Cubic critical, narrow range

Percentage Accurate: 55.8% → 91.4%

Time: 5.2s

Alternatives: 11

Speedup: 3.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.1:\\ \;\;\;\;\frac{\frac{b \cdot b - \sqrt{t\_0 \cdot t\_0}}{\left(-b\right) - \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \mathsf{fma}\left(-1.0546875, \frac{a \cdot c}{b \cdot b}, -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}\right), -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.1)
     (/ (/ (- (* b b) (sqrt (* t_0 t_0))) (- (- b) (sqrt t_0))) (* 3.0 a))
     (/
      (fma
       a
       (fma
        (/
         (* (* (* c c) c) (fma -1.0546875 (/ (* a c) (* b b)) -0.5625))
         (* (* b b) (* b b)))
        a
        (/ (* (* c c) -0.375) (* b b)))
       (* -0.5 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.1) {
		tmp = (((b * b) - sqrt((t_0 * t_0))) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(a, fma(((((c * c) * c) * fma(-1.0546875, ((a * c) / (b * b)), -0.5625)) / ((b * b) * (b * b))), a, (((c * c) * -0.375) / (b * b))), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1.1)
		tmp = Float64(Float64(Float64(Float64(b * b) - sqrt(Float64(t_0 * t_0))) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(a, fma(Float64(Float64(Float64(Float64(c * c) * c) * fma(-1.0546875, Float64(Float64(a * c) / Float64(b * b)), -0.5625)) / Float64(Float64(b * b) * Float64(b * b))), a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b))), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.1], N[(N[(N[(N[(b * b), $MachinePrecision] - N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(-1.0546875 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5625), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.1000000000000001

   1. Initial program 82.0%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 83.5%

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

   if -1.1000000000000001 < (/.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 51.7%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 92.6%

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

   11. Taylor expanded in c around 0

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

       1. lower-*.f64 N/A

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

       2. pow3 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. negate-sub N/A

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

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 92.6

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

   13. Applied rewrites 92.6%

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

Alternative 2: 89.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.5

   1. Initial program 81.0%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 82.6%

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

   if -0.5 < (/.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 51.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 92.8%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 90.3%

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

Alternative 3: 88.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.5

   1. Initial program 81.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. frac-add N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 80.6%

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

   if -0.5 < (/.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 51.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 92.8%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 90.3%

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

Alternative 4: 88.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.5

   1. Initial program 81.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. frac-add N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 80.6%

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

   if -0.5 < (/.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 51.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 90.1%

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

Alternative 5: 84.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.5

   1. Initial program 81.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. frac-add N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 80.6%

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

   if -0.5 < (/.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 51.2%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 84.9

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

   4. Applied rewrites 84.9%

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

Alternative 6: 84.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.5

   1. Initial program 81.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 81.0%

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

   if -0.5 < (/.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 51.2%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 84.9

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

   4. Applied rewrites 84.9%

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

Alternative 7: 84.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.5

   1. Initial program 81.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 81.0%

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

   if -0.5 < (/.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 51.2%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 84.9

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 84.7

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

   7. Applied rewrites 84.7%

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

Alternative 8: 84.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.5

   1. Initial program 81.0%

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

   3. Applied rewrites 81.0%

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

   if -0.5 < (/.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 51.2%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 84.9

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 84.7

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

   7. Applied rewrites 84.7%

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

Alternative 9: 84.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.5

   1. Initial program 81.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 81.2

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

   3. Applied rewrites 81.2%

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

   if -0.5 < (/.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 51.2%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 84.9

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 84.7

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

   7. Applied rewrites 84.7%

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

Alternative 10: 81.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-*.f64 81.1

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

4. Applied rewrites 81.1%

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

5. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. negate-sub N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 81.0

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

7. Applied rewrites 81.0%

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

Alternative 11: 64.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

2. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.1

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

4. Applied rewrites 64.1%

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

Reproduce

herbie shell --seed 2025119 
(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)))