Cubic critical, narrow range

Percentage Accurate: 54.8% → 91.5%

Time: 11.2s

Alternatives: 10

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.8% 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: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;b \leq 3.72:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \left(\mathsf{fma}\left(a \cdot c, -0.5625, \left(b \cdot b\right) \cdot -0.375\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -3.0 c (/ (* b b) a)) a)))
   (if (<= b 3.72)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* a 3.0))))
     (fma
      (/
       (fma
        (* -1.0546875 (pow c 4.0))
        (* a a)
        (* (* (fma (* a c) -0.5625 (* (* b b) -0.375)) (* c c)) (* b b)))
       (pow b 7.0))
      a
      (* (/ c b) -0.5)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, c, ((b * b) / a)) * a;
	double tmp;
	if (b <= 3.72) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(a * 3.0));
	} else {
		tmp = fma((fma((-1.0546875 * pow(c, 4.0)), (a * a), ((fma((a * c), -0.5625, ((b * b) * -0.375)) * (c * c)) * (b * b))) / pow(b, 7.0)), a, ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-3.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (b <= 3.72)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(a * 3.0))));
	else
		tmp = fma(Float64(fma(Float64(-1.0546875 * (c ^ 4.0)), Float64(a * a), Float64(Float64(fma(Float64(a * c), -0.5625, Float64(Float64(b * b) * -0.375)) * Float64(c * c)) * Float64(b * b))) / (b ^ 7.0)), a, Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[b, 3.72], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[(a * 3.0), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-1.0546875 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(N[(a * c), $MachinePrecision] * -0.5625 + N[(N[(b * b), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7200000000000002

   1. Initial program 84.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 84.2%

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

   if 3.7200000000000002 < b

   1. Initial program 49.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \frac{-1}{2} \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + \frac{-1}{2} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right) + \frac{-3}{8} \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 93.9%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \mathsf{fma}\left(\left(-0.375 \cdot \left(b \cdot b\right)\right) \cdot c, c, \left({c}^{3} \cdot a\right) \cdot -0.5625\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]

   9. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{-135}{128} \cdot {c}^{4}, a \cdot a, \left({c}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot c\right) + \frac{-3}{8} \cdot {b}^{2}\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 93.9%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \left(\mathsf{fma}\left(a \cdot c, -0.5625, \left(b \cdot b\right) \cdot -0.375\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.72:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-1.0546875 \cdot {c}^{4}, a \cdot a, \left(\mathsf{fma}\left(a \cdot c, -0.5625, \left(b \cdot b\right) \cdot -0.375\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;b \leq 3.72:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-0.5625 \cdot a\right) \cdot \frac{c}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -3.0 c (/ (* b b) a)) a)))
   (if (<= b 3.72)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* a 3.0))))
     (fma
      (* (- (* (* -0.5625 a) (/ c (pow b 5.0))) (/ 0.375 (pow b 3.0))) (* c c))
      a
      (* (/ c b) -0.5)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, c, ((b * b) / a)) * a;
	double tmp;
	if (b <= 3.72) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(a * 3.0));
	} else {
		tmp = fma(((((-0.5625 * a) * (c / pow(b, 5.0))) - (0.375 / pow(b, 3.0))) * (c * c)), a, ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-3.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (b <= 3.72)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(a * 3.0))));
	else
		tmp = fma(Float64(Float64(Float64(Float64(-0.5625 * a) * Float64(c / (b ^ 5.0))) - Float64(0.375 / (b ^ 3.0))) * Float64(c * c)), a, Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[b, 3.72], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[(a * 3.0), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.5625 * a), $MachinePrecision] * N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.375 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7200000000000002

   1. Initial program 84.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 84.2%

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

   if 3.7200000000000002 < b

   1. Initial program 49.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \frac{-1}{2} \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + \frac{-1}{2} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 91.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.72:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(-0.5625 \cdot a\right) \cdot \frac{c}{{b}^{5}} - \frac{0.375}{{b}^{3}}\right) \cdot \left(c \cdot c\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a\\ \mathbf{if}\;b \leq 3.72:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{{b}^{3}} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -3.0 c (/ (* b b) a)) a)))
   (if (<= b 3.72)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* a 3.0))))
     (*
      (fma
       (fma (* c -0.5625) (* a (/ a (pow b 5.0))) (* (/ a (pow b 3.0)) -0.375))
       c
       (/ -0.5 b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-3.0, c, ((b * b) / a)) * a;
	double tmp;
	if (b <= 3.72) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(a * 3.0));
	} else {
		tmp = fma(fma((c * -0.5625), (a * (a / pow(b, 5.0))), ((a / pow(b, 3.0)) * -0.375)), c, (-0.5 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-3.0, c, Float64(Float64(b * b) / a)) * a)
	tmp = 0.0
	if (b <= 3.72)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(a * 3.0))));
	else
		tmp = Float64(fma(fma(Float64(c * -0.5625), Float64(a * Float64(a / (b ^ 5.0))), Float64(Float64(a / (b ^ 3.0)) * -0.375)), c, Float64(-0.5 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * c + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[b, 3.72], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[(a * 3.0), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * -0.5625), $MachinePrecision] * N[(a * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * c + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.7200000000000002

   1. Initial program 84.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 83.7

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

   5. Applied rewrites 83.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 84.2%

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

   if 3.7200000000000002 < b

   1. Initial program 49.6%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.72:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c \cdot -0.5625, a \cdot \frac{a}{{b}^{5}}, \frac{a}{{b}^{3}} \cdot -0.375\right), c, \frac{-0.5}{b}\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 9

   1. Initial program 81.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 82.0%

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

   if 9 < b

   1. Initial program 48.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 87.7

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

   5. Applied rewrites 87.7%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot a, c \cdot \frac{c}{{b}^{3}}, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 9

   1. Initial program 81.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 82.0%

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

   if 9 < b

   1. Initial program 48.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \frac{-1}{2} \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + \frac{-1}{2} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.7

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

   8. Applied rewrites 87.7%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(-3, c, \frac{b \cdot b}{a}\right) \cdot a}\right) \cdot \left(-a \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1e-6)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1e-6) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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)) <= -1e-6)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	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], -1e-6], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $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)) < -9.99999999999999955e-7

   1. Initial program 73.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-abs-rev N/A

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

      10. fabs-fabs N/A

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

      11. fabs-fabs N/A

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

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

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   4. Applied rewrites 73.7%

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

   if -9.99999999999999955e-7 < (/.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 32.1%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 83.6

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

   5. Applied rewrites 83.6%

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

Alternative 7: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.375 \cdot a}{b}, \frac{c \cdot c}{b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.0)
   (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) (* 3.0 a))
   (/ (fma (/ (* -0.375 a) b) (/ (* c c) b) (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.0) {
		tmp = (-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / (3.0 * a);
	} else {
		tmp = fma(((-0.375 * a) / b), ((c * c) / b), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(-0.375 * a) / b), Float64(Float64(c * c) / b), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.375 * a), $MachinePrecision] / b), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7

   1. Initial program 82.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-abs-rev N/A

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

      10. fabs-fabs N/A

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

      11. fabs-fabs N/A

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

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

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   4. Applied rewrites 82.9%

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

   if 7 < b

   1. Initial program 48.9%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) + \frac{-1}{2} \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right) \cdot a} + \frac{-1}{2} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{6.328125}{b}, \frac{{c}^{3} \cdot -0.5625}{{b}^{5}}\right), a, \frac{\left(c \cdot c\right) \cdot -0.375}{{b}^{3}}\right), a, \frac{c}{b} \cdot -0.5\right)} \]

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.2

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

   8. Applied rewrites 87.2%

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

Alternative 8: 64.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 62.7

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

5. Applied rewrites 62.7%

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

Alternative 9: 64.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 62.7

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

5. Applied rewrites 62.7%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
6. Step-by-step derivation

7. Applied rewrites 62.7%

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

Alternative 10: 3.2% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 57.2

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

5. Applied rewrites 57.2%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

7. Applied rewrites 57.0%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 57.2%

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

11. Taylor expanded in a around 0

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

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

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

    2. metadata-eval N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot b - \color{blue}{\frac{-1}{3}} \cdot b}{a} \]

    3. +-inverses N/A

       \[\leadsto \frac{\color{blue}{0}}{a} \]

    4. div0 3.2

       \[\leadsto \color{blue}{0} \]

13. Applied rewrites 3.2%

    \[\leadsto \color{blue}{0} \]

14. Final simplification 3.2%

    \[\leadsto 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024359 
(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)))