Cubic critical, narrow range

Percentage Accurate: 55.6% → 92.3%

Time: 7.6s

Alternatives: 14

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: 55.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -5:\\ \;\;\;\;\frac{\frac{\left(b \cdot b\right) \cdot \left(-b\right) + t\_1 \cdot t\_0}{\mathsf{fma}\left(b, b, t\_0 - \left(-b\right) \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, a \cdot \mathsf{fma}\left(-0.375, \frac{c \cdot c}{b \cdot b}, a \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(-1.0546875 \cdot \frac{a \cdot c}{{b}^{6}} - 0.5625 \cdot {b}^{-4}\right)\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -5.0)
     (/
      (/ (+ (* (* b b) (- b)) (* t_1 t_0)) (fma b b (- t_0 (* (- b) t_1))))
      (* 3.0 a))
     (/
      (fma
       -0.5
       c
       (*
        a
        (fma
         -0.375
         (/ (* c c) (* b b))
         (*
          a
          (*
           (* (* c c) c)
           (-
            (* -1.0546875 (/ (* a c) (pow b 6.0)))
            (* 0.5625 (pow b -4.0))))))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -5.0) {
		tmp = ((((b * b) * -b) + (t_1 * t_0)) / fma(b, b, (t_0 - (-b * t_1)))) / (3.0 * a);
	} else {
		tmp = fma(-0.5, c, (a * fma(-0.375, ((c * c) / (b * b)), (a * (((c * c) * c) * ((-1.0546875 * ((a * c) / pow(b, 6.0))) - (0.5625 * pow(b, -4.0)))))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -5.0)
		tmp = Float64(Float64(Float64(Float64(Float64(b * b) * Float64(-b)) + Float64(t_1 * t_0)) / fma(b, b, Float64(t_0 - Float64(Float64(-b) * t_1)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(-0.5, c, Float64(a * fma(-0.375, Float64(Float64(c * c) / Float64(b * b)), Float64(a * Float64(Float64(Float64(c * c) * c) * Float64(Float64(-1.0546875 * Float64(Float64(a * c) / (b ^ 6.0))) - Float64(0.5625 * (b ^ -4.0)))))))) / 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]}, Block[{t$95$1 = N[Sqrt[t$95$0], $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], -5.0], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * (-b)), $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(t$95$0 - N[((-b) * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c + N[(a * N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(-1.0546875 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5625 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5

   1. Initial program 55.6%

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

   3. Applied rewrites 57.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

   5. Applied rewrites 56.5%

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

   if -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 55.6%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 90.9%

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

   8. Taylor expanded in c around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, c, a \cdot \mathsf{fma}\left(\frac{-3}{8}, \frac{c \cdot c}{b \cdot b}, a \cdot \left({c}^{3} \cdot \left(\frac{-135}{128} \cdot \frac{a \cdot c}{{b}^{6}} - \frac{9}{16} \cdot \frac{1}{{b}^{4}}\right)\right)\right)\right)}{b} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-flip N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval 90.9

          \[\leadsto \frac{\mathsf{fma}\left(-0.5, c, a \cdot \mathsf{fma}\left(-0.375, \frac{c \cdot c}{b \cdot b}, a \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(-1.0546875 \cdot \frac{a \cdot c}{{b}^{6}} - 0.5625 \cdot {b}^{-4}\right)\right)\right)\right)}{b} \]

   10. Applied rewrites 90.9%

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

Alternative 2: 92.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \left(b \cdot b\right) \cdot b\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -5:\\ \;\;\;\;\frac{\frac{\left(b \cdot b\right) \cdot \left(-b\right) + t\_1 \cdot t\_0}{\mathsf{fma}\left(b, b, t\_0 - \left(-b\right) \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, c \cdot \mathsf{fma}\left(-1.0546875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{t\_2 \cdot t\_2}, -0.5625 \cdot \frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right) - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)) (t_2 (* (* b b) b)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -5.0)
     (/
      (/ (+ (* (* b b) (- b)) (* t_1 t_0)) (fma b b (- t_0 (* (- b) t_1))))
      (* 3.0 a))
     (/
      (*
       c
       (-
        (*
         c
         (fma
          -0.375
          (/ a (* b b))
          (*
           c
           (fma
            -1.0546875
            (/ (* (* (* a a) a) c) (* t_2 t_2))
            (* -0.5625 (/ (* a a) (* (* b b) (* b b))))))))
        0.5))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double t_2 = (b * b) * b;
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -5.0) {
		tmp = ((((b * b) * -b) + (t_1 * t_0)) / fma(b, b, (t_0 - (-b * t_1)))) / (3.0 * a);
	} else {
		tmp = (c * ((c * fma(-0.375, (a / (b * b)), (c * fma(-1.0546875, ((((a * a) * a) * c) / (t_2 * t_2)), (-0.5625 * ((a * a) / ((b * b) * (b * b)))))))) - 0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = sqrt(t_0)
	t_2 = Float64(Float64(b * 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)) <= -5.0)
		tmp = Float64(Float64(Float64(Float64(Float64(b * b) * Float64(-b)) + Float64(t_1 * t_0)) / fma(b, b, Float64(t_0 - Float64(Float64(-b) * t_1)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * fma(-0.375, Float64(a / Float64(b * b)), Float64(c * fma(-1.0546875, Float64(Float64(Float64(Float64(a * a) * a) * c) / Float64(t_2 * t_2)), Float64(-0.5625 * Float64(Float64(a * a) / Float64(Float64(b * b) * Float64(b * b)))))))) - 0.5)) / 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]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(b * b), $MachinePrecision] * b), $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], -5.0], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * (-b)), $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(t$95$0 - N[((-b) * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(-0.375 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * N[(-1.0546875 * N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * c), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5

   1. Initial program 55.6%

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

   3. Applied rewrites 57.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

   5. Applied rewrites 56.5%

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

   if -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 55.6%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in c around 0

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

   7. Applied rewrites 90.8%

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

Alternative 3: 92.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot b\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -5:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, c \cdot \mathsf{fma}\left(-1.0546875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot c}{t\_1 \cdot t\_1}, -0.5625 \cdot \frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)\right) - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (* (* b b) b)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -5.0)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (/
      (*
       c
       (-
        (*
         c
         (fma
          -0.375
          (/ a (* b b))
          (*
           c
           (fma
            -1.0546875
            (/ (* (* (* a a) a) c) (* t_1 t_1))
            (* -0.5625 (/ (* a a) (* (* b b) (* b b))))))))
        0.5))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = (b * b) * b;
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -5.0) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = (c * ((c * fma(-0.375, (a / (b * b)), (c * fma(-1.0546875, ((((a * a) * a) * c) / (t_1 * t_1)), (-0.5625 * ((a * a) / ((b * b) * (b * b)))))))) - 0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = Float64(Float64(b * 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)) <= -5.0)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * fma(-0.375, Float64(a / Float64(b * b)), Float64(c * fma(-1.0546875, Float64(Float64(Float64(Float64(a * a) * a) * c) / Float64(t_1 * t_1)), Float64(-0.5625 * Float64(Float64(a * a) / Float64(Float64(b * b) * Float64(b * b)))))))) - 0.5)) / 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]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * b), $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], -5.0], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(-0.375 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * N[(-1.0546875 * N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * c), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -5

   1. Initial program 55.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 57.1%

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

   if -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 55.6%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in c around 0

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

   7. Applied rewrites 90.8%

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

Alternative 4: 91.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(a \cdot c\right)}^{4}\\ t_1 := \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}\\ t_2 := \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}\\ t_3 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, t\_2, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, t\_1, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, t\_0, 5.0625 \cdot t\_0\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, t\_2, \mathsf{fma}\left(4.5, t\_1, 5.0625 \cdot \frac{t\_0}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, t\_3 - \left(-b\right) \cdot \sqrt{t\_3}\right)}}{3 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* a c) 4.0))
        (t_1 (/ (* (* a a) (* c c)) (* b b)))
        (t_2 (/ (* (* (* a a) a) (* (* c c) c)) (pow b 4.0)))
        (t_3 (fma (* -3.0 a) c (* b b))))
   (/
    (/
     (*
      b
      (fma
       -3.0
       (* a c)
       (fma
        -1.6875
        t_2
        (fma
         -1.5
         (* a c)
         (fma
          -1.125
          t_1
          (fma
           -0.5
           (/ (fma 1.265625 t_0 (* 5.0625 t_0)) (pow b 6.0))
           (fma 3.375 t_2 (fma 4.5 t_1 (* 5.0625 (/ t_0 (pow b 6.0)))))))))))
     (fma b b (- t_3 (* (- b) (sqrt t_3)))))
    (* 3.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = pow((a * c), 4.0);
	double t_1 = ((a * a) * (c * c)) / (b * b);
	double t_2 = (((a * a) * a) * ((c * c) * c)) / pow(b, 4.0);
	double t_3 = fma((-3.0 * a), c, (b * b));
	return ((b * fma(-3.0, (a * c), fma(-1.6875, t_2, fma(-1.5, (a * c), fma(-1.125, t_1, fma(-0.5, (fma(1.265625, t_0, (5.0625 * t_0)) / pow(b, 6.0)), fma(3.375, t_2, fma(4.5, t_1, (5.0625 * (t_0 / pow(b, 6.0))))))))))) / fma(b, b, (t_3 - (-b * sqrt(t_3))))) / (3.0 * a);
}

Julia

function code(a, b, c)
	t_0 = Float64(a * c) ^ 4.0
	t_1 = Float64(Float64(Float64(a * a) * Float64(c * c)) / Float64(b * b))
	t_2 = Float64(Float64(Float64(Float64(a * a) * a) * Float64(Float64(c * c) * c)) / (b ^ 4.0))
	t_3 = fma(Float64(-3.0 * a), c, Float64(b * b))
	return Float64(Float64(Float64(b * fma(-3.0, Float64(a * c), fma(-1.6875, t_2, fma(-1.5, Float64(a * c), fma(-1.125, t_1, fma(-0.5, Float64(fma(1.265625, t_0, Float64(5.0625 * t_0)) / (b ^ 6.0)), fma(3.375, t_2, fma(4.5, t_1, Float64(5.0625 * Float64(t_0 / (b ^ 6.0))))))))))) / fma(b, b, Float64(t_3 - Float64(Float64(-b) * sqrt(t_3))))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(b * N[(-3.0 * N[(a * c), $MachinePrecision] + N[(-1.6875 * t$95$2 + N[(-1.5 * N[(a * c), $MachinePrecision] + N[(-1.125 * t$95$1 + N[(-0.5 * N[(N[(1.265625 * t$95$0 + N[(5.0625 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(3.375 * t$95$2 + N[(4.5 * t$95$1 + N[(5.0625 * N[(t$95$0 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(t$95$3 - N[((-b) * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 55.6%

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

3. Applied rewrites 57.5%

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

4. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(-3 \cdot \left(a \cdot c\right) + \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-3}{2} \cdot \left(a \cdot c\right) + \left(\frac{-9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(\frac{-1}{2} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(\frac{27}{8} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{9}{2} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{81}{16} \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

6. Applied rewrites 91.4%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, {\left(a \cdot c\right)}^{4}, 5.0625 \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(4.5, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 5.0625 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]
7. Add Preprocessing

Alternative 5: 90.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \frac{a \cdot a}{\left(b \cdot b\right) \cdot b}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \left(-1 \cdot \left(c \cdot \mathsf{fma}\left(-2.25, t\_1, 1.125 \cdot t\_1\right)\right) - -1.5 \cdot \frac{a}{b}\right)\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (/ (* a a) (* (* b b) b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (/
      1.0
      (/
       (fma
        -2.0
        b
        (*
         c
         (- (* -1.0 (* c (fma -2.25 t_1 (* 1.125 t_1)))) (* -1.5 (/ a b)))))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = (a * a) / ((b * b) * b);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = 1.0 / (fma(-2.0, b, (c * ((-1.0 * (c * fma(-2.25, t_1, (1.125 * t_1)))) - (-1.5 * (a / b))))) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = Float64(Float64(a * a) / Float64(Float64(b * 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)) <= -1.5)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(c * Float64(Float64(-1.0 * Float64(c * fma(-2.25, t_1, Float64(1.125 * t_1)))) - Float64(-1.5 * Float64(a / b))))) / c));
	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]}, Block[{t$95$1 = N[(N[(a * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * 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], -1.5], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(c * N[(N[(-1.0 * N[(c * N[(-2.25 * t$95$1 + N[(1.125 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.5

   1. Initial program 55.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 57.1%

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

   if -1.5 < (/.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 55.6%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 87.7%

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

   6. Applied rewrites 87.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{b}{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}}} \]

   7. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 88.2%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \left(-1 \cdot \left(c \cdot \mathsf{fma}\left(-2.25, \frac{a \cdot a}{\left(b \cdot b\right) \cdot b}, 1.125 \cdot \frac{a \cdot a}{\left(b \cdot b\right) \cdot b}\right)\right) - -1.5 \cdot \frac{a}{b}\right)\right)}{\color{blue}{c}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \frac{c}{\left(b \cdot b\right) \cdot b}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, a \cdot \mathsf{fma}\left(-1, a \cdot \mathsf{fma}\left(-2.25, t\_1, 1.125 \cdot t\_1\right), 1.5 \cdot \frac{1}{b}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (/ c (* (* b b) b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (/
      1.0
      (fma
       -2.0
       (/ b c)
       (*
        a
        (fma -1.0 (* a (fma -2.25 t_1 (* 1.125 t_1))) (* 1.5 (/ 1.0 b)))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = c / ((b * b) * b);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = 1.0 / fma(-2.0, (b / c), (a * fma(-1.0, (a * fma(-2.25, t_1, (1.125 * t_1))), (1.5 * (1.0 / b)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = Float64(c / Float64(Float64(b * 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)) <= -1.5)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / fma(-2.0, Float64(b / c), Float64(a * fma(-1.0, Float64(a * fma(-2.25, t_1, Float64(1.125 * t_1))), Float64(1.5 * Float64(1.0 / 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]}, Block[{t$95$1 = N[(c / N[(N[(b * b), $MachinePrecision] * 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], -1.5], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(a * N[(-1.0 * N[(a * N[(-2.25 * t$95$1 + N[(1.125 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.5

   1. Initial program 55.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 57.1%

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

   if -1.5 < (/.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 55.6%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 87.7%

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

   6. Applied rewrites 87.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{b}{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}}} \]

   7. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

   9. Applied rewrites 88.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \color{blue}{\frac{b}{c}}, a \cdot \mathsf{fma}\left(-1, a \cdot \mathsf{fma}\left(-2.25, \frac{c}{\left(b \cdot b\right) \cdot b}, 1.125 \cdot \frac{c}{\left(b \cdot b\right) \cdot b}\right), 1.5 \cdot \frac{1}{b}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, a \cdot \frac{\mathsf{fma}\left(-0.5625, \frac{a \cdot \left(\left(c \cdot c\right) \cdot c\right)}{b \cdot b}, -0.375 \cdot \left(c \cdot c\right)\right)}{b \cdot b}\right)}{b}\\ \end{array} \end{array} \]

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)) -1.5)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (/
      (fma
       -0.5
       c
       (*
        a
        (/
         (fma -0.5625 (/ (* a (* (* c c) c)) (* b b)) (* -0.375 (* c c)))
         (* b b))))
      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)) <= -1.5) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(-0.5, c, (a * (fma(-0.5625, ((a * ((c * c) * c)) / (b * b)), (-0.375 * (c * c))) / (b * b)))) / 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)) <= -1.5)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(-0.5, c, Float64(a * Float64(fma(-0.5625, Float64(Float64(a * Float64(Float64(c * c) * c)) / Float64(b * b)), Float64(-0.375 * Float64(c * c))) / Float64(b * b)))) / 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], -1.5], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c + N[(a * N[(N[(-0.5625 * N[(N[(a * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.5

   1. Initial program 55.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 57.1%

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

   if -1.5 < (/.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 55.6%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 90.9%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 87.8

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

   10. Applied rewrites 87.8%

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

Alternative 8: 89.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, c, a \cdot \mathsf{fma}\left(-0.5625, \frac{a \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.375 \cdot \frac{c \cdot c}{b \cdot b}\right)\right)}{b}\\ \end{array} \end{array} \]

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)) -1.5)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (/
      (fma
       -0.5
       c
       (*
        a
        (fma
         -0.5625
         (/ (* a (* (* c c) c)) (* (* b b) (* b b)))
         (* -0.375 (/ (* c c) (* b b))))))
      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)) <= -1.5) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(-0.5, c, (a * fma(-0.5625, ((a * ((c * c) * c)) / ((b * b) * (b * b))), (-0.375 * ((c * c) / (b * b)))))) / 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)) <= -1.5)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(-0.5, c, Float64(a * fma(-0.5625, Float64(Float64(a * Float64(Float64(c * c) * c)) / Float64(Float64(b * b) * Float64(b * b))), Float64(-0.375 * Float64(Float64(c * c) / Float64(b * b)))))) / 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], -1.5], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c + N[(a * N[(-0.5625 * N[(N[(a * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.5

   1. Initial program 55.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 57.1%

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

   if -1.5 < (/.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 55.6%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 87.8%

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

Alternative 9: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.018:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \end{array} \]

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)) -0.018)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (/ 1.0 (/ (fma -2.0 b (* 1.5 (/ (* a c) b))) c)))))

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)) <= -0.018) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = 1.0 / (fma(-2.0, b, (1.5 * ((a * c) / b))) / c);
	}
	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)) <= -0.018)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(1.5 * Float64(Float64(a * c) / b))) / c));
	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], -0.018], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0179999999999999986

   1. Initial program 55.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 57.1%

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

   if -0.0179999999999999986 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.6%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 87.7%

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

   6. Applied rewrites 87.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{b}{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}}} \]

   7. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 82.0

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}} \]

   9. Applied rewrites 82.0%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{\color{blue}{c}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.018:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{1 - \frac{\left(a \cdot 3\right) \cdot c}{b \cdot b}}, \left|b\right|, -b\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.018)
   (/
    (fma (sqrt (- 1.0 (/ (* (* a 3.0) c) (* b b)))) (fabs b) (- b))
    (* 3.0 a))
   (/ 1.0 (/ (fma -2.0 b (* 1.5 (/ (* a c) b))) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.018) {
		tmp = fma(sqrt((1.0 - (((a * 3.0) * c) / (b * b)))), fabs(b), -b) / (3.0 * a);
	} else {
		tmp = 1.0 / (fma(-2.0, b, (1.5 * ((a * c) / b))) / c);
	}
	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)) <= -0.018)
		tmp = Float64(fma(sqrt(Float64(1.0 - Float64(Float64(Float64(a * 3.0) * c) / Float64(b * b)))), abs(b), Float64(-b)) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(1.5 * Float64(Float64(a * c) / b))) / c));
	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], -0.018], N[(N[(N[Sqrt[N[(1.0 - N[(N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[b], $MachinePrecision] + (-b)), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0179999999999999986

   1. Initial program 55.6%

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

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

      8. pow2 N/A

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

      9. sub-to-mult N/A

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

      10. sqrt-prod N/A

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

      11. lower-fma.f64 N/A

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

   3. Applied rewrites 56.3%

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

   if -0.0179999999999999986 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.6%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 87.7%

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

   6. Applied rewrites 87.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{b}{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}}} \]

   7. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 82.0

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}} \]

   9. Applied rewrites 82.0%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{\color{blue}{c}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.018:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.018)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ 1.0 (/ (fma -2.0 b (* 1.5 (/ (* a c) b))) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.018) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = 1.0 / (fma(-2.0, b, (1.5 * ((a * c) / b))) / c);
	}
	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)) <= -0.018)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(1.5 * Float64(Float64(a * c) / b))) / c));
	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], -0.018], 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[(1.0 / N[(N[(-2.0 * b + N[(1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0179999999999999986

   1. Initial program 55.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 55.7

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

   3. Applied rewrites 55.7%

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

   if -0.0179999999999999986 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 55.6%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 87.7%

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

   6. Applied rewrites 87.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{b}{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}}} \]

   7. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 82.0

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}} \]

   9. Applied rewrites 82.0%

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{\color{blue}{c}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (/ (fma -2.0 b (* 1.5 (/ (* a c) b))) c)))

C

double code(double a, double b, double c) {
	return 1.0 / (fma(-2.0, b, (1.5 * ((a * c) / b))) / c);
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(fma(-2.0, b, Float64(1.5 * Float64(Float64(a * c) / b))) / c))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(-2.0 * b + N[(1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 87.7%

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

6. Applied rewrites 87.6%

   \[\leadsto \frac{1}{\color{blue}{\frac{b}{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}}} \]

7. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 82.0

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}} \]

9. Applied rewrites 82.0%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{\color{blue}{c}}} \]
10. Add Preprocessing

Alternative 13: 82.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \frac{a}{b}\right)} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (fma -2.0 (/ b c) (* 1.5 (/ a b)))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(-2.0, (b / c), (1.5 * (a / b)));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(-2.0, Float64(b / c), Float64(1.5 * Float64(a / b))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.6%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 87.7%

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

6. Applied rewrites 87.6%

   \[\leadsto \frac{1}{\color{blue}{\frac{b}{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -0.5625, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)\right)}}} \]

7. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 82.0

      \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \frac{a}{b}\right)} \]

9. Applied rewrites 82.0%

   \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \color{blue}{\frac{b}{c}}, 1.5 \cdot \frac{a}{b}\right)} \]
10. Add Preprocessing

Alternative 14: 64.3% 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 55.6%

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

2. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.3

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

4. Applied rewrites 64.3%

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

Reproduce

herbie shell --seed 2025136 
(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)))