Quadratic roots, narrow range

Percentage Accurate: 55.7% → 92.2%

Time: 14.0s

Alternatives: 11

Speedup: 3.6×

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

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot c\right) \cdot \left(a \cdot a\right)\\ t_1 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_2 := \sqrt{t\_1}\\ t_3 := \mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right)\\ t_4 := \mathsf{fma}\left(16, t\_0, 32 \cdot t\_0\right) - \left(t\_3 \cdot t\_3\right) \cdot 0.25\\ t_5 := \mathsf{fma}\left(-64, \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right), \left(t\_4 \cdot t\_3\right) \cdot -0.5\right)\\ \mathbf{if}\;b \leq 0.08:\\ \;\;\;\;\frac{\left(t\_1 - b \cdot b\right) \cdot \frac{0.5}{a}}{t\_2 + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(t\_2, b, t\_1\right)\right) \cdot \frac{-1}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(0.25, t\_4 \cdot t\_4, \left(t\_5 \cdot t\_3\right) \cdot 0.5\right)}{{b}^{6}}, -\mathsf{fma}\left(0.5, t\_3, \mathsf{fma}\left(0.5, \frac{t\_5}{{b}^{4}}, \frac{t\_4}{b \cdot b} \cdot 0.5\right)\right)\right) \cdot b}\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c c) (* a a)))
        (t_1 (fma (* c -4.0) a (* b b)))
        (t_2 (sqrt t_1))
        (t_3 (fma -8.0 (* a c) (* (* a c) -4.0)))
        (t_4 (- (fma 16.0 t_0 (* 32.0 t_0)) (* (* t_3 t_3) 0.25)))
        (t_5 (fma -64.0 (* (* (* c c) c) (* (* a a) a)) (* (* t_4 t_3) -0.5))))
   (if (<= b 0.08)
     (/ (* (- t_1 (* b b)) (/ 0.5 a)) (+ t_2 b))
     (/
      0.5
      (*
       (*
        (fma b b (fma t_2 b t_1))
        (/
         -1.0
         (*
          (fma
           0.5
           (/ (fma 0.25 (* t_4 t_4) (* (* t_5 t_3) 0.5)) (pow b 6.0))
           (-
            (fma
             0.5
             t_3
             (fma 0.5 (/ t_5 (pow b 4.0)) (* (/ t_4 (* b b)) 0.5)))))
          b)))
       a)))))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) * (a * a);
	double t_1 = fma((c * -4.0), a, (b * b));
	double t_2 = sqrt(t_1);
	double t_3 = fma(-8.0, (a * c), ((a * c) * -4.0));
	double t_4 = fma(16.0, t_0, (32.0 * t_0)) - ((t_3 * t_3) * 0.25);
	double t_5 = fma(-64.0, (((c * c) * c) * ((a * a) * a)), ((t_4 * t_3) * -0.5));
	double tmp;
	if (b <= 0.08) {
		tmp = ((t_1 - (b * b)) * (0.5 / a)) / (t_2 + b);
	} else {
		tmp = 0.5 / ((fma(b, b, fma(t_2, b, t_1)) * (-1.0 / (fma(0.5, (fma(0.25, (t_4 * t_4), ((t_5 * t_3) * 0.5)) / pow(b, 6.0)), -fma(0.5, t_3, fma(0.5, (t_5 / pow(b, 4.0)), ((t_4 / (b * b)) * 0.5)))) * b))) * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) * Float64(a * a))
	t_1 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_2 = sqrt(t_1)
	t_3 = fma(-8.0, Float64(a * c), Float64(Float64(a * c) * -4.0))
	t_4 = Float64(fma(16.0, t_0, Float64(32.0 * t_0)) - Float64(Float64(t_3 * t_3) * 0.25))
	t_5 = fma(-64.0, Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) * a)), Float64(Float64(t_4 * t_3) * -0.5))
	tmp = 0.0
	if (b <= 0.08)
		tmp = Float64(Float64(Float64(t_1 - Float64(b * b)) * Float64(0.5 / a)) / Float64(t_2 + b));
	else
		tmp = Float64(0.5 / Float64(Float64(fma(b, b, fma(t_2, b, t_1)) * Float64(-1.0 / Float64(fma(0.5, Float64(fma(0.25, Float64(t_4 * t_4), Float64(Float64(t_5 * t_3) * 0.5)) / (b ^ 6.0)), Float64(-fma(0.5, t_3, fma(0.5, Float64(t_5 / (b ^ 4.0)), Float64(Float64(t_4 / Float64(b * b)) * 0.5))))) * b))) * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(-8.0 * N[(a * c), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(16.0 * t$95$0 + N[(32.0 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$3 * t$95$3), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-64.0 * N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 * t$95$3), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.08], N[(N[(N[(t$95$1 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 + b), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(N[(b * b + N[(t$95$2 * b + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(N[(0.5 * N[(N[(0.25 * N[(t$95$4 * t$95$4), $MachinePrecision] + N[(N[(t$95$5 * t$95$3), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + (-N[(0.5 * t$95$3 + N[(0.5 * N[(t$95$5 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 / N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0800000000000000017

   1. Initial program 88.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   6. Applied rewrites 90.3%

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

   if 0.0800000000000000017 < b

   1. Initial program 51.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   6. Applied rewrites 52.4%

      \[\leadsto \frac{0.5}{\color{blue}{\left(\frac{-1}{\left(b \cdot b\right) \cdot b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} \cdot \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)\right)\right)} \cdot a} \]

   7. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{1}{2}}{\left(\frac{-1}{\color{blue}{b \cdot \left(\frac{1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} - \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{2}}\right)\right)\right)}} \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)\right)\right) \cdot a} \]

   9. Applied rewrites 94.2%

      \[\leadsto \frac{0.5}{\left(\frac{-1}{\color{blue}{b \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(0.25, \left(\mathsf{fma}\left(16, \left(a \cdot a\right) \cdot \left(c \cdot c\right), 32 \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)\right) - 0.25 \cdot \left(\mathsf{fma}\left(-8, a \cdot c, -4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(-8, a \cdot c, -4 \cdot \left(a \cdot c\right)\right)\right)\right) \cdot \left(\mathsf{fma}\left(16, \left(a \cdot a\right) \cdot \left(c \cdot c\right), 32 \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)\right) - 0.25 \cdot \left(\mathsf{fma}\left(-8, a \cdot c, -4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(-8, a \cdot c, -4 \cdot \left(a \cdot c\right)\right)\right)\right), 0.5 \cdot \left(\mathsf{fma}\left(-8, a \cdot c, -4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(-64, \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right), -0.5 \cdot \left(\mathsf{fma}\left(-8, a \cdot c, -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(16, \left(a \cdot a\right) \cdot \left(c \cdot c\right), 32 \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)\right) - 0.25 \cdot \left(\mathsf{fma}\left(-8, a \cdot c, -4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(-8, a \cdot c, -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right)\right)\right)}{{b}^{6}}, -\mathsf{fma}\left(0.5, \mathsf{fma}\left(-8, a \cdot c, -4 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-64, \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right), -0.5 \cdot \left(\mathsf{fma}\left(-8, a \cdot c, -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(16, \left(a \cdot a\right) \cdot \left(c \cdot c\right), 32 \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)\right) - 0.25 \cdot \left(\mathsf{fma}\left(-8, a \cdot c, -4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(-8, a \cdot c, -4 \cdot \left(a \cdot c\right)\right)\right)\right)\right)\right)}{{b}^{4}}, 0.5 \cdot \frac{\mathsf{fma}\left(16, \left(a \cdot a\right) \cdot \left(c \cdot c\right), 32 \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)\right) - 0.25 \cdot \left(\mathsf{fma}\left(-8, a \cdot c, -4 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(-8, a \cdot c, -4 \cdot \left(a \cdot c\right)\right)\right)}{b \cdot b}\right)\right)\right)}} \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)\right)\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.08:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}, b, \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\right)\right) \cdot \frac{-1}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(0.25, \left(\mathsf{fma}\left(16, \left(c \cdot c\right) \cdot \left(a \cdot a\right), 32 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right) - \left(\mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right) \cdot \mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right)\right) \cdot 0.25\right) \cdot \left(\mathsf{fma}\left(16, \left(c \cdot c\right) \cdot \left(a \cdot a\right), 32 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right) - \left(\mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right) \cdot \mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right)\right) \cdot 0.25\right), \left(\mathsf{fma}\left(-64, \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right), \left(\left(\mathsf{fma}\left(16, \left(c \cdot c\right) \cdot \left(a \cdot a\right), 32 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right) - \left(\mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right) \cdot \mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right)\right) \cdot 0.25\right) \cdot \mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right)\right) \cdot -0.5\right) \cdot \mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right)\right) \cdot 0.5\right)}{{b}^{6}}, -\mathsf{fma}\left(0.5, \mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right), \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-64, \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right), \left(\left(\mathsf{fma}\left(16, \left(c \cdot c\right) \cdot \left(a \cdot a\right), 32 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right) - \left(\mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right) \cdot \mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right)\right) \cdot 0.25\right) \cdot \mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right)\right) \cdot -0.5\right)}{{b}^{4}}, \frac{\mathsf{fma}\left(16, \left(c \cdot c\right) \cdot \left(a \cdot a\right), 32 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right) - \left(\mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right) \cdot \mathsf{fma}\left(-8, a \cdot c, \left(a \cdot c\right) \cdot -4\right)\right) \cdot 0.25}{b \cdot b} \cdot 0.5\right)\right)\right) \cdot b}\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.08:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\frac{\mathsf{fma}\left(-1.5 \cdot \left(a \cdot a\right), \left(a \cdot c\right) \cdot c, \mathsf{fma}\left(2.5 \cdot \left(c \cdot c\right), \left(a \cdot a\right) \cdot a, \mathsf{fma}\left(-0.5 \cdot c, a \cdot a, \mathsf{fma}\left(\left(0.5 \cdot a\right) \cdot b, b, \left(a \cdot c\right) \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\right)}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)} - \frac{0.5}{c}\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= b 0.08)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/
      0.5
      (*
       (-
        (/
         (fma
          (* -1.5 (* a a))
          (* (* a c) c)
          (fma
           (* 2.5 (* c c))
           (* (* a a) a)
           (*
            (fma (* -0.5 c) (* a a) (fma (* (* 0.5 a) b) b (* (* a c) a)))
            (* b b))))
         (* (* (* (* b b) b) b) (* b b)))
        (/ 0.5 c))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (b <= 0.08) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = 0.5 / (((fma((-1.5 * (a * a)), ((a * c) * c), fma((2.5 * (c * c)), ((a * a) * a), (fma((-0.5 * c), (a * a), fma(((0.5 * a) * b), b, ((a * c) * a))) * (b * b)))) / ((((b * b) * b) * b) * (b * b))) - (0.5 / c)) * b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.08)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(0.5 / Float64(Float64(Float64(fma(Float64(-1.5 * Float64(a * a)), Float64(Float64(a * c) * c), fma(Float64(2.5 * Float64(c * c)), Float64(Float64(a * a) * a), Float64(fma(Float64(-0.5 * c), Float64(a * a), fma(Float64(Float64(0.5 * a) * b), b, Float64(Float64(a * c) * a))) * Float64(b * b)))) / Float64(Float64(Float64(Float64(b * b) * b) * b) * Float64(b * b))) - Float64(0.5 / c)) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.08], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(N[(N[(N[(-1.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(a * c), $MachinePrecision] * c), $MachinePrecision] + N[(N[(2.5 * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[(-0.5 * c), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(0.5 * a), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(a * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0800000000000000017

   1. Initial program 88.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   6. Applied rewrites 90.3%

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

   if 0.0800000000000000017 < b

   1. Initial program 51.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 94.2%

      \[\leadsto \frac{0.5}{\color{blue}{\left(\left(\left(\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{c}{{b}^{4}}, -0.5, \mathsf{fma}\left(\frac{\left({a}^{4} \cdot {c}^{4}\right) \cdot 20}{\left({b}^{6} \cdot a\right) \cdot \left(c \cdot c\right)}, 0.125, \mathsf{fma}\left(\frac{a}{b \cdot b}, 0.5, \left(a \cdot a\right) \cdot \frac{c}{{b}^{4}}\right)\right)\right) - \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{6}}\right) - \frac{\left(\left(0.5 \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot a}{{b}^{6}}\right) - \frac{0.5}{c}\right) \cdot b}} \]

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 94.2%

       \[\leadsto \frac{0.5}{\left(\frac{\mathsf{fma}\left(2.5 \cdot \left(\left(a \cdot a\right) \cdot a\right), c \cdot c, \left(b \cdot b\right) \cdot \mathsf{fma}\left(-0.5, \left(a \cdot a\right) \cdot c, \mathsf{fma}\left(0.5, a \cdot \left(b \cdot b\right), \left(a \cdot a\right) \cdot c\right)\right)\right) - 1.5 \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right)\right)}{{b}^{6}} - \frac{0.5}{c}\right) \cdot b} \]

   12. Applied rewrites 94.2%

       \[\leadsto \color{blue}{\frac{0.5}{\left(\frac{\mathsf{fma}\left(-1.5 \cdot \left(a \cdot a\right), \left(c \cdot a\right) \cdot c, \mathsf{fma}\left(\left(c \cdot c\right) \cdot 2.5, \left(a \cdot a\right) \cdot a, \mathsf{fma}\left(-0.5 \cdot c, a \cdot a, \mathsf{fma}\left(\left(0.5 \cdot a\right) \cdot b, b, \left(c \cdot a\right) \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\right)}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)} - \frac{0.5}{c}\right) \cdot b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.08:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left(\frac{\mathsf{fma}\left(-1.5 \cdot \left(a \cdot a\right), \left(a \cdot c\right) \cdot c, \mathsf{fma}\left(2.5 \cdot \left(c \cdot c\right), \left(a \cdot a\right) \cdot a, \mathsf{fma}\left(-0.5 \cdot c, a \cdot a, \mathsf{fma}\left(\left(0.5 \cdot a\right) \cdot b, b, \left(a \cdot c\right) \cdot a\right)\right) \cdot \left(b \cdot b\right)\right)\right)}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)} - \frac{0.5}{c}\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.08:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot b} \cdot 0.5, c, \frac{a}{b} \cdot 0.5\right), c, -0.5 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= b 0.08)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/
      0.5
      (/
       (fma
        (fma (* (/ (* a a) (* (* b b) b)) 0.5) c (* (/ a b) 0.5))
        c
        (* -0.5 b))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (b <= 0.08) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = 0.5 / (fma(fma((((a * a) / ((b * b) * b)) * 0.5), c, ((a / b) * 0.5)), c, (-0.5 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.08)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(0.5 / Float64(fma(fma(Float64(Float64(Float64(a * a) / Float64(Float64(b * b) * b)) * 0.5), c, Float64(Float64(a / b) * 0.5)), c, Float64(-0.5 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.08], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * c + N[(N[(a / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * c + N[(-0.5 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0800000000000000017

   1. Initial program 88.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   6. Applied rewrites 90.3%

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

   if 0.0800000000000000017 < b

   1. Initial program 51.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 91.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.08:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot b} \cdot 0.5, c, \frac{a}{b} \cdot 0.5\right), c, -0.5 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 6.5:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{\left(b \cdot b\right) \cdot b} \cdot 0.5, a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= b 6.5)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/
      0.5
      (fma
       (fma (* (/ c (* (* b b) b)) 0.5) a (/ 0.5 b))
       a
       (* (/ b c) -0.5))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (b <= 6.5) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = 0.5 / fma(fma(((c / ((b * b) * b)) * 0.5), a, (0.5 / b)), a, ((b / c) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 6.5)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(0.5 / fma(fma(Float64(Float64(c / Float64(Float64(b * b) * b)) * 0.5), a, Float64(0.5 / b)), a, Float64(Float64(b / c) * -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 6.5], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(N[(N[(c / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * a + N[(0.5 / b), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 6.5

   1. Initial program 82.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   6. Applied rewrites 84.1%

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

   if 6.5 < b

   1. Initial program 48.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 93.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.5:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{\left(b \cdot b\right) \cdot b} \cdot 0.5, a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.08:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{\left(b \cdot b\right) \cdot b} \cdot 0.5, a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= b 0.08)
     (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) (* 2.0 a)))
     (/
      0.5
      (fma
       (fma (* (/ c (* (* b b) b)) 0.5) a (/ 0.5 b))
       a
       (* (/ b c) -0.5))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (b <= 0.08) {
		tmp = (t_0 - (b * b)) / ((sqrt(t_0) + b) * (2.0 * a));
	} else {
		tmp = 0.5 / fma(fma(((c / ((b * b) * b)) * 0.5), a, (0.5 / b)), a, ((b / c) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.08)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * Float64(2.0 * a)));
	else
		tmp = Float64(0.5 / fma(fma(Float64(Float64(c / Float64(Float64(b * b) * b)) * 0.5), a, Float64(0.5 / b)), a, Float64(Float64(b / c) * -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.08], N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] * N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(N[(N[(c / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * a + N[(0.5 / b), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0800000000000000017

   1. Initial program 88.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   6. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}\right)}} \]

   if 0.0800000000000000017 < b

   1. Initial program 51.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 91.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.08:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{\left(b \cdot b\right) \cdot b} \cdot 0.5, a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.08:\\ \;\;\;\;\frac{\frac{1}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{\left(b \cdot b\right) \cdot b} \cdot 0.5, a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.08)
   (/ (/ 1.0 (/ -1.0 (- b (sqrt (fma (* c -4.0) a (* b b)))))) (* 2.0 a))
   (/
    0.5
    (fma (fma (* (/ c (* (* b b) b)) 0.5) a (/ 0.5 b)) a (* (/ b c) -0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.08) {
		tmp = (1.0 / (-1.0 / (b - sqrt(fma((c * -4.0), a, (b * b)))))) / (2.0 * a);
	} else {
		tmp = 0.5 / fma(fma(((c / ((b * b) * b)) * 0.5), a, (0.5 / b)), a, ((b / c) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.08)
		tmp = Float64(Float64(1.0 / Float64(-1.0 / Float64(b - sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))))) / Float64(2.0 * a));
	else
		tmp = Float64(0.5 / fma(fma(Float64(Float64(c / Float64(Float64(b * b) * b)) * 0.5), a, Float64(0.5 / b)), a, Float64(Float64(b / c) * -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.08], N[(N[(1.0 / N[(-1.0 / N[(b - N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(N[(N[(c / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * a + N[(0.5 / b), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0800000000000000017

   1. Initial program 88.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 88.9%

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

   if 0.0800000000000000017 < b

   1. Initial program 51.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 51.7%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 91.8%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.08:\\ \;\;\;\;\frac{\frac{1}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{\left(b \cdot b\right) \cdot b} \cdot 0.5, a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 44:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 44.0)
   (/ (- (sqrt (fma b b (* a (* c -4.0)))) b) (* 2.0 a))
   (/ 0.5 (fma (/ a b) 0.5 (* (/ b c) -0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 44.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -4.0)))) - b) / (2.0 * a);
	} else {
		tmp = 0.5 / fma((a / b), 0.5, ((b / c) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 44.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -4.0)))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(0.5 / fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 44.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 44

   1. Initial program 80.4%

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

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} \]

      2. sub-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{b \cdot b} + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}{2 \cdot a} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)}}}{2 \cdot a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right) \cdot c}\right)\right)}}{2 \cdot a} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{neg}\left(\color{blue}{\left(4 \cdot a\right)} \cdot c\right)\right)}}{2 \cdot a} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \mathsf{neg}\left(\color{blue}{4 \cdot \left(a \cdot c\right)}\right)\right)}}{2 \cdot a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(4\right)\right) \cdot \color{blue}{\left(c \cdot a\right)}\right)}}{2 \cdot a} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot c\right) \cdot a}\right)}}{2 \cdot a} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot c\right) \cdot a}\right)}}{2 \cdot a} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(4\right)\right) \cdot c\right)} \cdot a\right)}}{2 \cdot a} \]

      13. metadata-eval 80.6

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

   4. Applied rewrites 80.6%

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

   if 44 < b

   1. Initial program 46.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 46.5%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.8

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

   7. Applied rewrites 89.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 44:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 44:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 44.0)
   (* (- (sqrt (fma (* c -4.0) a (* b b))) b) (/ 0.5 a))
   (/ 0.5 (fma (/ a b) 0.5 (* (/ b c) -0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 44.0) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = 0.5 / fma((a / b), 0.5, ((b / c) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 44.0)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(0.5 / fma(Float64(a / b), 0.5, Float64(Float64(b / c) * -0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 44.0], N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(a / b), $MachinePrecision] * 0.5 + N[(N[(b / c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 44

   1. Initial program 80.4%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}} \]

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 80.5

         \[\leadsto \color{blue}{\frac{0.5}{a}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \]

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

          \[\leadsto \frac{\frac{1}{2}}{a} \cdot \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \]

      13. lower--.f64 80.5

          \[\leadsto \frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right)} \]

   4. Applied rewrites 80.5%

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

   if 44 < b

   1. Initial program 46.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 46.5%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.8

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

   7. Applied rewrites 89.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 44:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\mathsf{fma}\left(\frac{a}{b}, 0.5, \frac{b}{c} \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied rewrites 55.2%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{-1}{b - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} \cdot a}} \]

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 82.5

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

7. Applied rewrites 82.5%

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

Alternative 10: 81.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. distribute-lft-out N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 81.9

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

5. Applied rewrites 81.9%

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

6. Final simplification 81.9%

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

Alternative 11: 64.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return -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 = -c / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

   4. lower-neg.f64 64.6

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

5. Applied rewrites 64.6%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))