Cubic critical, narrow range

Percentage Accurate: 54.7% → 91.7%

Time: 14.5s

Alternatives: 13

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: 54.7% 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: 91.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot t\_0\\ t_2 := \mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)\\ t_3 := b + \sqrt{t\_2}\\ t_4 := c \cdot \left(c \cdot c\right)\\ \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\frac{t\_2}{t\_3} - \frac{b \cdot b}{t\_3}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(c \cdot t\_4\right) \cdot \left(a \cdot 6.328125\right)}{b \cdot \left(b \cdot t\_1\right)}, -0.16666666666666666, \frac{t\_4 \cdot -0.5625}{t\_1}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{t\_0}\right), a, \frac{c}{b \cdot -2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b)))
        (t_1 (* (* b b) t_0))
        (t_2 (fma a (* -3.0 c) (* b b)))
        (t_3 (+ b (sqrt t_2)))
        (t_4 (* c (* c c))))
   (if (<= b 1.1)
     (/ (- (/ t_2 t_3) (/ (* b b) t_3)) (* a 3.0))
     (fma
      (fma
       a
       (fma
        (/ (* (* c t_4) (* a 6.328125)) (* b (* b t_1)))
        -0.16666666666666666
        (/ (* t_4 -0.5625) t_1))
       (/ (* (* c c) -0.375) t_0))
      a
      (/ c (* b -2.0))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	double t_2 = fma(a, (-3.0 * c), (b * b));
	double t_3 = b + sqrt(t_2);
	double t_4 = c * (c * c);
	double tmp;
	if (b <= 1.1) {
		tmp = ((t_2 / t_3) - ((b * b) / t_3)) / (a * 3.0);
	} else {
		tmp = fma(fma(a, fma((((c * t_4) * (a * 6.328125)) / (b * (b * t_1))), -0.16666666666666666, ((t_4 * -0.5625) / t_1)), (((c * c) * -0.375) / t_0)), a, (c / (b * -2.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(b * b) * t_0)
	t_2 = fma(a, Float64(-3.0 * c), Float64(b * b))
	t_3 = Float64(b + sqrt(t_2))
	t_4 = Float64(c * Float64(c * c))
	tmp = 0.0
	if (b <= 1.1)
		tmp = Float64(Float64(Float64(t_2 / t_3) - Float64(Float64(b * b) / t_3)) / Float64(a * 3.0));
	else
		tmp = fma(fma(a, fma(Float64(Float64(Float64(c * t_4) * Float64(a * 6.328125)) / Float64(b * Float64(b * t_1))), -0.16666666666666666, Float64(Float64(t_4 * -0.5625) / t_1)), Float64(Float64(Float64(c * c) * -0.375) / t_0)), a, Float64(c / Float64(b * -2.0)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(b + N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.1], N[(N[(N[(t$95$2 / t$95$3), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(N[(c * t$95$4), $MachinePrecision] * N[(a * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(N[(t$95$4 * -0.5625), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * a + N[(c / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001

   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 84.3%

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

   if 1.1000000000000001 < b

   1. Initial program 47.8%

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

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

   7. Applied egg-rr 95.5%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}} - \frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 6.328125\right)}{b \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), a, \frac{c}{b \cdot -2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot t\_0\\ t_2 := \mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)\\ t_3 := b + \sqrt{t\_2}\\ t_4 := c \cdot \left(c \cdot c\right)\\ \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\frac{t\_2}{t\_3} - \frac{b \cdot b}{t\_3}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{b}, c, a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(c \cdot t\_4\right) \cdot \left(a \cdot 6.328125\right)}{b \cdot \left(b \cdot t\_1\right)}, -0.16666666666666666, \frac{t\_4 \cdot -0.5625}{t\_1}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{t\_0}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b)))
        (t_1 (* (* b b) t_0))
        (t_2 (fma a (* -3.0 c) (* b b)))
        (t_3 (+ b (sqrt t_2)))
        (t_4 (* c (* c c))))
   (if (<= b 1.1)
     (/ (- (/ t_2 t_3) (/ (* b b) t_3)) (* a 3.0))
     (fma
      (/ -0.5 b)
      c
      (*
       a
       (fma
        a
        (fma
         (/ (* (* c t_4) (* a 6.328125)) (* b (* b t_1)))
         -0.16666666666666666
         (/ (* t_4 -0.5625) t_1))
        (/ (* (* c c) -0.375) t_0)))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	double t_2 = fma(a, (-3.0 * c), (b * b));
	double t_3 = b + sqrt(t_2);
	double t_4 = c * (c * c);
	double tmp;
	if (b <= 1.1) {
		tmp = ((t_2 / t_3) - ((b * b) / t_3)) / (a * 3.0);
	} else {
		tmp = fma((-0.5 / b), c, (a * fma(a, fma((((c * t_4) * (a * 6.328125)) / (b * (b * t_1))), -0.16666666666666666, ((t_4 * -0.5625) / t_1)), (((c * c) * -0.375) / t_0))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(b * b) * t_0)
	t_2 = fma(a, Float64(-3.0 * c), Float64(b * b))
	t_3 = Float64(b + sqrt(t_2))
	t_4 = Float64(c * Float64(c * c))
	tmp = 0.0
	if (b <= 1.1)
		tmp = Float64(Float64(Float64(t_2 / t_3) - Float64(Float64(b * b) / t_3)) / Float64(a * 3.0));
	else
		tmp = fma(Float64(-0.5 / b), c, Float64(a * fma(a, fma(Float64(Float64(Float64(c * t_4) * Float64(a * 6.328125)) / Float64(b * Float64(b * t_1))), -0.16666666666666666, Float64(Float64(t_4 * -0.5625) / t_1)), Float64(Float64(Float64(c * c) * -0.375) / t_0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(b + N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.1], N[(N[(N[(t$95$2 / t$95$3), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / b), $MachinePrecision] * c + N[(a * N[(a * N[(N[(N[(N[(c * t$95$4), $MachinePrecision] * N[(a * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(N[(t$95$4 * -0.5625), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001

   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 84.3%

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

   if 1.1000000000000001 < b

   1. Initial program 47.8%

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

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

   7. Applied egg-rr 95.4%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}} - \frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{b}, c, a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 6.328125\right)}{b \cdot \left(b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)\\ t_1 := b + \sqrt{t\_0}\\ \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\frac{t\_0}{t\_1} - \frac{b \cdot b}{t\_1}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(c \cdot c\right), \frac{-0.375}{b \cdot b}, \mathsf{fma}\left(-0.5625, \frac{a \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, c \cdot -0.5\right)\right)}{b}\\ \end{array} \end{array} \]

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 1.1)
     (/ (- (/ t_0 t_1) (/ (* b b) t_1)) (* a 3.0))
     (/
      (fma
       (* a (* c c))
       (/ -0.375 (* b b))
       (fma
        -0.5625
        (/ (* a (* a (* c (* c c)))) (* (* b b) (* b b)))
        (* 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 tmp;
	if (b <= 1.1) {
		tmp = ((t_0 / t_1) - ((b * b) / t_1)) / (a * 3.0);
	} else {
		tmp = fma((a * (c * c)), (-0.375 / (b * b)), fma(-0.5625, ((a * (a * (c * (c * c)))) / ((b * b) * (b * b))), (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))
	tmp = 0.0
	if (b <= 1.1)
		tmp = Float64(Float64(Float64(t_0 / t_1) - Float64(Float64(b * b) / t_1)) / Float64(a * 3.0));
	else
		tmp = Float64(fma(Float64(a * Float64(c * c)), Float64(-0.375 / Float64(b * b)), fma(-0.5625, Float64(Float64(a * Float64(a * Float64(c * Float64(c * c)))) / Float64(Float64(b * b) * Float64(b * b))), 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]}, If[LessEqual[b, 1.1], 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[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(-0.375 / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * N[(a * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001

   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 84.3%

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

   if 1.1000000000000001 < b

   1. Initial program 47.8%

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

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

      1. /-lowering-/.f64 N/A

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

   8. Simplified 93.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}} - \frac{b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(c \cdot c\right), \frac{-0.375}{b \cdot b}, \mathsf{fma}\left(-0.5625, \frac{a \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, c \cdot -0.5\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{-1}{\frac{a \cdot 3}{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(c \cdot c\right), \frac{-0.375}{b \cdot b}, \mathsf{fma}\left(-0.5625, \frac{a \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, c \cdot -0.5\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.15)
   (/ -1.0 (/ (* a 3.0) (- b (sqrt (fma a (* -3.0 c) (* b b))))))
   (/
    (fma
     (* a (* c c))
     (/ -0.375 (* b b))
     (fma
      -0.5625
      (/ (* a (* a (* c (* c c)))) (* (* b b) (* b b)))
      (* c -0.5)))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.15) {
		tmp = -1.0 / ((a * 3.0) / (b - sqrt(fma(a, (-3.0 * c), (b * b)))));
	} else {
		tmp = fma((a * (c * c)), (-0.375 / (b * b)), fma(-0.5625, ((a * (a * (c * (c * c)))) / ((b * b) * (b * b))), (c * -0.5))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.15)
		tmp = Float64(-1.0 / Float64(Float64(a * 3.0) / Float64(b - sqrt(fma(a, Float64(-3.0 * c), Float64(b * b))))));
	else
		tmp = Float64(fma(Float64(a * Float64(c * c)), Float64(-0.375 / Float64(b * b)), fma(-0.5625, Float64(Float64(a * Float64(a * Float64(c * Float64(c * c)))) / Float64(Float64(b * b) * Float64(b * b))), Float64(c * -0.5))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.15], N[(-1.0 / N[(N[(a * 3.0), $MachinePrecision] / N[(b - N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(-0.375 / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * N[(a * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1499999999999999

   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 83.7%

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

   if 1.1499999999999999 < b

   1. Initial program 47.8%

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

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

      1. /-lowering-/.f64 N/A

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

   8. Simplified 93.3%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{-1}{\frac{a \cdot 3}{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot \left(c \cdot c\right), \frac{-0.375}{b \cdot b}, \mathsf{fma}\left(-0.5625, \frac{a \cdot \left(a \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, c \cdot -0.5\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{-1}{\frac{a \cdot 3}{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(c, c \cdot -0.375, \frac{a \cdot \left(\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625\right)}{b \cdot b}\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.1)
   (/ -1.0 (/ (* a 3.0) (- b (sqrt (fma a (* -3.0 c) (* b b))))))
   (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 <= 1.1) {
		tmp = -1.0 / ((a * 3.0) / (b - sqrt(fma(a, (-3.0 * c), (b * b)))));
	} 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 <= 1.1)
		tmp = Float64(-1.0 / Float64(Float64(a * 3.0) / Float64(b - sqrt(fma(a, Float64(-3.0 * c), Float64(b * b))))));
	else
		tmp = fma(a, Float64(fma(c, Float64(c * -0.375), Float64(Float64(a * Float64(Float64(c * 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, 1.1], N[(-1.0 / N[(N[(a * 3.0), $MachinePrecision] / N[(b - N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(c * N[(c * -0.375), $MachinePrecision] + N[(N[(a * N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * -0.5625), $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 < 1.1000000000000001

   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 83.7%

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

   if 1.1000000000000001 < b

   1. Initial program 47.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. /-lowering-/.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 93.2%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{-1}{\frac{a \cdot 3}{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(c, c \cdot -0.375, \frac{a \cdot \left(\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625\right)}{b \cdot b}\right)}{b \cdot \left(b \cdot b\right)}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{-1}{\frac{a \cdot 3}{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.1)
   (/ -1.0 (/ (* a 3.0) (- b (sqrt (fma a (* -3.0 c) (* b b))))))
   (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.1) {
		tmp = -1.0 / ((a * 3.0) / (b - sqrt(fma(a, (-3.0 * c), (b * b)))));
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * b)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.1)
		tmp = Float64(-1.0 / Float64(Float64(a * 3.0) / Float64(b - sqrt(fma(a, Float64(-3.0 * c), Float64(b * b))))));
	else
		tmp = Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.1], N[(-1.0 / N[(N[(a * 3.0), $MachinePrecision] / N[(b - N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001

   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 83.7%

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

   if 1.1000000000000001 < b

   1. Initial program 47.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. /-lowering-/.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 88.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{-1}{\frac{a \cdot 3}{b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.1], N[(N[(N[(b - N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001

   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 83.7%

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

   if 1.1000000000000001 < b

   1. Initial program 47.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. /-lowering-/.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 88.2%

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

Alternative 8: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.22:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.22)
   (/ (- (sqrt (fma a (* -3.0 c) (* b b))) b) (* a 3.0))
   (/ (fma a (/ (* (* c c) -0.375) (* b b)) (* c -0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.22) {
		tmp = (sqrt(fma(a, (-3.0 * c), (b * b))) - b) / (a * 3.0);
	} else {
		tmp = fma(a, (((c * c) * -0.375) / (b * b)), (c * -0.5)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.22)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-3.0 * c), Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)), Float64(c * -0.5)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.22], N[(N[(N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.21999999999999997

   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. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. *-lowering-*.f64 83.6

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

   4. Applied egg-rr 83.6%

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

   if 1.21999999999999997 < b

   1. Initial program 47.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. /-lowering-/.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 88.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.22:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}, c \cdot -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.1], N[(N[(N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(a * N[(-0.375 * N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001

   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. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. *-lowering-*.f64 83.6

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

   4. Applied egg-rr 83.6%

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

   if 1.1000000000000001 < b

   1. Initial program 47.8%

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

   3. Taylor expanded in c around 0

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r* N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-in N/A

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

      8. *-lowering-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. associate-*l/ N/A

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

      11. associate-*r* N/A

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

      12. associate-*r/ N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. associate-*l* N/A

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

   5. Simplified 88.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{fma}\left(a, -0.375 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{-0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.1], N[(N[(N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-0.375 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001

   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. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. associate-*l* N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. *-lowering-*.f64 83.6

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

   4. Applied egg-rr 83.6%

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

   if 1.1000000000000001 < b

   1. Initial program 47.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. /-lowering-/.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

   5. Simplified 88.1%

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

   6. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

   8. Simplified 87.9%

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

   9. Taylor expanded in a around 0

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-/r* N/A

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

       4. associate-/l* N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. div-sub N/A

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

       8. /-lowering-/.f64 N/A

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

       11. accelerator-lowering-fma.f64 N/A

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

       12. associate-/l* N/A

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

       13. *-lowering-*.f64 N/A

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

       14. /-lowering-/.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 88.0

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

   11. Simplified 88.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{\mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{\mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.1)
   (* (- b (sqrt (fma a (* -3.0 c) (* b b)))) (/ -0.3333333333333333 a))
   (* c (/ (fma -0.375 (* a (/ c (* b b))) -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.1) {
		tmp = (b - sqrt(fma(a, (-3.0 * c), (b * b)))) * (-0.3333333333333333 / a);
	} else {
		tmp = c * (fma(-0.375, (a * (c / (b * b))), -0.5) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.1)
		tmp = Float64(Float64(b - sqrt(fma(a, Float64(-3.0 * c), Float64(b * b)))) * Float64(-0.3333333333333333 / a));
	else
		tmp = Float64(c * Float64(fma(-0.375, Float64(a * Float64(c / Float64(b * b))), -0.5) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.1], N[(N[(b - N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-0.375 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1000000000000001

   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 83.6%

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

   if 1.1000000000000001 < b

   1. Initial program 47.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. /-lowering-/.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

   5. Simplified 88.1%

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

   6. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

   8. Simplified 87.9%

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

   9. Taylor expanded in a around 0

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-/r* N/A

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

       4. associate-/l* N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. div-sub N/A

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

       8. /-lowering-/.f64 N/A

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

       11. accelerator-lowering-fma.f64 N/A

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

       12. associate-/l* N/A

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

       13. *-lowering-*.f64 N/A

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

       14. /-lowering-/.f64 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f64 88.0

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

   11. Simplified 88.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{\mathsf{fma}\left(-0.375, a \cdot \frac{c}{b \cdot b}, -0.5\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. associate-*r/ N/A

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

   14. metadata-eval N/A

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

   15. distribute-neg-frac N/A

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

   16. metadata-eval N/A

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

   17. /-lowering-/.f64 N/A

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

   18. *-commutative N/A

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

   19. associate-*r* N/A

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

5. Simplified 83.9%

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

6. Taylor expanded in c around inf

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

8. Simplified 83.7%

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

9. Taylor expanded in a around 0

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

    1. unpow3 N/A

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

    2. unpow2 N/A

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

    3. associate-/r* N/A

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

    4. associate-/l* N/A

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

    5. associate-*r/ N/A

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

    6. metadata-eval N/A

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

    7. div-sub N/A

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

    8. /-lowering-/.f64 N/A

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

    9. sub-neg N/A

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

    10. metadata-eval N/A

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

    11. accelerator-lowering-fma.f64 N/A

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

    12. associate-/l* N/A

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

    13. *-lowering-*.f64 N/A

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

    14. /-lowering-/.f64 N/A

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

    15. unpow2 N/A

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

    16. *-lowering-*.f64 83.8

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

11. Simplified 83.8%

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

12. Final simplification 83.8%

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

Alternative 13: 64.9% 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 52.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. *-lowering-*.f64 N/A

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

   2. /-lowering-/.f64 66.9

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

5. Simplified 66.9%

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

Reproduce

herbie shell --seed 2024201 
(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)))