Cubic critical, narrow range

Percentage Accurate: 55.2% → 99.5%

Time: 13.3s

Alternatives: 16

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: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 55.4

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

5. Simplified 55.4%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unpow1 N/A

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

   9. sqrt-pow1 N/A

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

   10. metadata-eval N/A

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

   11. pow1/2 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. +-commutative N/A

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

7. Applied egg-rr 57.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-in N/A

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

   6. *-lft-identity N/A

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

   7. lift-*.f64 N/A

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

   8. associate--l+ N/A

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

   9. lower-fma.f64 N/A

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

9. Applied egg-rr 99.2%

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

11. Applied egg-rr 99.5%

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

12. Final simplification 99.5%

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

Alternative 2: 89.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(a \cdot -3\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \frac{c \cdot -0.375}{b \cdot \left(b \cdot b\right)}, \frac{0.5}{-b}\right), \frac{b}{c} \cdot 0.6666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= b 11.5)
     (/ (- (* b b) t_0) (* (* a -3.0) (+ b (sqrt t_0))))
     (/
      -0.3333333333333333
      (fma
       a
       (fma a (/ (* c -0.375) (* b (* b b))) (/ 0.5 (- b)))
       (* (/ b c) 0.6666666666666666))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double tmp;
	if (b <= 11.5) {
		tmp = ((b * b) - t_0) / ((a * -3.0) * (b + sqrt(t_0)));
	} else {
		tmp = -0.3333333333333333 / fma(a, fma(a, ((c * -0.375) / (b * (b * b))), (0.5 / -b)), ((b / c) * 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	tmp = 0.0
	if (b <= 11.5)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(a * -3.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(-0.3333333333333333 / fma(a, fma(a, Float64(Float64(c * -0.375) / Float64(b * Float64(b * b))), Float64(0.5 / Float64(-b))), Float64(Float64(b / c) * 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 11.5], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(a * -3.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(a * N[(a * N[(N[(c * -0.375), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / (-b)), $MachinePrecision]), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 11.5

   1. Initial program 81.7%

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

   4. Applied egg-rr 81.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. metadata-eval N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

   6. Applied egg-rr 84.0%

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

   if 11.5 < b

   1. Initial program 47.7%

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

   4. Applied egg-rr 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.7

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

   6. Applied egg-rr 47.7%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   9. Simplified 92.8%

      \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \frac{-0.375 \cdot c}{b \cdot \left(b \cdot b\right)}, -\frac{0.5}{b}\right), \frac{b}{c} \cdot 0.6666666666666666\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \frac{c \cdot -0.375}{b \cdot \left(b \cdot b\right)}, \frac{0.5}{-b}\right), \frac{b}{c} \cdot 0.6666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 15:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(a \cdot -3\right) \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{b \cdot \mathsf{fma}\left(-0.5, \frac{a}{b \cdot b}, \frac{0.6666666666666666}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= b 15.0)
     (/ (- (* b b) t_0) (* (* a -3.0) (+ b (sqrt t_0))))
     (/
      -0.3333333333333333
      (* b (fma -0.5 (/ a (* b b)) (/ 0.6666666666666666 c)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double tmp;
	if (b <= 15.0) {
		tmp = ((b * b) - t_0) / ((a * -3.0) * (b + sqrt(t_0)));
	} else {
		tmp = -0.3333333333333333 / (b * fma(-0.5, (a / (b * b)), (0.6666666666666666 / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	tmp = 0.0
	if (b <= 15.0)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(a * -3.0) * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(-0.3333333333333333 / Float64(b * fma(-0.5, Float64(a / Float64(b * b)), Float64(0.6666666666666666 / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 15.0], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(a * -3.0), $MachinePrecision] * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(b * N[(-0.5 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 15

   1. Initial program 81.4%

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

   4. Applied egg-rr 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. metadata-eval N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

   6. Applied egg-rr 83.8%

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

   if 15 < b

   1. Initial program 47.4%

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

   4. Applied egg-rr 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.4

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

   6. Applied egg-rr 47.4%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 88.3

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

   9. Simplified 88.3%

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

Alternative 4: 85.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \mathbf{if}\;b \leq 15:\\ \;\;\;\;\frac{\left(b \cdot b - t\_0\right) \cdot -0.3333333333333333}{a \cdot \left(b + \sqrt{t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{b \cdot \mathsf{fma}\left(-0.5, \frac{a}{b \cdot b}, \frac{0.6666666666666666}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma c (* a -3.0) (* b b))))
   (if (<= b 15.0)
     (/ (* (- (* b b) t_0) -0.3333333333333333) (* a (+ b (sqrt t_0))))
     (/
      -0.3333333333333333
      (* b (fma -0.5 (/ a (* b b)) (/ 0.6666666666666666 c)))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(c, (a * -3.0), (b * b));
	double tmp;
	if (b <= 15.0) {
		tmp = (((b * b) - t_0) * -0.3333333333333333) / (a * (b + sqrt(t_0)));
	} else {
		tmp = -0.3333333333333333 / (b * fma(-0.5, (a / (b * b)), (0.6666666666666666 / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(c, Float64(a * -3.0), Float64(b * b))
	tmp = 0.0
	if (b <= 15.0)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) * -0.3333333333333333) / Float64(a * Float64(b + sqrt(t_0))));
	else
		tmp = Float64(-0.3333333333333333 / Float64(b * fma(-0.5, Float64(a / Float64(b * b)), Float64(0.6666666666666666 / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 15.0], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] / N[(a * N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(b * N[(-0.5 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 15

   1. Initial program 81.4%

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

   4. Applied egg-rr 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 81.3

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

   6. Applied egg-rr 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   8. Applied egg-rr 83.7%

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

   if 15 < b

   1. Initial program 47.4%

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

   4. Applied egg-rr 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.4

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

   6. Applied egg-rr 47.4%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 88.3

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

   9. Simplified 88.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15:\\ \;\;\;\;\frac{\left(b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right) \cdot -0.3333333333333333}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{b \cdot \mathsf{fma}\left(-0.5, \frac{a}{b \cdot b}, \frac{0.6666666666666666}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 55.4

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

5. Simplified 55.4%

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

   1. lift-neg.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unpow1 N/A

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

   9. sqrt-pow1 N/A

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

   10. metadata-eval N/A

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

   11. pow1/2 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. +-commutative N/A

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

7. Applied egg-rr 57.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-in N/A

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

   6. *-lft-identity N/A

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

   7. lift-*.f64 N/A

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

   8. associate--l+ N/A

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

   9. lower-fma.f64 N/A

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

9. Applied egg-rr 99.2%

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

11. Applied egg-rr 99.4%

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

12. Final simplification 99.4%

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

Alternative 6: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{-3}{b \cdot b}, c \cdot a, 1\right)}, b, -b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{b \cdot \mathsf{fma}\left(-0.5, \frac{a}{b \cdot b}, \frac{0.6666666666666666}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15.0)
   (/ (fma (sqrt (fma (/ -3.0 (* b b)) (* c a) 1.0)) b (- b)) (* a 3.0))
   (/
    -0.3333333333333333
    (* b (fma -0.5 (/ a (* b b)) (/ 0.6666666666666666 c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15.0) {
		tmp = fma(sqrt(fma((-3.0 / (b * b)), (c * a), 1.0)), b, -b) / (a * 3.0);
	} else {
		tmp = -0.3333333333333333 / (b * fma(-0.5, (a / (b * b)), (0.6666666666666666 / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15.0)
		tmp = Float64(fma(sqrt(fma(Float64(-3.0 / Float64(b * b)), Float64(c * a), 1.0)), b, Float64(-b)) / Float64(a * 3.0));
	else
		tmp = Float64(-0.3333333333333333 / Float64(b * fma(-0.5, Float64(a / Float64(b * b)), Float64(0.6666666666666666 / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15.0], N[(N[(N[Sqrt[N[(N[(-3.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * a), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * b + (-b)), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(b * N[(-0.5 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 15

   1. Initial program 81.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 81.4

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

   5. Simplified 81.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow1 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. +-commutative N/A

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

   7. Applied egg-rr 83.5%

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

   9. Applied egg-rr 82.2%

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

   if 15 < b

   1. Initial program 47.4%

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

   4. Applied egg-rr 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.4

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

   6. Applied egg-rr 47.4%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 88.3

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

   9. Simplified 88.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(\frac{-3}{b \cdot b}, c \cdot a, 1\right)}, b, -b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{b \cdot \mathsf{fma}\left(-0.5, \frac{a}{b \cdot b}, \frac{0.6666666666666666}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot \frac{-3}{b \cdot b}, 1\right)}, b, -b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{b \cdot \mathsf{fma}\left(-0.5, \frac{a}{b \cdot b}, \frac{0.6666666666666666}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15.0)
   (/ (fma (sqrt (fma a (* c (/ -3.0 (* b b))) 1.0)) b (- b)) (* a 3.0))
   (/
    -0.3333333333333333
    (* b (fma -0.5 (/ a (* b b)) (/ 0.6666666666666666 c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15.0) {
		tmp = fma(sqrt(fma(a, (c * (-3.0 / (b * b))), 1.0)), b, -b) / (a * 3.0);
	} else {
		tmp = -0.3333333333333333 / (b * fma(-0.5, (a / (b * b)), (0.6666666666666666 / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15.0)
		tmp = Float64(fma(sqrt(fma(a, Float64(c * Float64(-3.0 / Float64(b * b))), 1.0)), b, Float64(-b)) / Float64(a * 3.0));
	else
		tmp = Float64(-0.3333333333333333 / Float64(b * fma(-0.5, Float64(a / Float64(b * b)), Float64(0.6666666666666666 / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15.0], N[(N[(N[Sqrt[N[(a * N[(c * N[(-3.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * b + (-b)), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(b * N[(-0.5 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 15

   1. Initial program 81.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 81.4

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

   5. Simplified 81.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow1 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. +-commutative N/A

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

   7. Applied egg-rr 82.2%

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

   if 15 < b

   1. Initial program 47.4%

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

   4. Applied egg-rr 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.4

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

   6. Applied egg-rr 47.4%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 88.3

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

   9. Simplified 88.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a, c \cdot \frac{-3}{b \cdot b}, 1\right)}, b, -b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{b \cdot \mathsf{fma}\left(-0.5, \frac{a}{b \cdot b}, \frac{0.6666666666666666}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15:\\ \;\;\;\;\mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(\frac{-3}{b \cdot b}, c \cdot a, 1\right)}, -b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{b \cdot \mathsf{fma}\left(-0.5, \frac{a}{b \cdot b}, \frac{0.6666666666666666}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15.0)
   (*
    (fma b (sqrt (fma (/ -3.0 (* b b)) (* c a) 1.0)) (- b))
    (/ 0.3333333333333333 a))
   (/
    -0.3333333333333333
    (* b (fma -0.5 (/ a (* b b)) (/ 0.6666666666666666 c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15.0) {
		tmp = fma(b, sqrt(fma((-3.0 / (b * b)), (c * a), 1.0)), -b) * (0.3333333333333333 / a);
	} else {
		tmp = -0.3333333333333333 / (b * fma(-0.5, (a / (b * b)), (0.6666666666666666 / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15.0)
		tmp = Float64(fma(b, sqrt(fma(Float64(-3.0 / Float64(b * b)), Float64(c * a), 1.0)), Float64(-b)) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(-0.3333333333333333 / Float64(b * fma(-0.5, Float64(a / Float64(b * b)), Float64(0.6666666666666666 / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15.0], N[(N[(b * N[Sqrt[N[(N[(-3.0 / N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(c * a), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(b * N[(-0.5 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 15

   1. Initial program 81.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 81.4

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

   5. Simplified 81.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unpow1 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. +-commutative N/A

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

   7. Applied egg-rr 83.5%

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

   9. Applied egg-rr 82.1%

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

   if 15 < b

   1. Initial program 47.4%

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

   4. Applied egg-rr 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.4

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

   6. Applied egg-rr 47.4%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 88.3

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

   9. Simplified 88.3%

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

Alternative 9: 85.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a, c \cdot \frac{-3}{b \cdot b}, 1\right)}, -b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{b \cdot \mathsf{fma}\left(-0.5, \frac{a}{b \cdot b}, \frac{0.6666666666666666}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15.0)
   (*
    (/ 0.3333333333333333 a)
    (fma b (sqrt (fma a (* c (/ -3.0 (* b b))) 1.0)) (- b)))
   (/
    -0.3333333333333333
    (* b (fma -0.5 (/ a (* b b)) (/ 0.6666666666666666 c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15.0) {
		tmp = (0.3333333333333333 / a) * fma(b, sqrt(fma(a, (c * (-3.0 / (b * b))), 1.0)), -b);
	} else {
		tmp = -0.3333333333333333 / (b * fma(-0.5, (a / (b * b)), (0.6666666666666666 / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15.0)
		tmp = Float64(Float64(0.3333333333333333 / a) * fma(b, sqrt(fma(a, Float64(c * Float64(-3.0 / Float64(b * b))), 1.0)), Float64(-b)));
	else
		tmp = Float64(-0.3333333333333333 / Float64(b * fma(-0.5, Float64(a / Float64(b * b)), Float64(0.6666666666666666 / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15.0], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(b * N[Sqrt[N[(a * N[(c * N[(-3.0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(b * N[(-0.5 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 15

   1. Initial program 81.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 81.4

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

   5. Simplified 81.4%

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

   7. Applied egg-rr 82.1%

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

   if 15 < b

   1. Initial program 47.4%

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

   4. Applied egg-rr 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.4

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

   6. Applied egg-rr 47.4%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 88.3

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

   9. Simplified 88.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(b, \sqrt{\mathsf{fma}\left(a, c \cdot \frac{-3}{b \cdot b}, 1\right)}, -b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{b \cdot \mathsf{fma}\left(-0.5, \frac{a}{b \cdot b}, \frac{0.6666666666666666}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{b \cdot \mathsf{fma}\left(-0.5, \frac{a}{b \cdot b}, \frac{0.6666666666666666}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 11.5)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/
    -0.3333333333333333
    (* b (fma -0.5 (/ a (* b b)) (/ 0.6666666666666666 c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 11.5) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.3333333333333333 / (b * fma(-0.5, (a / (b * b)), (0.6666666666666666 / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 11.5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.3333333333333333 / Float64(b * fma(-0.5, Float64(a / Float64(b * b)), Float64(0.6666666666666666 / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 11.5], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(b * N[(-0.5 * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 11.5

   1. Initial program 81.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 81.7

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

   4. Applied egg-rr 81.7%

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

   if 11.5 < b

   1. Initial program 47.7%

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

   4. Applied egg-rr 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.7

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

   6. Applied egg-rr 47.7%

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

   7. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 88.1

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

   9. Simplified 88.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{b \cdot \mathsf{fma}\left(-0.5, \frac{a}{b \cdot b}, \frac{0.6666666666666666}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{b}{c} \cdot 0.6666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ -0.3333333333333333 (fma -0.5 (/ a b) (* (/ b c) 0.6666666666666666)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.3333333333333333 / fma(-0.5, (a / b), ((b / c) * 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.3333333333333333 / fma(-0.5, Float64(a / b), Float64(Float64(b / c) * 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(-0.5 * N[(a / b), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 15

   1. Initial program 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 81.5

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

   4. Applied egg-rr 81.5%

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

   if 15 < b

   1. Initial program 47.4%

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

   4. Applied egg-rr 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.4

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

   6. Applied egg-rr 47.4%

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

   7. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.2

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

   9. Simplified 88.2%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{b}{c} \cdot 0.6666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{b}{c} \cdot 0.6666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 11.5)
   (/ (- (sqrt (fma a (* c -3.0) (* b b))) b) (* a 3.0))
   (/ -0.3333333333333333 (fma -0.5 (/ a b) (* (/ b c) 0.6666666666666666)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 11.5) {
		tmp = (sqrt(fma(a, (c * -3.0), (b * b))) - b) / (a * 3.0);
	} else {
		tmp = -0.3333333333333333 / fma(-0.5, (a / b), ((b / c) * 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 11.5)
		tmp = Float64(Float64(sqrt(fma(a, Float64(c * -3.0), Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.3333333333333333 / fma(-0.5, Float64(a / b), Float64(Float64(b / c) * 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 11.5], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(-0.5 * N[(a / b), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 11.5

   1. Initial program 81.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 81.7

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

   4. Applied egg-rr 81.7%

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

   if 11.5 < b

   1. Initial program 47.7%

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

   4. Applied egg-rr 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.7

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

   6. Applied egg-rr 47.7%

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

   7. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.1

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

   9. Simplified 88.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{b}{c} \cdot 0.6666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{b}{c} \cdot 0.6666666666666666\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 11.5)
   (* (/ -0.3333333333333333 a) (- b (sqrt (fma a (* c -3.0) (* b b)))))
   (/ -0.3333333333333333 (fma -0.5 (/ a b) (* (/ b c) 0.6666666666666666)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 11.5) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt(fma(a, (c * -3.0), (b * b))));
	} else {
		tmp = -0.3333333333333333 / fma(-0.5, (a / b), ((b / c) * 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 11.5)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))));
	else
		tmp = Float64(-0.3333333333333333 / fma(-0.5, Float64(a / b), Float64(Float64(b / c) * 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 11.5], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(-0.5 * N[(a / b), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 11.5

   1. Initial program 81.7%

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

   4. Applied egg-rr 81.7%

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

   if 11.5 < b

   1. Initial program 47.7%

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

   4. Applied egg-rr 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 47.7

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

   6. Applied egg-rr 47.7%

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

   7. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.1

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

   9. Simplified 88.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11.5:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{b}{c} \cdot 0.6666666666666666\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

4. Applied egg-rr 55.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

   8. lift-/.f64 N/A

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

   9. clear-num N/A

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

   10. associate-*l/ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 55.4

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

6. Applied egg-rr 55.4%

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

7. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 81.5

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

9. Simplified 81.5%

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

Alternative 15: 64.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (c * -0.5) / 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 * (-0.5d0)) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 64.1

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

5. Simplified 64.1%

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

Alternative 16: 64.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * (-0.5 / 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 * ((-0.5d0) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

3. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 64.1

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

5. Simplified 64.1%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 64.1

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

7. Applied egg-rr 64.1%

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

8. Final simplification 64.1%

   \[\leadsto c \cdot \frac{-0.5}{b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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)))