Cubic critical, narrow range

Percentage Accurate: 54.5% → 92.2%

Time: 14.9s

Alternatives: 11

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.5% 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

real(8) function code(a, b, c)
    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: 92.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)\\ t_1 := b + \sqrt{t\_0}\\ t_2 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 0.62:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1} - \frac{b \cdot b}{t\_1}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c, c \cdot \frac{-0.375}{t\_2}, a \cdot \mathsf{fma}\left(c, \left(c \cdot c\right) \cdot \frac{-0.5625}{\left(b \cdot b\right) \cdot t\_2}, \frac{a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot 6.328125\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot t\_2\right)\right)} \cdot -0.16666666666666666\right)\right), a, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* -3.0 c) (* b b)))
        (t_1 (+ b (sqrt t_0)))
        (t_2 (* b (* b b))))
   (if (<= b 0.62)
     (/ (- (/ t_0 t_1) (/ (* b b) t_1)) (* a 3.0))
     (fma
      (fma
       c
       (* c (/ -0.375 t_2))
       (*
        a
        (fma
         c
         (* (* c c) (/ -0.5625 (* (* b b) t_2)))
         (*
          (/
           (* a (* (* c (* c (* c c))) 6.328125))
           (* b (* (* b b) (* b t_2))))
          -0.16666666666666666))))
      a
      (/ (* c -0.5) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (-3.0 * c), (b * b));
	double t_1 = b + sqrt(t_0);
	double t_2 = b * (b * b);
	double tmp;
	if (b <= 0.62) {
		tmp = ((t_0 / t_1) - ((b * b) / t_1)) / (a * 3.0);
	} else {
		tmp = fma(fma(c, (c * (-0.375 / t_2)), (a * fma(c, ((c * c) * (-0.5625 / ((b * b) * t_2))), (((a * ((c * (c * (c * c))) * 6.328125)) / (b * ((b * b) * (b * t_2)))) * -0.16666666666666666)))), a, ((c * -0.5) / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(-3.0 * c), Float64(b * b))
	t_1 = Float64(b + sqrt(t_0))
	t_2 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (b <= 0.62)
		tmp = Float64(Float64(Float64(t_0 / t_1) - Float64(Float64(b * b) / t_1)) / Float64(a * 3.0));
	else
		tmp = fma(fma(c, Float64(c * Float64(-0.375 / t_2)), Float64(a * fma(c, Float64(Float64(c * c) * Float64(-0.5625 / Float64(Float64(b * b) * t_2))), Float64(Float64(Float64(a * Float64(Float64(c * Float64(c * Float64(c * c))) * 6.328125)) / Float64(b * Float64(Float64(b * b) * Float64(b * t_2)))) * -0.16666666666666666)))), a, Float64(Float64(c * -0.5) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.62], N[(N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(c * N[(-0.375 / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(a * N[(c * N[(N[(c * c), $MachinePrecision] * N[(-0.5625 / N[(N[(b * b), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(b * N[(N[(b * b), $MachinePrecision] * N[(b * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.619999999999999996

   1. Initial program 85.1%

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

   4. Applied egg-rr 85.9%

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

   if 0.619999999999999996 < b

   1. Initial program 47.7%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 94.6%

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

   7. Applied egg-rr 94.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.62:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}} - \frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c, c \cdot \frac{-0.375}{b \cdot \left(b \cdot b\right)}, a \cdot \mathsf{fma}\left(c, \left(c \cdot c\right) \cdot \frac{-0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot 6.328125\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot -0.16666666666666666\right)\right), a, \frac{c \cdot -0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot \left(b \cdot b\right)\right)\\ t_1 := \mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)\\ t_2 := b + \sqrt{t\_1}\\ \mathbf{if}\;b \leq 0.62:\\ \;\;\;\;\frac{\frac{t\_1}{t\_2} - \frac{b \cdot b}{t\_2}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c}{\left(b \cdot b\right) \cdot -2.6666666666666665}, \frac{c \cdot \left(a \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)\right)}{t\_0}\right), \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(-1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot b\right) \cdot t\_0}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b (* b b))))
        (t_1 (fma a (* -3.0 c) (* b b)))
        (t_2 (+ b (sqrt t_1))))
   (if (<= b 0.62)
     (/ (- (/ t_1 t_2) (/ (* b b) t_2)) (* a 3.0))
     (/
      (fma
       c
       -0.5
       (fma
        a
        (fma
         c
         (/ c (* (* b b) -2.6666666666666665))
         (/ (* c (* a (* (* c c) -0.5625))) t_0))
        (/
         (* (* c (* c (* c c))) (* -1.0546875 (* a (* a a))))
         (* (* b b) t_0))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * (b * b));
	double t_1 = fma(a, (-3.0 * c), (b * b));
	double t_2 = b + sqrt(t_1);
	double tmp;
	if (b <= 0.62) {
		tmp = ((t_1 / t_2) - ((b * b) / t_2)) / (a * 3.0);
	} else {
		tmp = fma(c, -0.5, fma(a, fma(c, (c / ((b * b) * -2.6666666666666665)), ((c * (a * ((c * c) * -0.5625))) / t_0)), (((c * (c * (c * c))) * (-1.0546875 * (a * (a * a)))) / ((b * b) * t_0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * Float64(b * b)))
	t_1 = fma(a, Float64(-3.0 * c), Float64(b * b))
	t_2 = Float64(b + sqrt(t_1))
	tmp = 0.0
	if (b <= 0.62)
		tmp = Float64(Float64(Float64(t_1 / t_2) - Float64(Float64(b * b) / t_2)) / Float64(a * 3.0));
	else
		tmp = Float64(fma(c, -0.5, fma(a, fma(c, Float64(c / Float64(Float64(b * b) * -2.6666666666666665)), Float64(Float64(c * Float64(a * Float64(Float64(c * c) * -0.5625))) / t_0)), Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(-1.0546875 * Float64(a * Float64(a * a)))) / Float64(Float64(b * b) * t_0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.62], N[(N[(N[(t$95$1 / t$95$2), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5 + N[(a * N[(c * N[(c / N[(N[(b * b), $MachinePrecision] * -2.6666666666666665), $MachinePrecision]), $MachinePrecision] + N[(N[(c * N[(a * N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0546875 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.619999999999999996

   1. Initial program 85.1%

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

   4. Applied egg-rr 85.9%

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

   if 0.619999999999999996 < b

   1. Initial program 47.7%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 94.6%

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

   6. Taylor expanded in b around inf

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

   8. Simplified 94.6%

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

   10. Applied egg-rr 94.6%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c}{\left(b \cdot b\right) \cdot -2.6666666666666665}, \frac{c \cdot \left(\left(\left(c \cdot c\right) \cdot -0.5625\right) \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right), \frac{\left(-1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right)\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.62:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}} - \frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c}{\left(b \cdot b\right) \cdot -2.6666666666666665}, \frac{c \cdot \left(a \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right), \frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(-1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)\\ t_1 := b + \sqrt{t\_0}\\ \mathbf{if}\;b \leq 3.95:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1} - \frac{b \cdot b}{t\_1}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(c, c \cdot -0.375, \frac{a \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)\right)}{b \cdot b}\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* -3.0 c) (* b b))) (t_1 (+ b (sqrt t_0))))
   (if (<= b 3.95)
     (/ (- (/ t_0 t_1) (/ (* b b) t_1)) (* a 3.0))
     (fma
      a
      (/
       (fma c (* c -0.375) (/ (* a (* c (* (* c c) -0.5625))) (* b b)))
       (* b (* b b)))
      (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(a, (-3.0 * c), (b * b));
	double t_1 = b + sqrt(t_0);
	double tmp;
	if (b <= 3.95) {
		tmp = ((t_0 / t_1) - ((b * b) / t_1)) / (a * 3.0);
	} else {
		tmp = fma(a, (fma(c, (c * -0.375), ((a * (c * ((c * c) * -0.5625))) / (b * b))) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(a, Float64(-3.0 * c), Float64(b * b))
	t_1 = Float64(b + sqrt(t_0))
	tmp = 0.0
	if (b <= 3.95)
		tmp = Float64(Float64(Float64(t_0 / t_1) - Float64(Float64(b * b) / t_1)) / Float64(a * 3.0));
	else
		tmp = fma(a, Float64(fma(c, Float64(c * -0.375), Float64(Float64(a * Float64(c * Float64(Float64(c * c) * -0.5625))) / Float64(b * b))) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3.95], N[(N[(N[(t$95$0 / t$95$1), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(c * N[(c * -0.375), $MachinePrecision] + N[(N[(a * N[(c * N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.9500000000000002

   1. Initial program 83.6%

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

   4. Applied egg-rr 84.6%

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

   if 3.9500000000000002 < b

   1. Initial program 45.6%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 95.5%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   8. Simplified 93.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.95:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}} - \frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(c, c \cdot -0.375, \frac{a \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)\right)}{b \cdot b}\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.95:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(c, c \cdot -0.375, \frac{a \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)\right)}{b \cdot b}\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.95)
   (/ (- (sqrt (fma b b (* a (* -3.0 c)))) b) (* a 3.0))
   (fma
    a
    (/
     (fma c (* c -0.375) (/ (* a (* c (* (* c c) -0.5625))) (* b b)))
     (* b (* b b)))
    (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.95) {
		tmp = (sqrt(fma(b, b, (a * (-3.0 * c)))) - b) / (a * 3.0);
	} else {
		tmp = fma(a, (fma(c, (c * -0.375), ((a * (c * ((c * c) * -0.5625))) / (b * b))) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.95)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(-3.0 * c)))) - b) / Float64(a * 3.0));
	else
		tmp = fma(a, Float64(fma(c, Float64(c * -0.375), Float64(Float64(a * Float64(c * Float64(Float64(c * c) * -0.5625))) / Float64(b * b))) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.95], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(c * N[(c * -0.375), $MachinePrecision] + N[(N[(a * N[(c * N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.9500000000000002

   1. Initial program 83.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 83.8

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

   4. Applied egg-rr 83.8%

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

   if 3.9500000000000002 < b

   1. Initial program 45.6%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 95.5%

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

   6. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   8. Simplified 93.6%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.95:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(c, c \cdot -0.375, \frac{a \cdot \left(c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)\right)}{b \cdot b}\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, \frac{c \cdot \left(a \cdot \left(a \cdot -0.5625\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.4)
   (/ (- (sqrt (fma b b (* a (* -3.0 c)))) b) (* a 3.0))
   (/
    (*
     c
     (fma
      c
      (fma
       -0.375
       (/ a (* b b))
       (/ (* c (* a (* a -0.5625))) (* (* b b) (* b b))))
      -0.5))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.4) {
		tmp = (sqrt(fma(b, b, (a * (-3.0 * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * fma(c, fma(-0.375, (a / (b * b)), ((c * (a * (a * -0.5625))) / ((b * b) * (b * b)))), -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.4)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(-3.0 * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * fma(c, fma(-0.375, Float64(a / Float64(b * b)), Float64(Float64(c * Float64(a * Float64(a * -0.5625))) / Float64(Float64(b * b) * Float64(b * b)))), -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.4], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(c * N[(-0.375 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(c * N[(a * N[(a * -0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.4000000000000004

   1. Initial program 83.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 83.8

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

   4. Applied egg-rr 83.8%

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

   if 4.4000000000000004 < b

   1. Initial program 45.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Simplified 93.6%

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

   6. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   8. Simplified 93.4%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(c, \mathsf{fma}\left(-0.375, \frac{a}{b \cdot b}, \frac{c \cdot \left(a \cdot \left(a \cdot -0.5625\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right), -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 10.0)
   (/ (- (sqrt (fma b b (* a (* -3.0 c)))) b) (* a 3.0))
   (fma a (/ (* -0.375 (* c c)) (* b (* b b))) (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 10.0) {
		tmp = (sqrt(fma(b, b, (a * (-3.0 * c)))) - b) / (a * 3.0);
	} else {
		tmp = fma(a, ((-0.375 * (c * c)) / (b * (b * b))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 10.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(-3.0 * c)))) - b) / Float64(a * 3.0));
	else
		tmp = fma(a, Float64(Float64(-0.375 * Float64(c * c)) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 10.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 10

   1. Initial program 83.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 83.7

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

   4. Applied egg-rr 83.7%

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

   if 10 < b

   1. Initial program 45.3%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 95.4%

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

   6. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 89.2

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

   8. Simplified 89.2%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 10.5)
   (/ (- (sqrt (fma b b (* a (* -3.0 c)))) b) (* a 3.0))
   (/ (fma a (/ (* -0.375 (* c c)) (* b b)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 10.5) {
		tmp = (sqrt(fma(b, b, (a * (-3.0 * c)))) - b) / (a * 3.0);
	} else {
		tmp = fma(a, ((-0.375 * (c * c)) / (b * b)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 10.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(-3.0 * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(a, Float64(Float64(-0.375 * Float64(c * c)) / Float64(b * b)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 10.5], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 10.5

   1. Initial program 83.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 83.7

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

   4. Applied egg-rr 83.7%

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

   if 10.5 < b

   1. Initial program 45.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Simplified 89.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(\frac{a}{b \cdot b}, c \cdot -0.375, -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 10.0)
   (/ (- (sqrt (fma b b (* a (* -3.0 c)))) b) (* a 3.0))
   (/ (* c (fma (/ a (* b b)) (* c -0.375) -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 10.0) {
		tmp = (sqrt(fma(b, b, (a * (-3.0 * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c * fma((a / (b * b)), (c * -0.375), -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 10.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(-3.0 * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * fma(Float64(a / Float64(b * b)), Float64(c * -0.375), -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 10.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * -0.375), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 10

   1. Initial program 83.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 83.7

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

   4. Applied egg-rr 83.7%

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

   if 10 < b

   1. Initial program 45.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Simplified 89.1%

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

   6. Taylor expanded in c around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 89.0

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

   8. Simplified 89.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(\frac{a}{b \cdot b}, c \cdot -0.375, -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(\frac{a}{b \cdot b}, c \cdot -0.375, -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 10.5)
   (* (/ -0.3333333333333333 a) (- b (sqrt (fma a (* -3.0 c) (* b b)))))
   (/ (* c (fma (/ a (* b b)) (* c -0.375) -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 10.5) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt(fma(a, (-3.0 * c), (b * b))));
	} else {
		tmp = (c * fma((a / (b * b)), (c * -0.375), -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 10.5)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(fma(a, Float64(-3.0 * c), Float64(b * b)))));
	else
		tmp = Float64(Float64(c * fma(Float64(a / Float64(b * b)), Float64(c * -0.375), -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 10.5], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * -0.375), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 10.5

   1. Initial program 83.5%

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

   4. Applied egg-rr 83.5%

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

   if 10.5 < b

   1. Initial program 45.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Simplified 89.1%

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

   6. Taylor expanded in c around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 89.0

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

   8. Simplified 89.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10.5:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(\frac{a}{b \cdot b}, c \cdot -0.375, -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Simplified 81.4%

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

6. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r* N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 81.3

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

8. Simplified 81.3%

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

9. Final simplification 81.3%

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

Alternative 11: 65.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 64.7

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

5. Simplified 64.7%

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

Reproduce

herbie shell --seed 2024207 
(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)))