Cubic critical, narrow range

Percentage Accurate: 55.2% → 99.2%

Time: 11.0s

Alternatives: 10

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-3 \cdot c, a, 0\right)}{a} \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right)}^{-1}}{3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  (/ (fma (* -3.0 c) a 0.0) a)
  (/ (pow (+ (sqrt (fma (* -3.0 c) a (* b b))) b) -1.0) 3.0)))

C

double code(double a, double b, double c) {
	return (fma((-3.0 * c), a, 0.0) / a) * (pow((sqrt(fma((-3.0 * c), a, (b * b))) + b), -1.0) / 3.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(-3.0 * c), a, 0.0) / a) * Float64((Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) + b) ^ -1.0) / 3.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-3.0 * c), $MachinePrecision] * a + 0.0), $MachinePrecision] / a), $MachinePrecision] * N[(N[Power[N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision], -1.0], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 57.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-*.f64 N/A

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

   11. lower--.f64 59.3

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

6. Applied rewrites 59.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/r/ N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

8. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\frac{{\left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right)}^{-1}}{3} \cdot \frac{\mathsf{fma}\left(c \cdot -3, a, 0\right)}{a}} \]

9. Final simplification 99.1%

   \[\leadsto \frac{\mathsf{fma}\left(-3 \cdot c, a, 0\right)}{a} \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right)}^{-1}}{3} \]
10. Add Preprocessing

Alternative 2: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0433:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 1.5, -2 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0433)
   (/ (/ (- (sqrt (fma b b (* a (* -3.0 c)))) b) a) 3.0)
   (/ 1.0 (/ (fma (* (/ c b) a) 1.5 (* -2.0 b)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0433) {
		tmp = ((sqrt(fma(b, b, (a * (-3.0 * c)))) - b) / a) / 3.0;
	} else {
		tmp = 1.0 / (fma(((c / b) * a), 1.5, (-2.0 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0433)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(-3.0 * c)))) - b) / a) / 3.0);
	else
		tmp = Float64(1.0 / Float64(fma(Float64(Float64(c / b) * a), 1.5, Float64(-2.0 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0433], N[(N[(N[(N[Sqrt[N[(b * b + N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * 1.5 + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0432999999999999982

   1. Initial program 83.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 83.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 83.3

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

   6. Applied rewrites 83.3%

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

   if -0.0432999999999999982 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 48.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 48.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. rgt-mult-inverse N/A

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

      6. unpow-1 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

   6. Applied rewrites 48.3%

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

   7. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 89.3

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, \color{blue}{-2 \cdot b}\right)}{c}} \]

   9. Applied rewrites 89.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, -2 \cdot b\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0433:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(-3 \cdot c\right)\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 1.5, -2 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0433:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 1.5, -2 \cdot b\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0433)
   (/ (- (sqrt (fma b b (* (* a -3.0) c))) b) (* 3.0 a))
   (/ 1.0 (/ (fma (* (/ c b) a) 1.5 (* -2.0 b)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0433) {
		tmp = (sqrt(fma(b, b, ((a * -3.0) * c))) - b) / (3.0 * a);
	} else {
		tmp = 1.0 / (fma(((c / b) * a), 1.5, (-2.0 * b)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0433)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -3.0) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(fma(Float64(Float64(c / b) * a), 1.5, Float64(-2.0 * b)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0433], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] * 1.5 + N[(-2.0 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0432999999999999982

   1. Initial program 83.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 83.2

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

   4. Applied rewrites 83.2%

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

   if -0.0432999999999999982 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 48.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 48.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. rgt-mult-inverse N/A

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

      6. unpow-1 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

   6. Applied rewrites 48.3%

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

   7. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 89.3

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, \color{blue}{-2 \cdot b}\right)}{c}} \]

   9. Applied rewrites 89.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(a \cdot \frac{c}{b}, 1.5, -2 \cdot b\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0433:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{c}{b} \cdot a, 1.5, -2 \cdot b\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0433:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0433)
   (/ (- (sqrt (fma b b (* (* a -3.0) c))) b) (* 3.0 a))
   (/ 0.3333333333333333 (fma -0.6666666666666666 (/ b c) (* (/ a b) 0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0433) {
		tmp = (sqrt(fma(b, b, ((a * -3.0) * c))) - b) / (3.0 * a);
	} else {
		tmp = 0.3333333333333333 / fma(-0.6666666666666666, (b / c), ((a / b) * 0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0433)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(a * -3.0) * c))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(0.3333333333333333 / fma(-0.6666666666666666, Float64(b / c), Float64(Float64(a / b) * 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0433], N[(N[(N[Sqrt[N[(b * b + N[(N[(a * -3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0432999999999999982

   1. Initial program 83.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 83.2

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

   4. Applied rewrites 83.2%

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

   if -0.0432999999999999982 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 48.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)}^{-1} \cdot a}} \]

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 95.8%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \mathsf{fma}\left(\frac{-0.75}{b}, \frac{\left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot c}{b}, \mathsf{fma}\left(\frac{-0.2222222222222222}{c}, \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot b}{c}, \frac{0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right), 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)}} \]

   8. Taylor expanded in c around 0

      \[\leadsto \frac{\frac{1}{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{-9}{16} \cdot \frac{{c}^{2}}{{b}^{5}}, \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right), a, \frac{\frac{1}{2}}{b}\right), a, \frac{b}{c} \cdot \frac{-2}{3}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 95.8%

       \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{-0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}, 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)} \]

   11. Taylor expanded in a around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 89.3

          \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \color{blue}{\frac{a}{b}}\right)} \]

   13. Applied rewrites 89.3%

       \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \frac{a}{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0433:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -3\right) \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0433:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0433)
   (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) (* 3.0 a))
   (/ 0.3333333333333333 (fma -0.6666666666666666 (/ b c) (* (/ a b) 0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0433) {
		tmp = (sqrt(fma((-3.0 * c), a, (b * b))) - b) / (3.0 * a);
	} else {
		tmp = 0.3333333333333333 / fma(-0.6666666666666666, (b / c), ((a / b) * 0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0433)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(0.3333333333333333 / fma(-0.6666666666666666, Float64(b / c), Float64(Float64(a / b) * 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0433], N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0432999999999999982

   1. Initial program 83.1%

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

   4. Applied rewrites 83.1%

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

   if -0.0432999999999999982 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 48.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)}^{-1} \cdot a}} \]

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 95.8%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \mathsf{fma}\left(\frac{-0.75}{b}, \frac{\left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot c}{b}, \mathsf{fma}\left(\frac{-0.2222222222222222}{c}, \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot b}{c}, \frac{0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right), 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)}} \]

   8. Taylor expanded in c around 0

      \[\leadsto \frac{\frac{1}{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{-9}{16} \cdot \frac{{c}^{2}}{{b}^{5}}, \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right), a, \frac{\frac{1}{2}}{b}\right), a, \frac{b}{c} \cdot \frac{-2}{3}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 95.8%

       \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{-0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}, 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)} \]

   11. Taylor expanded in a around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 89.3

          \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \color{blue}{\frac{a}{b}}\right)} \]

   13. Applied rewrites 89.3%

       \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \frac{a}{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0433:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0433:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0433)
   (* (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) a) 0.3333333333333333)
   (/ 0.3333333333333333 (fma -0.6666666666666666 (/ b c) (* (/ a b) 0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0433) {
		tmp = ((sqrt(fma((-3.0 * c), a, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = 0.3333333333333333 / fma(-0.6666666666666666, (b / c), ((a / b) * 0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0433)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(0.3333333333333333 / fma(-0.6666666666666666, Float64(b / c), Float64(Float64(a / b) * 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0433], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(0.3333333333333333 / N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0432999999999999982

   1. Initial program 83.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 83.1%

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

   if -0.0432999999999999982 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 48.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)}^{-1} \cdot a}} \]

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 95.8%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \mathsf{fma}\left(\frac{-0.75}{b}, \frac{\left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot c}{b}, \mathsf{fma}\left(\frac{-0.2222222222222222}{c}, \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot b}{c}, \frac{0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right), 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)}} \]

   8. Taylor expanded in c around 0

      \[\leadsto \frac{\frac{1}{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{-9}{16} \cdot \frac{{c}^{2}}{{b}^{5}}, \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right), a, \frac{\frac{1}{2}}{b}\right), a, \frac{b}{c} \cdot \frac{-2}{3}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 95.8%

       \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{-0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}, 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)} \]

   11. Taylor expanded in a around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 89.3

          \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \color{blue}{\frac{a}{b}}\right)} \]

   13. Applied rewrites 89.3%

       \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \frac{a}{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0433:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0433:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -0.0433)
   (* (/ 0.3333333333333333 a) (- (sqrt (fma (* -3.0 c) a (* b b))) b))
   (/ 0.3333333333333333 (fma -0.6666666666666666 (/ b c) (* (/ a b) 0.5)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -0.0433) {
		tmp = (0.3333333333333333 / a) * (sqrt(fma((-3.0 * c), a, (b * b))) - b);
	} else {
		tmp = 0.3333333333333333 / fma(-0.6666666666666666, (b / c), ((a / b) * 0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -0.0433)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b));
	else
		tmp = Float64(0.3333333333333333 / fma(-0.6666666666666666, Float64(b / c), Float64(Float64(a / b) * 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.0433], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0432999999999999982

   1. Initial program 83.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval 83.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-neg.f64 N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 83.0

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

   4. Applied rewrites 83.0%

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

   if -0.0432999999999999982 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 48.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

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

      5. clear-num N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. frac-times N/A

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

   4. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)}^{-1} \cdot a}} \]

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 95.8%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \mathsf{fma}\left(\frac{-0.75}{b}, \frac{\left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot c}{b}, \mathsf{fma}\left(\frac{-0.2222222222222222}{c}, \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot b}{c}, \frac{0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right), 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)}} \]

   8. Taylor expanded in c around 0

      \[\leadsto \frac{\frac{1}{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{-9}{16} \cdot \frac{{c}^{2}}{{b}^{5}}, \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right), a, \frac{\frac{1}{2}}{b}\right), a, \frac{b}{c} \cdot \frac{-2}{3}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 95.8%

       \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{-0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}, 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)} \]

   11. Taylor expanded in a around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 89.3

          \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \color{blue}{\frac{a}{b}}\right)} \]

   13. Applied rewrites 89.3%

       \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \frac{a}{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.0433:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(-3 \cdot c, a, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot a}}{3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (fma (* -3.0 c) a 0.0) (* (+ (sqrt (fma (* -3.0 c) a (* b b))) b) a))
  3.0))

C

double code(double a, double b, double c) {
	return (fma((-3.0 * c), a, 0.0) / ((sqrt(fma((-3.0 * c), a, (b * b))) + b) * a)) / 3.0;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(-3.0 * c), a, 0.0) / Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) + b) * a)) / 3.0)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-3.0 * c), $MachinePrecision] * a + 0.0), $MachinePrecision] / N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]

Derivation

1. Initial program 57.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

4. Applied rewrites 57.3%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-*.f64 N/A

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

   11. lower--.f64 59.3

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

6. Applied rewrites 59.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

8. Applied rewrites 99.1%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(c \cdot -3, a, 0\right)}{a \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -3, a, b \cdot b\right)} + b\right)}}}{3} \]

9. Final simplification 99.1%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(-3 \cdot c, a, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} + b\right) \cdot a}}{3} \]
10. Add Preprocessing

Alternative 9: 82.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 0.3333333333333333 (fma -0.6666666666666666 (/ b c) (* (/ a b) 0.5))))

C

double code(double a, double b, double c) {
	return 0.3333333333333333 / fma(-0.6666666666666666, (b / c), ((a / b) * 0.5));
}

Julia

function code(a, b, c)
	return Float64(0.3333333333333333 / fma(-0.6666666666666666, Float64(b / c), Float64(Float64(a / b) * 0.5)))
end

Wolfram

code[a_, b_, c_] := N[(0.3333333333333333 / N[(-0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.3%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-+.f64 N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. frac-times N/A

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

4. Applied rewrites 57.2%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{\left(\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b\right)}^{-1} \cdot a}} \]

5. Taylor expanded in a around 0

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

7. Applied rewrites 90.7%

   \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \mathsf{fma}\left(\frac{-0.75}{b}, \frac{\left(\frac{c}{{b}^{3}} \cdot -0.375\right) \cdot c}{b}, \mathsf{fma}\left(\frac{-0.2222222222222222}{c}, \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right) \cdot b}{c}, \frac{0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right), 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)}} \]

8. Taylor expanded in c around 0

   \[\leadsto \frac{\frac{1}{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{-9}{16} \cdot \frac{{c}^{2}}{{b}^{5}}, \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right), a, \frac{\frac{1}{2}}{b}\right), a, \frac{b}{c} \cdot \frac{-2}{3}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 90.7%

    \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-a, \frac{-0.5625 \cdot \left(c \cdot c\right)}{{b}^{5}}, 0.375 \cdot \frac{c}{{b}^{3}}\right), a, \frac{0.5}{b}\right), a, \frac{b}{c} \cdot -0.6666666666666666\right)} \]

11. Taylor expanded in a around 0

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

    1. lower-fma.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-/.f64 81.3

       \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \color{blue}{\frac{a}{b}}\right)} \]

13. Applied rewrites 81.3%

    \[\leadsto \frac{0.3333333333333333}{\color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, 0.5 \cdot \frac{a}{b}\right)}} \]

14. Final simplification 81.3%

    \[\leadsto \frac{0.3333333333333333}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c}, \frac{a}{b} \cdot 0.5\right)} \]
15. Add Preprocessing

Alternative 10: 64.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 63.0

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

5. Applied rewrites 63.0%

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

Reproduce

herbie shell --seed 2024297 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))