Quadratic roots, narrow range

Percentage Accurate: 56.0% → 90.9%

Time: 12.2s

Alternatives: 16

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: 56.0% 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: 90.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{{b}^{4}}\\ \frac{{a}^{-1}}{\left(\left({\left(b \cdot b\right)}^{-1} - \frac{\mathsf{fma}\left(\left(a \cdot c\right) \cdot c, a, \mathsf{fma}\left(\frac{{a}^{4} \cdot {c}^{4}}{a \cdot a} \cdot \frac{20}{c \cdot c}, -0.25, \left(2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)\right)}{{b}^{6}}\right) - \mathsf{fma}\left(a \cdot t\_0, -2, \mathsf{fma}\left(a, t\_0, \frac{{a}^{-1}}{c}\right)\right)\right) \cdot b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ c (pow b 4.0))))
   (/
    (pow a -1.0)
    (*
     (-
      (-
       (pow (* b b) -1.0)
       (/
        (fma
         (* (* a c) c)
         a
         (fma
          (* (/ (* (pow a 4.0) (pow c 4.0)) (* a a)) (/ 20.0 (* c c)))
          -0.25
          (* (* 2.0 (* a a)) (* c c))))
        (pow b 6.0)))
      (fma (* a t_0) -2.0 (fma a t_0 (/ (pow a -1.0) c))))
     b))))

C

double code(double a, double b, double c) {
	double t_0 = c / pow(b, 4.0);
	return pow(a, -1.0) / (((pow((b * b), -1.0) - (fma(((a * c) * c), a, fma((((pow(a, 4.0) * pow(c, 4.0)) / (a * a)) * (20.0 / (c * c))), -0.25, ((2.0 * (a * a)) * (c * c)))) / pow(b, 6.0))) - fma((a * t_0), -2.0, fma(a, t_0, (pow(a, -1.0) / c)))) * b);
}

Julia

function code(a, b, c)
	t_0 = Float64(c / (b ^ 4.0))
	return Float64((a ^ -1.0) / Float64(Float64(Float64((Float64(b * b) ^ -1.0) - Float64(fma(Float64(Float64(a * c) * c), a, fma(Float64(Float64(Float64((a ^ 4.0) * (c ^ 4.0)) / Float64(a * a)) * Float64(20.0 / Float64(c * c))), -0.25, Float64(Float64(2.0 * Float64(a * a)) * Float64(c * c)))) / (b ^ 6.0))) - fma(Float64(a * t_0), -2.0, fma(a, t_0, Float64((a ^ -1.0) / c)))) * b))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]}, N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[Power[N[(b * b), $MachinePrecision], -1.0], $MachinePrecision] - N[(N[(N[(N[(a * c), $MachinePrecision] * c), $MachinePrecision] * a + N[(N[(N[(N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(20.0 / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25 + N[(N[(2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * t$95$0), $MachinePrecision] * -2.0 + N[(a * t$95$0 + N[(N[Power[a, -1.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 55.7%

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

4. Applied rewrites 55.7%

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

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 92.1%

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

8. Final simplification 92.1%

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

Alternative 2: 90.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot -2}{{b}^{5}}, -c, \frac{a}{{b}^{3}}\right), c, {b}^{-1}\right), c, \frac{-b}{a}\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (pow a -1.0)
  (/
   (fma
    (fma
     (fma (/ (* (* a a) -2.0) (pow b 5.0)) (- c) (/ a (pow b 3.0)))
     c
     (pow b -1.0))
    c
    (/ (- b) a))
   c)))

C

double code(double a, double b, double c) {
	return pow(a, -1.0) / (fma(fma(fma((((a * a) * -2.0) / pow(b, 5.0)), -c, (a / pow(b, 3.0))), c, pow(b, -1.0)), c, (-b / a)) / c);
}

Julia

function code(a, b, c)
	return Float64((a ^ -1.0) / Float64(fma(fma(fma(Float64(Float64(Float64(a * a) * -2.0) / (b ^ 5.0)), Float64(-c), Float64(a / (b ^ 3.0))), c, (b ^ -1.0)), c, Float64(Float64(-b) / a)) / c))
end

Wolfram

code[a_, b_, c_] := N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * (-c) + N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[Power[b, -1.0], $MachinePrecision]), $MachinePrecision] * c + N[((-b) / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

4. Applied rewrites 55.7%

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

5. Taylor expanded in c around 0

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

7. Applied rewrites 92.0%

   \[\leadsto \frac{{a}^{-1}}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.25}{a}, \frac{\left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right) \cdot b}{a}, \mathsf{fma}\left(\frac{a \cdot a}{{b}^{5}}, 2, \frac{\frac{-a}{{b}^{3}} \cdot a}{\left(-b\right) \cdot b}\right)\right), -c, \frac{a}{{b}^{3}}\right), c, \frac{1}{b}\right), c, \frac{-b}{a}\right)}{c}}} \]

8. Taylor expanded in a around 0

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

10. Applied rewrites 92.0%

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

11. Final simplification 92.0%

    \[\leadsto \frac{{a}^{-1}}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot -2}{{b}^{5}}, -c, \frac{a}{{b}^{3}}\right), c, {b}^{-1}\right), c, \frac{-b}{a}\right)}{c}} \]
12. Add Preprocessing

Alternative 3: 90.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-2 \cdot a, {c}^{3}, \left(\left(-c\right) \cdot c\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (/
   (fma
    (* -5.0 (* a a))
    (pow c 4.0)
    (* (fma (* -2.0 a) (pow c 3.0) (* (* (- c) c) (* b b))) (* b b)))
   (pow b 7.0))
  a
  (/ (- c) b)))

C

double code(double a, double b, double c) {
	return fma((fma((-5.0 * (a * a)), pow(c, 4.0), (fma((-2.0 * a), pow(c, 3.0), ((-c * c) * (b * b))) * (b * b))) / pow(b, 7.0)), a, (-c / b));
}

Julia

function code(a, b, c)
	return fma(Float64(fma(Float64(-5.0 * Float64(a * a)), (c ^ 4.0), Float64(fma(Float64(-2.0 * a), (c ^ 3.0), Float64(Float64(Float64(-c) * c) * Float64(b * b))) * Float64(b * b))) / (b ^ 7.0)), a, Float64(Float64(-c) / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-5.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision] + N[(N[(N[(-2.0 * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision] + N[(N[((-c) * c), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 91.9%

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

6. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\frac{-5 \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(-2 \cdot \left(a \cdot {c}^{3}\right) + -1 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
7. Step-by-step derivation

8. Applied rewrites 91.9%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-2 \cdot a, {c}^{3}, \left(\left(-c\right) \cdot c\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
9. Add Preprocessing

Alternative 4: 89.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.0546:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, -t\_0\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(a, \mathsf{fma}\left(-2, a \cdot \left(\frac{c}{{b}^{3}} \cdot -0.5\right), {b}^{-1}\right), \frac{-b}{c}\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= b 0.0546)
     (/ (* (fma b b (- t_0)) (/ 0.5 a)) (- (- b) (sqrt t_0)))
     (pow
      (fma
       a
       (fma -2.0 (* a (* (/ c (pow b 3.0)) -0.5)) (pow b -1.0))
       (/ (- b) c))
      -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (b <= 0.0546) {
		tmp = (fma(b, b, -t_0) * (0.5 / a)) / (-b - sqrt(t_0));
	} else {
		tmp = pow(fma(a, fma(-2.0, (a * ((c / pow(b, 3.0)) * -0.5)), pow(b, -1.0)), (-b / c)), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.0546)
		tmp = Float64(Float64(fma(b, b, Float64(-t_0)) * Float64(0.5 / a)) / Float64(Float64(-b) - sqrt(t_0)));
	else
		tmp = fma(a, fma(-2.0, Float64(a * Float64(Float64(c / (b ^ 3.0)) * -0.5)), (b ^ -1.0)), Float64(Float64(-b) / c)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0546], N[(N[(N[(b * b + (-t$95$0)), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(a * N[(-2.0 * N[(a * N[(N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[Power[b, -1.0], $MachinePrecision]), $MachinePrecision] + N[((-b) / c), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0546000000000000027

   1. Initial program 84.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. clear-num N/A

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

   4. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 86.8%

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

   if 0.0546000000000000027 < b

   1. Initial program 52.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 52.6

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

   5. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.5

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 52.5

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

   7. Applied rewrites 52.5%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-rgt-out N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 91.7

          \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(-2, a \cdot \left(\frac{c}{{b}^{3}} \cdot -0.5\right), \frac{1}{b}\right), \frac{\color{blue}{-b}}{c}\right)} \]

   10. Applied rewrites 91.7%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0546:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, -\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(a, \mathsf{fma}\left(-2, a \cdot \left(\frac{c}{{b}^{3}} \cdot -0.5\right), {b}^{-1}\right), \frac{-b}{c}\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{{a}^{-1}}{\frac{\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(-1, \frac{\frac{b \cdot b}{a}}{c}, 1\right) + a \cdot c\right) - \mathsf{fma}\left(c \cdot c, \left(a \cdot a\right) \cdot -3, \left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{{b}^{6}} \cdot b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (pow a -1.0)
  (*
   (/
    (-
     (* (* b b) (+ (* (* b b) (fma -1.0 (/ (/ (* b b) a) c) 1.0)) (* a c)))
     (fma (* c c) (* (* a a) -3.0) (* (* a a) (* c c))))
    (pow b 6.0))
   b)))

C

double code(double a, double b, double c) {
	return pow(a, -1.0) / (((((b * b) * (((b * b) * fma(-1.0, (((b * b) / a) / c), 1.0)) + (a * c))) - fma((c * c), ((a * a) * -3.0), ((a * a) * (c * c)))) / pow(b, 6.0)) * b);
}

Julia

function code(a, b, c)
	return Float64((a ^ -1.0) / Float64(Float64(Float64(Float64(Float64(b * b) * Float64(Float64(Float64(b * b) * fma(-1.0, Float64(Float64(Float64(b * b) / a) / c), 1.0)) + Float64(a * c))) - fma(Float64(c * c), Float64(Float64(a * a) * -3.0), Float64(Float64(a * a) * Float64(c * c)))) / (b ^ 6.0)) * b))
end

Wolfram

code[a_, b_, c_] := N[(N[Power[a, -1.0], $MachinePrecision] / N[(N[(N[(N[(N[(b * b), $MachinePrecision] * N[(N[(N[(b * b), $MachinePrecision] * N[(-1.0 * N[(N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision] / c), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -3.0), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

4. Applied rewrites 55.7%

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

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 92.1%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 91.5%

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

11. Final simplification 91.5%

    \[\leadsto \frac{{a}^{-1}}{\frac{\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \mathsf{fma}\left(-1, \frac{\frac{b \cdot b}{a}}{c}, 1\right) + a \cdot c\right) - \mathsf{fma}\left(c \cdot c, \left(a \cdot a\right) \cdot -3, \left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)}{{b}^{6}} \cdot b} \]
12. Add Preprocessing

Alternative 6: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.0546:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, -t\_0\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-2 \cdot c, \frac{a \cdot a}{{b}^{3}} \cdot -0.5, \frac{a}{b}\right), -b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= b 0.0546)
     (/ (* (fma b b (- t_0)) (/ 0.5 a)) (- (- b) (sqrt t_0)))
     (pow
      (/
       (fma c (fma (* -2.0 c) (* (/ (* a a) (pow b 3.0)) -0.5) (/ a b)) (- b))
       c)
      -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (b <= 0.0546) {
		tmp = (fma(b, b, -t_0) * (0.5 / a)) / (-b - sqrt(t_0));
	} else {
		tmp = pow((fma(c, fma((-2.0 * c), (((a * a) / pow(b, 3.0)) * -0.5), (a / b)), -b) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.0546)
		tmp = Float64(Float64(fma(b, b, Float64(-t_0)) * Float64(0.5 / a)) / Float64(Float64(-b) - sqrt(t_0)));
	else
		tmp = Float64(fma(c, fma(Float64(-2.0 * c), Float64(Float64(Float64(a * a) / (b ^ 3.0)) * -0.5), Float64(a / b)), Float64(-b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0546], N[(N[(N[(b * b + (-t$95$0)), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(c * N[(N[(-2.0 * c), $MachinePrecision] * N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0546000000000000027

   1. Initial program 84.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. clear-num N/A

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

   4. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 86.8%

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

   if 0.0546000000000000027 < b

   1. Initial program 52.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 52.6

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

   5. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.5

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 52.5

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

   7. Applied rewrites 52.5%

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

   8. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 91.8%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0546:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, -\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(-2 \cdot c, \frac{a \cdot a}{{b}^{3}} \cdot -0.5, \frac{a}{b}\right), -b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.0546:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, -t\_0\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left({b}^{-3} \cdot \mathsf{fma}\left(\left(1 - \frac{\frac{b \cdot b}{a}}{c}\right) \cdot b, b, c \cdot a\right)\right) \cdot a\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= b 0.0546)
     (/ (* (fma b b (- t_0)) (/ 0.5 a)) (- (- b) (sqrt t_0)))
     (pow
      (* (* (pow b -3.0) (fma (* (- 1.0 (/ (/ (* b b) a) c)) b) b (* c a))) a)
      -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (b <= 0.0546) {
		tmp = (fma(b, b, -t_0) * (0.5 / a)) / (-b - sqrt(t_0));
	} else {
		tmp = pow(((pow(b, -3.0) * fma(((1.0 - (((b * b) / a) / c)) * b), b, (c * a))) * a), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.0546)
		tmp = Float64(Float64(fma(b, b, Float64(-t_0)) * Float64(0.5 / a)) / Float64(Float64(-b) - sqrt(t_0)));
	else
		tmp = Float64(Float64((b ^ -3.0) * fma(Float64(Float64(1.0 - Float64(Float64(Float64(b * b) / a) / c)) * b), b, Float64(c * a))) * a) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0546], N[(N[(N[(b * b + (-t$95$0)), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(N[Power[b, -3.0], $MachinePrecision] * N[(N[(N[(1.0 - N[(N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0546000000000000027

   1. Initial program 84.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. clear-num N/A

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

   4. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 86.8%

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

   if 0.0546000000000000027 < b

   1. Initial program 52.7%

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

   4. Applied rewrites 52.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 91.6%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 91.4%

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

       1. lift-/.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow-1 N/A

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

       4. associate-/l/ N/A

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

       5. lower-/.f64 N/A

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

   12. Applied rewrites 91.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0546:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, -\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left({b}^{-3} \cdot \mathsf{fma}\left(\left(1 - \frac{\frac{b \cdot b}{a}}{c}\right) \cdot b, b, c \cdot a\right)\right) \cdot a\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.3:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, -t\_0\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-2 \cdot \left(a \cdot a\right), c, \left(-a\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}}, c, \frac{-1}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= b 0.3)
     (/ (* (fma b b (- t_0)) (/ 0.5 a)) (- (- b) (sqrt t_0)))
     (*
      (fma
       (/ (fma (* -2.0 (* a a)) c (* (- a) (* b b))) (pow b 5.0))
       c
       (/ -1.0 b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (b <= 0.3) {
		tmp = (fma(b, b, -t_0) * (0.5 / a)) / (-b - sqrt(t_0));
	} else {
		tmp = fma((fma((-2.0 * (a * a)), c, (-a * (b * b))) / pow(b, 5.0)), c, (-1.0 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.3)
		tmp = Float64(Float64(fma(b, b, Float64(-t_0)) * Float64(0.5 / a)) / Float64(Float64(-b) - sqrt(t_0)));
	else
		tmp = Float64(fma(Float64(fma(Float64(-2.0 * Float64(a * a)), c, Float64(Float64(-a) * Float64(b * b))) / (b ^ 5.0)), c, Float64(-1.0 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.3], N[(N[(N[(b * b + (-t$95$0)), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * c + N[((-a) * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * c + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.299999999999999989

   1. Initial program 83.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. clear-num N/A

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

   4. Applied rewrites 83.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 84.9%

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

   if 0.299999999999999989 < b

   1. Initial program 51.6%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 91.8%

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

Alternative 9: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 0.3:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, -t\_0\right) \cdot \frac{0.5}{a}}{\left(-b\right) - \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-\mathsf{fma}\left(b, b, a \cdot c\right), b \cdot b, \left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right)\right)}{{b}^{5}} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= b 0.3)
     (/ (* (fma b b (- t_0)) (/ 0.5 a)) (- (- b) (sqrt t_0)))
     (*
      (/
       (fma (- (fma b b (* a c))) (* b b) (* (* -2.0 (* a a)) (* c c)))
       (pow b 5.0))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (b <= 0.3) {
		tmp = (fma(b, b, -t_0) * (0.5 / a)) / (-b - sqrt(t_0));
	} else {
		tmp = (fma(-fma(b, b, (a * c)), (b * b), ((-2.0 * (a * a)) * (c * c))) / pow(b, 5.0)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 0.3)
		tmp = Float64(Float64(fma(b, b, Float64(-t_0)) * Float64(0.5 / a)) / Float64(Float64(-b) - sqrt(t_0)));
	else
		tmp = Float64(Float64(fma(Float64(-fma(b, b, Float64(a * c))), Float64(b * b), Float64(Float64(-2.0 * Float64(a * a)) * Float64(c * c))) / (b ^ 5.0)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.3], N[(N[(N[(b * b + (-t$95$0)), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[(b * b + N[(a * c), $MachinePrecision]), $MachinePrecision]) * N[(b * b), $MachinePrecision] + N[(N[(-2.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.299999999999999989

   1. Initial program 83.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. clear-num N/A

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

   4. Applied rewrites 83.2%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 84.9%

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

   if 0.299999999999999989 < b

   1. Initial program 51.6%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 91.5%

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

Alternative 10: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;b \leq 135:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= b 135.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (pow (/ (fma a (/ c b) (- b)) c) -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (b <= 135.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = pow((fma(a, (c / b), -b) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (b <= 135.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 135.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 135

   1. Initial program 77.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. clear-num N/A

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

   4. Applied rewrites 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 79.2%

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

   if 135 < b

   1. Initial program 46.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 46.5

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

   5. Applied rewrites 46.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 46.4

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 46.4

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

   7. Applied rewrites 46.4%

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

   8. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 89.3

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

   10. Applied rewrites 89.3%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 135:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) - b \cdot b}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\\ \mathbf{if}\;b \leq 135:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a c) -4.0 (* b b))))
   (if (<= b 135.0)
     (/ (- (* b b) t_0) (* (* 2.0 a) (- (- b) (sqrt t_0))))
     (pow (/ (fma a (/ c b) (- b)) c) -1.0))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * c), -4.0, (b * b));
	double tmp;
	if (b <= 135.0) {
		tmp = ((b * b) - t_0) / ((2.0 * a) * (-b - sqrt(t_0)));
	} else {
		tmp = pow((fma(a, (c / b), -b) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(a * c), -4.0, Float64(b * b))
	tmp = 0.0
	if (b <= 135.0)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(2.0 * a) * Float64(Float64(-b) - sqrt(t_0))));
	else
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 135.0], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(2.0 * a), $MachinePrecision] * N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 135

   1. Initial program 77.6%

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

   4. Applied rewrites 77.7%

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

   6. Applied rewrites 79.2%

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

   if 135 < b

   1. Initial program 46.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 46.5

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

   5. Applied rewrites 46.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 46.4

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 46.4

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

   7. Applied rewrites 46.4%

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

   8. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 89.3

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

   10. Applied rewrites 89.3%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 135:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.31:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.31)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 c) a)))) (* 2.0 a))
   (pow (/ (fma a (/ c b) (- b)) c) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.31) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * c) * a)))) / (2.0 * a);
	} else {
		tmp = pow((fma(a, (c / b), -b) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.31)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a)))) / Float64(2.0 * a));
	else
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.31], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.309999999999999998

   1. Initial program 83.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 83.8

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

   4. Applied rewrites 83.8%

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

   if 0.309999999999999998 < b

   1. Initial program 51.6%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 51.4

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

   5. Applied rewrites 51.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 51.4

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 51.4

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

   7. Applied rewrites 51.4%

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

   8. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 86.3

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

   10. Applied rewrites 86.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.31:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.31:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.31)
   (* (/ 0.5 a) (- (sqrt (fma (* -4.0 c) a (* b b))) b))
   (pow (/ (fma a (/ c b) (- b)) c) -1.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.31) {
		tmp = (0.5 / a) * (sqrt(fma((-4.0 * c), a, (b * b))) - b);
	} else {
		tmp = pow((fma(a, (c / b), -b) / c), -1.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.31)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) - b));
	else
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / c) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.31], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.309999999999999998

   1. Initial program 83.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 83.7

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

      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 83.7

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

   4. Applied rewrites 83.7%

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

   if 0.309999999999999998 < b

   1. Initial program 51.6%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 51.4

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

   5. Applied rewrites 51.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 51.4

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 51.4

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

   7. Applied rewrites 51.4%

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

   8. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 86.3

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

   10. Applied rewrites 86.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.31:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (pow (/ (fma a (/ c b) (- b)) c) -1.0))

C

double code(double a, double b, double c) {
	return pow((fma(a, (c / b), -b) / c), -1.0);
}

Julia

function code(a, b, c)
	return Float64(fma(a, Float64(c / b), Float64(-b)) / c) ^ -1.0
end

Wolfram

code[a_, b_, c_] := N[Power[N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / c), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in c around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 55.6

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

5. Applied rewrites 55.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 55.5

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-neg.f64 N/A

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

   8. unsub-neg N/A

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

   9. lower--.f64 55.5

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

7. Applied rewrites 55.5%

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

8. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f64 82.9

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

10. Applied rewrites 82.9%

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

11. Final simplification 82.9%

    \[\leadsto {\left(\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{c}\right)}^{-1} \]
12. Add Preprocessing

Alternative 15: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(-1, \frac{b}{c}, \frac{a}{b}\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (pow (fma -1.0 (/ b c) (/ a b)) -1.0))

C

double code(double a, double b, double c) {
	return pow(fma(-1.0, (b / c), (a / b)), -1.0);
}

Julia

function code(a, b, c)
	return fma(-1.0, Float64(b / c), Float64(a / b)) ^ -1.0
end

Wolfram

code[a_, b_, c_] := N[Power[N[(-1.0 * N[(b / c), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in c around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 55.6

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

5. Applied rewrites 55.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 55.5

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-neg.f64 N/A

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

   8. unsub-neg N/A

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

   9. lower--.f64 55.5

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

7. Applied rewrites 55.5%

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

8. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 82.9

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

10. Applied rewrites 82.9%

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

11. Final simplification 82.9%

    \[\leadsto {\left(\mathsf{fma}\left(-1, \frac{b}{c}, \frac{a}{b}\right)\right)}^{-1} \]
12. Add Preprocessing

Alternative 16: 63.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 64.0

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

5. Applied rewrites 64.0%

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

Reproduce

herbie shell --seed 2024307 
(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)))