\;\frac{\left(a \cdot 3\right) \cdot \frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{\left(a \cdot a\right) \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c, c \cdot \left(-0.5625 \cdot {b}^{-4}\right), \mathsf{fma}\left(\left(\frac{c}{b \cdot b} \cdot c\right) \cdot -0.375, a, \frac{-1.0546875}{{b}^{6} \cdot a} \cdot {\left(c \cdot a\right)}^{4}\right)\right)}{b}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -70.0)
     (/ (* (* a 3.0) (/ (- t_0 (* b b)) (+ (sqrt t_0) b))) (* (* a a) 9.0))
     (fma
      -0.5
      (/ c b)
      (/
       (fma
        (* (* (* a a) c) c)
        (* c (* -0.5625 (pow b -4.0)))
        (fma
         (* (* (/ c (* b b)) c) -0.375)
         a
         (* (/ -1.0546875 (* (pow b 6.0) a)) (pow (* c a) 4.0))))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -70.0) {
		tmp = ((a * 3.0) * ((t_0 - (b * b)) / (sqrt(t_0) + b))) / ((a * a) * 9.0);
	} else {
		tmp = fma(-0.5, (c / b), (fma((((a * a) * c) * c), (c * (-0.5625 * pow(b, -4.0))), fma((((c / (b * b)) * c) * -0.375), a, ((-1.0546875 / (pow(b, 6.0) * a)) * pow((c * a), 4.0)))) / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -70.0)
		tmp = Float64(Float64(Float64(a * 3.0) * Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b))) / Float64(Float64(a * a) * 9.0));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(fma(Float64(Float64(Float64(a * a) * c) * c), Float64(c * Float64(-0.5625 * (b ^ -4.0))), fma(Float64(Float64(Float64(c / Float64(b * b)) * c) * -0.375), a, Float64(Float64(-1.0546875 / Float64((b ^ 6.0) * a)) * (Float64(c * a) ^ 4.0)))) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -70.0], N[(N[(N[(a * 3.0), $MachinePrecision] * N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * a), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * N[(c * N[(-0.5625 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * -0.375), $MachinePrecision] * a + N[(N[(-1.0546875 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -70

   1. Initial program 55.2%

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

      1. lift-/.f64 N/A

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

      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. div-add N/A

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

      4. common-denominator N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\frac{\left(a \cdot 3\right) \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right)}{\left(a \cdot a\right) \cdot 9}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(a \cdot 3\right) \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right)}}{\left(a \cdot a\right) \cdot 9} \]

      2. flip-- N/A

         \[\leadsto \frac{\left(a \cdot 3\right) \cdot \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}}{\left(a \cdot a\right) \cdot 9} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{\left(a \cdot 3\right) \cdot \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}}{\left(a \cdot a\right) \cdot 9} \]

   5. Applied rewrites 56.7%

      \[\leadsto \frac{\left(a \cdot 3\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}}}{\left(a \cdot a\right) \cdot 9} \]

   if -70 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.8%

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

   6. Applied rewrites 90.8%

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

   8. Applied rewrites 90.8%

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

Alternative 2: 92.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -70:\\ \;\;\;\;\frac{\left(a \cdot 3\right) \cdot \frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{\left(a \cdot a\right) \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left({\left(c \cdot a\right)}^{4}, \frac{-1.0546875}{{b}^{6} \cdot a}, \mathsf{fma}\left(\left(\frac{c}{b \cdot b} \cdot c\right) \cdot -0.375, a, \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot \left(c \cdot \left(-0.5625 \cdot {b}^{-4}\right)\right)\right)\right)\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 c) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -70.0)
     (/ (* (* a 3.0) (/ (- t_0 (* b b)) (+ (sqrt t_0) b))) (* (* a a) 9.0))
     (/
      (fma
       c
       -0.5
       (fma
        (pow (* c a) 4.0)
        (/ -1.0546875 (* (pow b 6.0) a))
        (fma
         (* (* (/ c (* b b)) c) -0.375)
         a
         (* (* (* (* a a) c) c) (* c (* -0.5625 (pow b -4.0)))))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * c), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -70.0) {
		tmp = ((a * 3.0) * ((t_0 - (b * b)) / (sqrt(t_0) + b))) / ((a * a) * 9.0);
	} else {
		tmp = fma(c, -0.5, fma(pow((c * a), 4.0), (-1.0546875 / (pow(b, 6.0) * a)), fma((((c / (b * b)) * c) * -0.375), a, ((((a * a) * c) * c) * (c * (-0.5625 * pow(b, -4.0))))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -70.0)
		tmp = Float64(Float64(Float64(a * 3.0) * Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b))) / Float64(Float64(a * a) * 9.0));
	else
		tmp = Float64(fma(c, -0.5, fma((Float64(c * a) ^ 4.0), Float64(-1.0546875 / Float64((b ^ 6.0) * a)), fma(Float64(Float64(Float64(c / Float64(b * b)) * c) * -0.375), a, Float64(Float64(Float64(Float64(a * a) * c) * c) * Float64(c * Float64(-0.5625 * (b ^ -4.0))))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -70.0], N[(N[(N[(a * 3.0), $MachinePrecision] * N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * a), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5 + N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(-1.0546875 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * -0.375), $MachinePrecision] * a + N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * N[(c * N[(-0.5625 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -70

   1. Initial program 55.2%

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

      1. lift-/.f64 N/A

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

      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. div-add N/A

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

      4. common-denominator N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\frac{\left(a \cdot 3\right) \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right)}{\left(a \cdot a\right) \cdot 9}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(a \cdot 3\right) \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right)}}{\left(a \cdot a\right) \cdot 9} \]

      2. flip-- N/A

         \[\leadsto \frac{\left(a \cdot 3\right) \cdot \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}}{\left(a \cdot a\right) \cdot 9} \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{\left(a \cdot 3\right) \cdot \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b}}}{\left(a \cdot a\right) \cdot 9} \]

   5. Applied rewrites 56.7%

      \[\leadsto \frac{\left(a \cdot 3\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b}}}{\left(a \cdot a\right) \cdot 9} \]

   if -70 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.8%

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

   6. Applied rewrites 90.8%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   8. Applied rewrites 90.8%

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

Alternative 3: 89.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.00265:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, a \cdot \mathsf{fma}\left(-0.5625, \frac{a \cdot {c}^{3}}{{b}^{5}}, -0.375 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.00265)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 3.0 a))
     (fma
      -0.5
      (/ c b)
      (*
       a
       (fma
        -0.5625
        (/ (* a (pow c 3.0)) (pow b 5.0))
        (* -0.375 (/ (pow c 2.0) (pow b 3.0)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.00265) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), (a * fma(-0.5625, ((a * pow(c, 3.0)) / pow(b, 5.0)), (-0.375 * (pow(c, 2.0) / pow(b, 3.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.00265)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(a * fma(-0.5625, Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0)), Float64(-0.375 * Float64((c ^ 2.0) / (b ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00265], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(a * N[(-0.5625 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.00265

   1. Initial program 55.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 56.7%

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

   if -0.00265 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 87.8%

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

Alternative 4: 88.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.00265:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot c}{{b}^{4}}, -0.375 \cdot \frac{a}{{b}^{2}}\right) - 0.5\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.00265)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 3.0 a))
     (/
      (*
       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) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.00265) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (3.0 * a);
	} else {
		tmp = (c * ((c * fma(-0.5625, ((pow(a, 2.0) * c) / pow(b, 4.0)), (-0.375 * (a / pow(b, 2.0))))) - 0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.00265)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(3.0 * a));
	else
		tmp = 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
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00265], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(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. 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)) < -0.00265

   1. Initial program 55.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 56.7%

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

   if -0.00265 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.8%

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

   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 87.7%

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

Alternative 5: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.00265:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.00265)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 3.0 a))
     (/ (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 t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.00265) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (3.0 * a);
	} 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)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.00265)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(3.0 * a));
	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_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00265], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-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)) < -0.00265

   1. Initial program 55.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 56.7%

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

   if -0.00265 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. 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 81.7%

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

   4. Applied rewrites 81.7%

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

Alternative 6: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.00265:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.00265)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 3.0 a))
     (fma -0.5 (/ c b) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.00265) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.00265)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00265], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $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)) < -0.00265

   1. Initial program 55.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 56.7%

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

   if -0.00265 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.7%

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

   7. Applied rewrites 81.7%

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

Alternative 7: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.00265:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -3.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.00265)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 3.0 a))
     (/ (* c (- (* -0.375 (/ (* a c) (pow b 2.0))) 0.5)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -3.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.00265) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (3.0 * a);
	} else {
		tmp = (c * ((-0.375 * ((a * c) / pow(b, 2.0))) - 0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -3.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.00265)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(3.0 * 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_] := Block[{t$95$0 = N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00265], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(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)) < -0.00265

   1. Initial program 55.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 56.7%

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

   if -0.00265 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.8%

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

   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 81.6%

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

   7. Applied rewrites 81.6%

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

Alternative 8: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.00265:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{2}} - 0.5\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.00265)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 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)) <= -0.00265) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * 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)) <= -0.00265)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * 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], -0.00265], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(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)) < -0.00265

   1. Initial program 55.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. add-flip N/A

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

      5. sub-flip N/A

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

      6. lift-*.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sqr-neg-rev N/A

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

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

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

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

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

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

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

      14. lift-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. lift-*.f64 N/A

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

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

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 55.3%

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

   if -0.00265 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   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 90.8%

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

   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 81.6%

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

   7. Applied rewrites 81.6%

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

Alternative 9: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -8e-8)
   (/ (+ (- 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)) <= -8e-8) {
		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)) <= -8e-8)
		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], -8e-8], 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)) < -8.0000000000000002e-8

   1. Initial program 55.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. add-flip N/A

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

      5. sub-flip N/A

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

      6. lift-*.f64 N/A

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

      7. sqr-neg-rev N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. sqr-neg-rev N/A

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

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

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

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

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

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

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

      14. lift-*.f64 N/A

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

      15. remove-double-neg N/A

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

      16. lift-*.f64 N/A

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

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

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

      18. lower-fma.f64 N/A

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

   3. Applied rewrites 55.3%

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

   if -8.0000000000000002e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.5%

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

   4. Applied rewrites 64.5%

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

Alternative 10: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -8e-8)
   (/ (- (sqrt (fma (* c -3.0) a (* b b))) b) (* a 3.0))
   (* -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)) <= -8e-8) {
		tmp = (sqrt(fma((c * -3.0), a, (b * b))) - b) / (a * 3.0);
	} 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)) <= -8e-8)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) / Float64(a * 3.0));
	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], -8e-8], N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $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)) < -8.0000000000000002e-8

   1. Initial program 55.2%

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

   3. Applied rewrites 55.2%

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

   if -8.0000000000000002e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.5%

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

   4. Applied rewrites 64.5%

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

Alternative 11: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -8e-8)
   (* (/ (- (sqrt (fma (* c -3.0) a (* b b))) b) a) 0.3333333333333333)
   (* -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)) <= -8e-8) {
		tmp = ((sqrt(fma((c * -3.0), a, (b * b))) - b) / a) * 0.3333333333333333;
	} 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)) <= -8e-8)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	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], -8e-8], N[(N[(N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $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)) < -8.0000000000000002e-8

   1. Initial program 55.2%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. mult-flip N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 55.2%

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

   if -8.0000000000000002e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.5%

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

   4. Applied rewrites 64.5%

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

Alternative 12: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -8 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -8e-8)
   (* (/ 0.3333333333333333 a) (- (sqrt (fma (* c -3.0) a (* b b))) b))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -8e-8) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma((c * -3.0), a, (b * b))) - b);
	} 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)) <= -8e-8)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(Float64(c * -3.0), a, Float64(b * b))) - b));
	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], -8e-8], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(c * -3.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $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)) < -8.0000000000000002e-8

   1. Initial program 55.2%

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

      1. lift-/.f64 N/A

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

      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. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 55.2%

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. add-flip N/A

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

      12. lower--.f64 N/A

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

   3. Applied rewrites 55.2%

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

   if -8.0000000000000002e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.2%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.5%

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

   4. Applied rewrites 64.5%

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

Alternative 13: 64.5% accurate, 3.2× speedup

Math

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

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 55.2%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 64.5%

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

4. Applied rewrites 64.5%

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

Reproduce

herbie shell --seed 2025192 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))