Quadratic roots, narrow range

Percentage Accurate: 55.1% → 90.8%

Time: 13.3s

Alternatives: 21

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: 55.1% 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: 90.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.01042:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2 \cdot \left(b \cdot b\right), {c}^{3}, -5 \cdot \left({c}^{4} \cdot a\right)\right)}{{b}^{7}}, a, \frac{c}{{b}^{3}} \cdot \left(-c\right)\right), a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -0.01042)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (fma
      (fma
       (/
        (fma (* -2.0 (* b b)) (pow c 3.0) (* -5.0 (* (pow c 4.0) a)))
        (pow b 7.0))
       a
       (* (/ c (pow b 3.0)) (- c)))
      a
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.01042) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = fma(fma((fma((-2.0 * (b * b)), pow(c, 3.0), (-5.0 * (pow(c, 4.0) * a))) / pow(b, 7.0)), a, ((c / pow(b, 3.0)) * -c)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a)) <= -0.01042)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = fma(fma(Float64(fma(Float64(-2.0 * Float64(b * b)), (c ^ 3.0), Float64(-5.0 * Float64((c ^ 4.0) * a))) / (b ^ 7.0)), a, Float64(Float64(c / (b ^ 3.0)) * Float64(-c))), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.01042], 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[(N[(N[(N[(N[(-2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision] + N[(-5.0 * N[(N[Power[c, 4.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * (-c)), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0104200000000000004

   1. Initial program 82.6%

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

   4. Applied rewrites 82.6%

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

   6. Applied rewrites 84.1%

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

   if -0.0104200000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right), a, -1 \cdot \frac{c}{b}\right)} \]

   5. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25 \cdot a, \frac{{c}^{4}}{{b}^{6}} \cdot \frac{20}{b}, \frac{{c}^{3} \cdot -2}{{b}^{5}}\right), a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-5 \cdot \left(a \cdot {c}^{4}\right) + -2 \cdot \left({b}^{2} \cdot {c}^{3}\right)}{{b}^{7}}, a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 95.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2 \cdot \left(b \cdot b\right), {c}^{3}, \left({c}^{4} \cdot a\right) \cdot -5\right)}{{b}^{7}}, a, \left(-c\right) \cdot \frac{c}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.01042:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2 \cdot \left(b \cdot b\right), {c}^{3}, -5 \cdot \left({c}^{4} \cdot a\right)\right)}{{b}^{7}}, a, \frac{c}{{b}^{3}} \cdot \left(-c\right)\right), a, \frac{-c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.032:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{\left(a \cdot a\right) \cdot \mathsf{fma}\left(c \cdot c, -3, c \cdot c\right)}{{b}^{6}}\right) - \mathsf{fma}\left(\frac{-1}{a}, \frac{-1}{c}, \left(\left(-a\right) \cdot c\right) \cdot {b}^{-4}\right)\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -0.032)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/
      (pow a -1.0)
      (*
       (-
        (-
         (/ 1.0 (* b b))
         (/ (* (* a a) (fma (* c c) -3.0 (* c c))) (pow b 6.0)))
        (fma (/ -1.0 a) (/ -1.0 c) (* (* (- a) c) (pow b -4.0))))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.032) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = pow(a, -1.0) / ((((1.0 / (b * b)) - (((a * a) * fma((c * c), -3.0, (c * c))) / pow(b, 6.0))) - fma((-1.0 / a), (-1.0 / c), ((-a * c) * pow(b, -4.0)))) * b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a)) <= -0.032)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(Float64(1.0 / Float64(b * b)) - Float64(Float64(Float64(a * a) * fma(Float64(c * c), -3.0, Float64(c * c))) / (b ^ 6.0))) - fma(Float64(-1.0 / a), Float64(-1.0 / c), Float64(Float64(Float64(-a) * c) * (b ^ -4.0)))) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.032], 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[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * -3.0 + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 / a), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision] + N[(N[((-a) * c), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.032000000000000001

   1. Initial program 83.8%

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

   4. Applied rewrites 83.9%

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

   6. Applied rewrites 85.2%

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

   if -0.032000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 49.6%

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 94.3%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{\left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-\left(-a \cdot c\right) \cdot c, a, \mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, -0.25, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(a \cdot \frac{c}{{b}^{4}}, -2, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right) \cdot b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{{a}^{2} \cdot \left(-5 \cdot {c}^{2} + \left(2 \cdot {c}^{2} + {c}^{2}\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(a \cdot \frac{c}{{b}^{4}}, -2, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right) \cdot b} \]
   9. Step-by-step derivation

   10. Applied rewrites 94.3%

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

   12. Applied rewrites 94.2%

       \[\leadsto \frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(c \cdot c, -3, c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{6}}\right) - \mathsf{fma}\left(\frac{-1}{a}, \frac{1}{-c}, -1 \cdot \left(\left(c \cdot a\right) \cdot {b}^{-4}\right)\right)\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.032:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{\left(a \cdot a\right) \cdot \mathsf{fma}\left(c \cdot c, -3, c \cdot c\right)}{{b}^{6}}\right) - \mathsf{fma}\left(\frac{-1}{a}, \frac{-1}{c}, \left(\left(-a\right) \cdot c\right) \cdot {b}^{-4}\right)\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.032:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{\left(a \cdot a\right) \cdot \mathsf{fma}\left(c \cdot c, -3, c \cdot c\right)}{{b}^{6}}\right) - \left(\frac{1}{\left(a \cdot a\right) \cdot c} - \frac{c}{{b}^{4}}\right) \cdot a\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -0.032)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/
      (pow a -1.0)
      (*
       (-
        (-
         (/ 1.0 (* b b))
         (/ (* (* a a) (fma (* c c) -3.0 (* c c))) (pow b 6.0)))
        (* (- (/ 1.0 (* (* a a) c)) (/ c (pow b 4.0))) a))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.032) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = pow(a, -1.0) / ((((1.0 / (b * b)) - (((a * a) * fma((c * c), -3.0, (c * c))) / pow(b, 6.0))) - (((1.0 / ((a * a) * c)) - (c / pow(b, 4.0))) * a)) * b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a)) <= -0.032)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(Float64(1.0 / Float64(b * b)) - Float64(Float64(Float64(a * a) * fma(Float64(c * c), -3.0, Float64(c * c))) / (b ^ 6.0))) - Float64(Float64(Float64(1.0 / Float64(Float64(a * a) * c)) - Float64(c / (b ^ 4.0))) * a)) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.032], 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[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * -3.0 + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] - N[(c / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.032000000000000001

   1. Initial program 83.8%

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

   4. Applied rewrites 83.9%

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

   6. Applied rewrites 85.2%

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

   if -0.032000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 49.6%

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 94.3%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{\left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-\left(-a \cdot c\right) \cdot c, a, \mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, -0.25, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(a \cdot \frac{c}{{b}^{4}}, -2, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right) \cdot b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{{a}^{2} \cdot \left(-5 \cdot {c}^{2} + \left(2 \cdot {c}^{2} + {c}^{2}\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(a \cdot \frac{c}{{b}^{4}}, -2, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right) \cdot b} \]
   9. Step-by-step derivation

   10. Applied rewrites 94.3%

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

   11. Taylor expanded in a around -inf

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

   13. Applied rewrites 94.1%

       \[\leadsto \frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(c \cdot c, -3, c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{6}}\right) - \left(-a\right) \cdot \left(\frac{c}{{b}^{4}} - \frac{1}{\left(a \cdot a\right) \cdot c}\right)\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.032:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{\left(a \cdot a\right) \cdot \mathsf{fma}\left(c \cdot c, -3, c \cdot c\right)}{{b}^{6}}\right) - \left(\frac{1}{\left(a \cdot a\right) \cdot c} - \frac{c}{{b}^{4}}\right) \cdot a\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.01042:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(1 - \frac{\frac{b \cdot b}{a}}{c}, b \cdot b, c \cdot a\right) \cdot \left(b \cdot b\right) - \mathsf{fma}\left(\left(a \cdot a\right) \cdot -5, c \cdot c, \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 3\right)}{{b}^{6}} \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -0.01042)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/
      (pow a -1.0)
      (*
       (/
        (-
         (* (fma (- 1.0 (/ (/ (* b b) a) c)) (* b b) (* c a)) (* b b))
         (fma (* (* a a) -5.0) (* c c) (* (* (* a a) (* c c)) 3.0)))
        (pow b 6.0))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.01042) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = pow(a, -1.0) / ((((fma((1.0 - (((b * b) / a) / c)), (b * b), (c * a)) * (b * b)) - fma(((a * a) * -5.0), (c * c), (((a * a) * (c * c)) * 3.0))) / pow(b, 6.0)) * b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a)) <= -0.01042)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(Float64(fma(Float64(1.0 - Float64(Float64(Float64(b * b) / a) / c)), Float64(b * b), Float64(c * a)) * Float64(b * b)) - fma(Float64(Float64(a * a) * -5.0), Float64(c * c), Float64(Float64(Float64(a * a) * Float64(c * c)) * 3.0))) / (b ^ 6.0)) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.01042], 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[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 - N[(N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a * a), $MachinePrecision] * -5.0), $MachinePrecision] * N[(c * c), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0104200000000000004

   1. Initial program 82.6%

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

   4. Applied rewrites 82.6%

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

   6. Applied rewrites 84.1%

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

   if -0.0104200000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   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%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 95.2%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{\left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-\left(-a \cdot c\right) \cdot c, a, \mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, -0.25, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(a \cdot \frac{c}{{b}^{4}}, -2, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right) \cdot b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{{a}^{2} \cdot \left(-5 \cdot {c}^{2} + \left(2 \cdot {c}^{2} + {c}^{2}\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(a \cdot \frac{c}{{b}^{4}}, -2, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right) \cdot b} \]
   9. Step-by-step derivation

   10. Applied rewrites 95.2%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 94.4%

       \[\leadsto \frac{{a}^{-1}}{\frac{\mathsf{fma}\left(1 - \frac{\frac{b \cdot b}{a}}{c}, b \cdot b, 1 \cdot \left(c \cdot a\right)\right) \cdot \left(b \cdot b\right) - \mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), c \cdot c, 3 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{{b}^{6}} \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.01042:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(1 - \frac{\frac{b \cdot b}{a}}{c}, b \cdot b, c \cdot a\right) \cdot \left(b \cdot b\right) - \mathsf{fma}\left(\left(a \cdot a\right) \cdot -5, c \cdot c, \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot 3\right)}{{b}^{6}} \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.01042:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\frac{\frac{-1}{a}}{c} - \left(\frac{-1}{b \cdot b} - \frac{c \cdot a}{{b}^{4}}\right)\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -0.01042)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/
      (pow a -1.0)
      (*
       (- (/ (/ -1.0 a) c) (- (/ -1.0 (* b b)) (/ (* c a) (pow b 4.0))))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -0.01042) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = pow(a, -1.0) / ((((-1.0 / a) / c) - ((-1.0 / (b * b)) - ((c * a) / pow(b, 4.0)))) * b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a)) <= -0.01042)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(Float64(-1.0 / a) / c) - Float64(Float64(-1.0 / Float64(b * b)) - Float64(Float64(c * a) / (b ^ 4.0)))) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.01042], 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[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(-1.0 / a), $MachinePrecision] / c), $MachinePrecision] - N[(N[(-1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0104200000000000004

   1. Initial program 82.6%

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

   4. Applied rewrites 82.6%

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

   6. Applied rewrites 84.1%

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

   if -0.0104200000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   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%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 93.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -0.01042:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\left(\frac{\frac{-1}{a}}{c} - \left(\frac{-1}{b \cdot b} - \frac{c \cdot a}{{b}^{4}}\right)\right) \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{\frac{1}{\frac{\sqrt{t\_0} + b}{b \cdot b - t\_0}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\frac{a}{{b}^{4}}, c, \frac{1}{b \cdot b}\right) \cdot c - \frac{1}{a}}{c} \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 85.0)
     (/ (pow a -1.0) (/ -2.0 (/ 1.0 (/ (+ (sqrt t_0) b) (- (* b b) t_0)))))
     (/
      (pow a -1.0)
      (*
       (/ (- (* (fma (/ a (pow b 4.0)) c (/ 1.0 (* b b))) c) (/ 1.0 a)) c)
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 85.0) {
		tmp = pow(a, -1.0) / (-2.0 / (1.0 / ((sqrt(t_0) + b) / ((b * b) - t_0))));
	} else {
		tmp = pow(a, -1.0) / ((((fma((a / pow(b, 4.0)), c, (1.0 / (b * b))) * c) - (1.0 / a)) / c) * b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 85.0)
		tmp = Float64((a ^ -1.0) / Float64(-2.0 / Float64(1.0 / Float64(Float64(sqrt(t_0) + b) / Float64(Float64(b * b) - t_0)))));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(Float64(fma(Float64(a / (b ^ 4.0)), c, Float64(1.0 / Float64(b * b))) * c) - Float64(1.0 / a)) / c) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 85.0], N[(N[Power[a, -1.0], $MachinePrecision] / N[(-2.0 / N[(1.0 / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(N[(N[(a / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * c + N[(1.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 85

   1. Initial program 83.0%

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

   4. Applied rewrites 83.0%

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

   6. Applied rewrites 84.6%

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

   if 85 < b

   1. Initial program 46.8%

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 95.0%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{\left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-\left(-a \cdot c\right) \cdot c, a, \mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, -0.25, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(a \cdot \frac{c}{{b}^{4}}, -2, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right) \cdot b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{{a}^{2} \cdot \left(-5 \cdot {c}^{2} + \left(2 \cdot {c}^{2} + {c}^{2}\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(a \cdot \frac{c}{{b}^{4}}, -2, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right) \cdot b} \]
   9. Step-by-step derivation

   10. Applied rewrites 95.0%

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

   11. Taylor expanded in c around 0

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

   13. Applied rewrites 93.4%

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

Alternative 7: 88.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{\frac{1}{\frac{\sqrt{t\_0} + b}{b \cdot b - t\_0}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(c \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 85.0)
     (/ (pow a -1.0) (/ -2.0 (/ 1.0 (/ (+ (sqrt t_0) b) (- (* b b) t_0)))))
     (/
      (-
       (/ (* (* (* (* c a) (* c c)) a) -2.0) (pow b 4.0))
       (fma (/ c b) (/ (* c a) b) c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 85.0) {
		tmp = pow(a, -1.0) / (-2.0 / (1.0 / ((sqrt(t_0) + b) / ((b * b) - t_0))));
	} else {
		tmp = ((((((c * a) * (c * c)) * a) * -2.0) / pow(b, 4.0)) - fma((c / b), ((c * a) / b), c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 85.0)
		tmp = Float64((a ^ -1.0) / Float64(-2.0 / Float64(1.0 / Float64(Float64(sqrt(t_0) + b) / Float64(Float64(b * b) - t_0)))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(c * a) * Float64(c * c)) * a) * -2.0) / (b ^ 4.0)) - fma(Float64(c / b), Float64(Float64(c * a) / b), c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 85.0], N[(N[Power[a, -1.0], $MachinePrecision] / N[(-2.0 / N[(1.0 / N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(c * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 85

   1. Initial program 83.0%

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

   4. Applied rewrites 83.0%

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

   6. Applied rewrites 84.6%

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

   if 85 < b

   1. Initial program 46.8%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 93.3%

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

   7. Applied rewrites 93.3%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{-2}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}{b \cdot b - \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(c \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(c \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 85.0)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/
      (-
       (/ (* (* (* (* c a) (* c c)) a) -2.0) (pow b 4.0))
       (fma (/ c b) (/ (* c a) b) c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 85.0) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = ((((((c * a) * (c * c)) * a) * -2.0) / pow(b, 4.0)) - fma((c / b), ((c * a) / b), c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 85.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(c * a) * Float64(c * c)) * a) * -2.0) / (b ^ 4.0)) - fma(Float64(c / b), Float64(Float64(c * a) / b), c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 85.0], 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[(N[(N[(N[(N[(N[(N[(c * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - N[(N[(c / b), $MachinePrecision] * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 85

   1. Initial program 83.0%

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

   4. Applied rewrites 83.0%

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

   6. Applied rewrites 84.6%

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

   if 85 < b

   1. Initial program 46.8%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 93.3%

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

   7. Applied rewrites 93.3%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\left(\left(c \cdot a\right) \cdot \left(c \cdot c\right)\right) \cdot a\right) \cdot -2}{{b}^{4}} - \mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot -2}{{b}^{4}} - \frac{a}{b \cdot b}, c, -1\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 85.0)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/
      (*
       (fma (- (/ (* (* (* a a) c) -2.0) (pow b 4.0)) (/ a (* b b))) c -1.0)
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 85.0) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = (fma((((((a * a) * c) * -2.0) / pow(b, 4.0)) - (a / (b * b))), c, -1.0) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 85.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(a * a) * c) * -2.0) / (b ^ 4.0)) - Float64(a / Float64(b * b))), c, -1.0) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 85.0], 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[(N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + -1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 85

   1. Initial program 83.0%

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

   4. Applied rewrites 83.0%

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

   6. Applied rewrites 84.6%

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

   if 85 < b

   1. Initial program 46.8%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 93.2%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot -2}{{b}^{4}} - \frac{a}{b \cdot b}, c, -1\right) \cdot c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{c}{b \cdot b} - \frac{1}{a}}{c} \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 85.0)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/ (pow a -1.0) (* (/ (- (/ c (* b b)) (/ 1.0 a)) c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 85.0) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = pow(a, -1.0) / ((((c / (b * b)) - (1.0 / a)) / c) * b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 85.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(Float64(c / Float64(b * b)) - Float64(1.0 / a)) / c) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 85.0], 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[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(1.0 / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 85

   1. Initial program 83.0%

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

   4. Applied rewrites 83.0%

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

   6. Applied rewrites 84.6%

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

   if 85 < b

   1. Initial program 46.8%

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 95.0%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{\left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-\left(-a \cdot c\right) \cdot c, a, \mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, -0.25, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(a \cdot \frac{c}{{b}^{4}}, -2, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right) \cdot b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{{a}^{2} \cdot \left(-5 \cdot {c}^{2} + \left(2 \cdot {c}^{2} + {c}^{2}\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(a \cdot \frac{c}{{b}^{4}}, -2, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right) \cdot b} \]
   9. Step-by-step derivation

   10. Applied rewrites 95.0%

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

   11. Taylor expanded in c around 0

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

   13. Applied rewrites 89.3%

       \[\leadsto \frac{{a}^{-1}}{\frac{\frac{c}{b \cdot b} - \frac{1}{a}}{c} \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{c}{b \cdot b} - \frac{1}{a}}{c} \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{a}{b \cdot b} - \frac{1}{c}}{a} \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 85.0)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/ (pow a -1.0) (* (/ (- (/ a (* b b)) (/ 1.0 c)) a) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 85.0) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = pow(a, -1.0) / ((((a / (b * b)) - (1.0 / c)) / a) * b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 85.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(Float64(a / Float64(b * b)) - Float64(1.0 / c)) / a) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 85.0], 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[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(1.0 / c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 85

   1. Initial program 83.0%

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

   4. Applied rewrites 83.0%

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

   6. Applied rewrites 84.6%

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

   if 85 < b

   1. Initial program 46.8%

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 95.0%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{\left(\left(\frac{1}{b \cdot b} - \frac{\mathsf{fma}\left(-\left(-a \cdot c\right) \cdot c, a, \mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, -0.25, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(a \cdot \frac{c}{{b}^{4}}, -2, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right) \cdot b}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{{a}^{-1}}{\left(\left(\frac{1}{b \cdot b} - \frac{{a}^{2} \cdot \left(-5 \cdot {c}^{2} + \left(2 \cdot {c}^{2} + {c}^{2}\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(a \cdot \frac{c}{{b}^{4}}, -2, \mathsf{fma}\left(a, \frac{c}{{b}^{4}}, \frac{\frac{1}{a}}{c}\right)\right)\right) \cdot b} \]
   9. Step-by-step derivation

   10. Applied rewrites 95.0%

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

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 89.2%

       \[\leadsto \frac{{a}^{-1}}{\frac{\frac{a}{b \cdot b} - \frac{1}{c}}{a} \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{a}{b \cdot b} - \frac{1}{c}}{a} \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{c}{b} - \frac{b}{a}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 85.0)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/ (pow a -1.0) (/ (- (/ c b) (/ b a)) c)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 85.0) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = pow(a, -1.0) / (((c / b) - (b / a)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 85.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(c / b) - Float64(b / a)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 85.0], 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[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 85

   1. Initial program 83.0%

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

   4. Applied rewrites 83.0%

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

   6. Applied rewrites 84.6%

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

   if 85 < b

   1. Initial program 46.8%

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 89.1

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

   7. Applied rewrites 89.1%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{\frac{\frac{c}{b} - \frac{b}{a}}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{c}{b} - \frac{b}{a}}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{a}{b} - \frac{b}{c}}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 85.0)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/ (pow a -1.0) (/ (- (/ a b) (/ b c)) a)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 85.0) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = pow(a, -1.0) / (((a / b) - (b / c)) / a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 85.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64((a ^ -1.0) / Float64(Float64(Float64(a / b) - Float64(b / c)) / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 85.0], 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[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(a / b), $MachinePrecision] - N[(b / c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 85

   1. Initial program 83.0%

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

   4. Applied rewrites 83.0%

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

   6. Applied rewrites 84.6%

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

   if 85 < b

   1. Initial program 46.8%

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 89.1

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

   7. Applied rewrites 89.1%

      \[\leadsto \frac{{a}^{-1}}{\color{blue}{\frac{\frac{a}{b} - \frac{b}{c}}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{-1}}{\frac{\frac{a}{b} - \frac{b}{c}}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{{b}^{3}}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 85.0)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (- (fma a (/ (* c c) (pow b 3.0)) (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 85.0) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = -fma(a, ((c * c) / pow(b, 3.0)), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 85.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(-fma(a, Float64(Float64(c * c) / (b ^ 3.0)), Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 85.0], 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[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if b < 85

   1. Initial program 83.0%

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

   4. Applied rewrites 83.0%

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

   6. Applied rewrites 84.6%

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

   if 85 < b

   1. Initial program 46.8%

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 89.0

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

   7. Applied rewrites 89.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{{b}^{3}}, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -8e-7)
   (/ (- (sqrt (fma (* -4.0 c) a (* b b))) b) (* 2.0 a))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -8e-7) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a)) <= -8e-7)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -8e-7], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -7.9999999999999996e-7

   1. Initial program 75.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 75.7

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 75.8

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

   4. Applied rewrites 75.8%

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

   if -7.9999999999999996e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 32.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 83.0

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

   5. Applied rewrites 83.0%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a)) -8e-7)
   (* (- (sqrt (fma (* -4.0 c) a (* b b))) b) (/ 0.5 a))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 4.0)))) - b) / (2.0 * a)) <= -8e-7) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(2.0 * a)) <= -8e-7)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -8e-7], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -7.9999999999999996e-7

   1. Initial program 75.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 75.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      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 75.7

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

   4. Applied rewrites 75.8%

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

   if -7.9999999999999996e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 32.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 83.0

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

   5. Applied rewrites 83.0%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a} \leq -8 \cdot 10^{-7}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;b \leq 108:\\ \;\;\;\;\frac{\left(t\_0 - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{t\_0} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 a) c (* b b))))
   (if (<= b 108.0)
     (/ (* (- t_0 (* b b)) (/ 0.5 a)) (+ (sqrt t_0) b))
     (/ (fma (/ c b) (/ (* c a) b) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * a), c, (b * b));
	double tmp;
	if (b <= 108.0) {
		tmp = ((t_0 - (b * b)) * (0.5 / a)) / (sqrt(t_0) + b);
	} else {
		tmp = fma((c / b), ((c * a) / b), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (b <= 108.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) * Float64(0.5 / a)) / Float64(sqrt(t_0) + b));
	else
		tmp = Float64(fma(Float64(c / b), Float64(Float64(c * a) / b), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 108.0], 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[(N[(N[(c / b), $MachinePrecision] * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 108

   1. Initial program 82.5%

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

   4. Applied rewrites 82.5%

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

   6. Applied rewrites 84.0%

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

   if 108 < b

   1. Initial program 46.2%

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

   3. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-*r/ N/A

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

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. div-sub N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-out N/A

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

      11. associate-/l* N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 89.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 108:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - b \cdot b\right) \cdot \frac{0.5}{a}}{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{t\_0 - b \cdot b}{\left(\sqrt{t\_0} + b\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))))
   (if (<= b 85.0)
     (/ (- t_0 (* b b)) (* (+ (sqrt t_0) b) (* 2.0 a)))
     (/ (fma (/ c b) (/ (* c a) b) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double tmp;
	if (b <= 85.0) {
		tmp = (t_0 - (b * b)) / ((sqrt(t_0) + b) * (2.0 * a));
	} else {
		tmp = fma((c / b), ((c * a) / b), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	tmp = 0.0
	if (b <= 85.0)
		tmp = Float64(Float64(t_0 - Float64(b * b)) / Float64(Float64(sqrt(t_0) + b) * Float64(2.0 * a)));
	else
		tmp = Float64(fma(Float64(c / b), Float64(Float64(c * a) / b), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 85.0], 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[(N[(N[(c / b), $MachinePrecision] * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 85

   1. Initial program 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. unsub-neg N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 82.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-div N/A

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

      5. flip-- N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. lift-*.f64 N/A

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

      12. lower--.f64 N/A

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

   6. Applied rewrites 84.5%

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

   if 85 < b

   1. Initial program 46.8%

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

   3. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-*r/ N/A

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

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. div-sub N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-out N/A

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

      11. associate-/l* N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 89.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right) - b \cdot b}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 85.0)
   (/ (- (sqrt (fma b b (* (* -4.0 c) a))) b) (* 2.0 a))
   (/ (fma (/ c b) (/ (* c a) b) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 85.0) {
		tmp = (sqrt(fma(b, b, ((-4.0 * c) * a))) - b) / (2.0 * a);
	} else {
		tmp = fma((c / b), ((c * a) / b), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 85.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(Float64(c / b), Float64(Float64(c * a) / b), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 85.0], N[(N[(N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 85

   1. Initial program 83.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 83.1

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

   4. Applied rewrites 83.1%

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

   if 85 < b

   1. Initial program 46.8%

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

   3. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-*r/ N/A

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

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. div-sub N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-out N/A

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

      11. associate-/l* N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 89.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 85.0)
   (/ (- (sqrt (fma (* -4.0 c) a (* b b))) b) (* 2.0 a))
   (/ (fma (/ c b) (/ (* c a) b) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 85.0) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) - b) / (2.0 * a);
	} else {
		tmp = fma((c / b), ((c * a) / b), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 85.0)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b) / Float64(2.0 * a));
	else
		tmp = Float64(fma(Float64(c / b), Float64(Float64(c * a) / b), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 85.0], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / b), $MachinePrecision] * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 85

   1. Initial program 83.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 83.0

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval 83.0

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

   4. Applied rewrites 83.0%

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

   if 85 < b

   1. Initial program 46.8%

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

   3. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. associate-*r/ N/A

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

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. div-sub N/A

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

      8. unsub-neg N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-out N/A

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

      11. associate-/l* N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

   5. Applied rewrites 89.0%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 85:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, \frac{c \cdot a}{b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 64.6% 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 58.5%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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 61.5

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

5. Applied rewrites 61.5%

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

Reproduce

herbie shell --seed 2024283 
(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)))