Cubic critical, narrow range

Percentage Accurate: 55.5% → 90.9%

Time: 5.3s

Alternatives: 9

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: 55.5% 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: 90.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot b\\ t_1 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.075:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_1 \cdot t\_1}{\left(-b\right) - t\_1}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot a\right)\right) \cdot -0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}}{b} + \frac{\mathsf{fma}\left(-0.5, c, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{\left(t\_0 \cdot t\_0\right) \cdot a}, -0.16666666666666666, \frac{\left(\left(c \cdot c\right) \cdot a\right) \cdot -0.375}{b \cdot b}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* b b) b)) (t_1 (sqrt (fma (* -3.0 a) c (* b b)))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.075)
     (/ (/ (- (* b b) (* t_1 t_1)) (- (- b) t_1)) (* 3.0 a))
     (+
      (/ (/ (* (* a (* (* (* c c) c) a)) -0.5625) (* (* b b) (* b b))) b)
      (/
       (fma
        -0.5
        c
        (fma
         (* (pow (* c a) 4.0) (/ 6.328125 (* (* t_0 t_0) a)))
         -0.16666666666666666
         (/ (* (* (* c c) a) -0.375) (* b b))))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) * b;
	double t_1 = sqrt(fma((-3.0 * a), c, (b * b)));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.075) {
		tmp = (((b * b) - (t_1 * t_1)) / (-b - t_1)) / (3.0 * a);
	} else {
		tmp = ((((a * (((c * c) * c) * a)) * -0.5625) / ((b * b) * (b * b))) / b) + (fma(-0.5, c, fma((pow((c * a), 4.0) * (6.328125 / ((t_0 * t_0) * a))), -0.16666666666666666, ((((c * c) * a) * -0.375) / (b * b)))) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) * b)
	t_1 = sqrt(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)) <= -0.075)
		tmp = Float64(Float64(Float64(Float64(b * b) - Float64(t_1 * t_1)) / Float64(Float64(-b) - t_1)) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * Float64(Float64(Float64(c * c) * c) * a)) * -0.5625) / Float64(Float64(b * b) * Float64(b * b))) / b) + Float64(fma(-0.5, c, fma(Float64((Float64(c * a) ^ 4.0) * Float64(6.328125 / Float64(Float64(t_0 * t_0) * a))), -0.16666666666666666, Float64(Float64(Float64(Float64(c * c) * a) * -0.375) / Float64(b * b)))) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $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], -0.075], N[(N[(N[(N[(b * b), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * -0.5625), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(N[(-0.5 * c + N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(6.328125 / N[(N[(t$95$0 * t$95$0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0749999999999999972

   1. Initial program 79.7%

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

      1. lift-+.f64 N/A

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

      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 79.7%

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

   if -0.0749999999999999972 < (/.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 49.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 93.8%

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

   6. Applied rewrites 93.8%

      \[\leadsto \frac{\frac{\left(a \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot a\right)\right) \cdot -0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}}{b} + \color{blue}{\frac{\mathsf{fma}\left(-0.5, c, \mathsf{fma}\left({\left(c \cdot a\right)}^{4} \cdot \frac{6.328125}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.16666666666666666, \frac{\left(\left(c \cdot c\right) \cdot a\right) \cdot -0.375}{b \cdot b}\right)\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.075:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0 \cdot t\_0}{\left(-b\right) - t\_0}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(-1.0546875 \cdot \frac{a \cdot c}{b \cdot b} - 0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}\right), -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -3.0 a) c (* b b)))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.075)
     (/ (/ (- (* b b) (* t_0 t_0)) (- (- b) t_0)) (* 3.0 a))
     (/
      (fma
       a
       (fma
        (/
         (* (* (* c c) c) (- (* -1.0546875 (/ (* a c) (* b b))) 0.5625))
         (* (* b b) (* b b)))
        a
        (/ (* -0.375 (* c c)) (* b b)))
       (* -0.5 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-3.0 * a), c, (b * b)));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.075) {
		tmp = (((b * b) - (t_0 * t_0)) / (-b - t_0)) / (3.0 * a);
	} else {
		tmp = fma(a, fma(((((c * c) * c) * ((-1.0546875 * ((a * c) / (b * b))) - 0.5625)) / ((b * b) * (b * b))), a, ((-0.375 * (c * c)) / (b * b))), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(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)) <= -0.075)
		tmp = Float64(Float64(Float64(Float64(b * b) - Float64(t_0 * t_0)) / Float64(Float64(-b) - t_0)) / Float64(3.0 * a));
	else
		tmp = Float64(fma(a, fma(Float64(Float64(Float64(Float64(c * c) * c) * Float64(Float64(-1.0546875 * Float64(Float64(a * c) / Float64(b * b))) - 0.5625)) / Float64(Float64(b * b) * Float64(b * b))), a, Float64(Float64(-0.375 * Float64(c * c)) / Float64(b * b))), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $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], -0.075], N[(N[(N[(N[(b * b), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(-1.0546875 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5625), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0749999999999999972

   1. Initial program 79.7%

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

      1. lift-+.f64 N/A

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

      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 79.7%

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

   if -0.0749999999999999972 < (/.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 49.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 93.8%

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

   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 93.9%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 93.9%

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

   11. Taylor expanded in c around 0

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

       1. lower-*.f64 N/A

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

       2. pow3 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 93.9

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

   13. Applied rewrites 93.9%

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

Alternative 3: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.075:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0 \cdot t\_0}{\left(-b\right) - t\_0}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(-0.5625, \frac{a \cdot \left(\left(c \cdot c\right) \cdot c\right)}{b \cdot b}, -0.375 \cdot \left(c \cdot c\right)\right)}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -3.0 a) c (* b b)))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.075)
     (/ (/ (- (* b b) (* t_0 t_0)) (- (- b) t_0)) (* 3.0 a))
     (/
      (fma
       a
       (/
        (fma -0.5625 (/ (* a (* (* c c) c)) (* b b)) (* -0.375 (* c c)))
        (* b b))
       (* -0.5 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-3.0 * a), c, (b * b)));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.075) {
		tmp = (((b * b) - (t_0 * t_0)) / (-b - t_0)) / (3.0 * a);
	} else {
		tmp = fma(a, (fma(-0.5625, ((a * ((c * c) * c)) / (b * b)), (-0.375 * (c * c))) / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(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)) <= -0.075)
		tmp = Float64(Float64(Float64(Float64(b * b) - Float64(t_0 * t_0)) / Float64(Float64(-b) - t_0)) / Float64(3.0 * a));
	else
		tmp = Float64(fma(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)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $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], -0.075], N[(N[(N[(N[(b * b), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(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] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0749999999999999972

   1. Initial program 79.7%

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

      1. lift-+.f64 N/A

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

      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 79.7%

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

   if -0.0749999999999999972 < (/.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 49.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 93.8%

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

   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 93.9%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 91.5

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

   10. Applied rewrites 91.5%

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

Alternative 4: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.075:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(-0.5625, \frac{a \cdot \left(\left(c \cdot c\right) \cdot c\right)}{b \cdot b}, -0.375 \cdot \left(c \cdot c\right)\right)}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.075)
   (/ (+ (- b) (sqrt (fma b b (* (* a -3.0) c)))) (* 3.0 a))
   (/
    (fma
     a
     (/
      (fma -0.5625 (/ (* a (* (* c c) c)) (* b b)) (* -0.375 (* c c)))
      (* b b))
     (* -0.5 c))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.075) {
		tmp = (-b + sqrt(fma(b, b, ((a * -3.0) * c)))) / (3.0 * a);
	} else {
		tmp = fma(a, (fma(-0.5625, ((a * ((c * c) * c)) / (b * b)), (-0.375 * (c * c))) / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.075)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(a * -3.0) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(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)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.075], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(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] + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0749999999999999972

   1. Initial program 79.7%

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

      1. lift-*.f64 N/A

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

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

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.8

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

   3. Applied rewrites 79.8%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if -0.0749999999999999972 < (/.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 49.2%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 93.8%

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

   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 93.9%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 91.5

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

   10. Applied rewrites 91.5%

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

Alternative 5: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.016:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.016)
   (/ (+ (- b) (sqrt (fma b b (* (* a -3.0) c)))) (* 3.0 a))
   (/ (fma (/ (* (* c c) a) (* b b)) -0.375 (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.016) {
		tmp = (-b + sqrt(fma(b, b, ((a * -3.0) * c)))) / (3.0 * a);
	} else {
		tmp = fma((((c * c) * a) / (b * b)), -0.375, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.016)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(a * -3.0) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(c * c) * a) / Float64(b * b)), -0.375, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.016], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.016

   1. Initial program 78.7%

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

      1. lift-*.f64 N/A

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

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

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 78.8

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

   3. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if -0.016 < (/.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 47.6%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 87.4

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

   4. Applied rewrites 87.4%

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

Alternative 6: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.016:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.016)
   (/ (+ (- b) (sqrt (fma b b (* (* a -3.0) c)))) (* 3.0 a))
   (/ (* (- (/ (* -0.375 (* c a)) (* b b)) 0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.016) {
		tmp = (-b + sqrt(fma(b, b, ((a * -3.0) * c)))) / (3.0 * a);
	} else {
		tmp = ((((-0.375 * (c * a)) / (b * b)) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.016)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(a * -3.0) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.375 * Float64(c * a)) / Float64(b * b)) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.016], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.375 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.016

   1. Initial program 78.7%

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

      1. lift-*.f64 N/A

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

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

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 78.8

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

   3. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if -0.016 < (/.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 47.6%

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

   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 94.4%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 87.3

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

   7. Applied rewrites 87.3%

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

Alternative 7: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.016:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.016)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (* (- (/ (* -0.375 (* c a)) (* b b)) 0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.016) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = ((((-0.375 * (c * a)) / (b * b)) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.016)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.375 * Float64(c * a)) / Float64(b * b)) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.016], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.375 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.016

   1. Initial program 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 78.8

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

   3. Applied rewrites 78.8%

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

   if -0.016 < (/.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 47.6%

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

   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 94.4%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 87.3

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

   7. Applied rewrites 87.3%

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

Alternative 8: 81.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((((-0.375d0) * (c * a)) / (b * b)) - 0.5d0) * c) / b
end function

Java

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

Python

def code(a, b, c):
	return ((((-0.375 * (c * a)) / (b * b)) - 0.5) * c) / b

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = ((((-0.375 * (c * a)) / (b * b)) - 0.5) * c) / b;
end

Wolfram

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

Derivation

1. Initial program 55.5%

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

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 90.7%

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

5. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 81.3

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

7. Applied rewrites 81.3%

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

Alternative 9: 64.2% 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 55.5%

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

2. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.2

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

4. Applied rewrites 64.2%

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

Reproduce

herbie shell --seed 2025120 
(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)))