Cubic critical, narrow range

Percentage Accurate: 55.3% → 99.3%

Time: 15.9s

Alternatives: 14

Speedup: 23.2×

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: 55.3% 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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (c * a) / (a * ((0.0 - b) - sqrt(((c * (a * -3.0)) + (b * 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 * a) / (a * ((0.0d0 - b) - sqrt(((c * (a * (-3.0d0))) + (b * b)))))
end function

Java

public static double code(double a, double b, double c) {
	return (c * a) / (a * ((0.0 - b) - Math.sqrt(((c * (a * -3.0)) + (b * b)))));
}

Python

def code(a, b, c):
	return (c * a) / (a * ((0.0 - b) - math.sqrt(((c * (a * -3.0)) + (b * b)))))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (c * a) / (a * ((0.0 - b) - sqrt(((c * (a * -3.0)) + (b * b)))));
end

Wolfram

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

Derivation

1. Initial program 56.5%

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

4. Applied egg-rr 56.5%

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

   1. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}{a}\right), -3\right) \]

   2. flip-- N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)} \cdot \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}{b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}}{a}\right), -3\right) \]

   3. associate-/l/ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)} \cdot \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}{a \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)}\right), -3\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)} \cdot \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right), \left(a \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right)\right), -3\right) \]

6. Applied egg-rr 58.0%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{b \cdot b - \left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}{-3}\right), \color{blue}{\left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b - \left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)\right), -3\right), \left(\color{blue}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   5. associate--r+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(b \cdot b - b \cdot b\right) - c \cdot \left(a \cdot -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   6. +-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - c \cdot \left(a \cdot -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(c \cdot \left(a \cdot -3\right)\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(c \cdot a\right) \cdot -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(c \cdot a\right), -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \color{blue}{\left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}\right)\right)\right) \]

   13. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(c \cdot a\right) \cdot -3\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{\frac{0 - \left(c \cdot a\right) \cdot -3}{-3}}{a \cdot \left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -3}\right)}} \]
9. Step-by-step derivation

   1. frac-2neg N/A

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

   2. distribute-neg-frac N/A

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

   3. sub0-neg N/A

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

   4. remove-double-neg N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(c \cdot a\right) \cdot -3}{-3}\right), \color{blue}{\left(\mathsf{neg}\left(a \cdot \left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -3}\right)\right)\right)}\right) \]

   6. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(c \cdot a\right) \cdot \frac{-3}{-3}\right), \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -3}\right)}\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(c \cdot a\right) \cdot 1\right), \left(\mathsf{neg}\left(a \cdot \color{blue}{\left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -3}\right)}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c \cdot a\right), 1\right), \left(\mathsf{neg}\left(\color{blue}{a \cdot \left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -3}\right)}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), 1\right), \left(\mathsf{neg}\left(\color{blue}{a} \cdot \left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -3}\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), 1\right), \left(\mathsf{neg}\left(\left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -3}\right) \cdot a\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), 1\right), \left(\left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -3}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), 1\right), \mathsf{*.f64}\left(\left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -3}\right), \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}\right)\right) \]

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 2: 92.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -15:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{1}{\frac{b - \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -3}}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{\frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}}{b} + a \cdot \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot t\_0} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot t\_0\right)}}{b}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -15.0)
     (/
      -0.3333333333333333
      (/ 1.0 (/ (- b (sqrt (+ (* b b) (* (* c a) -3.0)))) a)))
     (+
      (/ (* c -0.5) b)
      (*
       a
       (+
        (/ (/ (* (* c c) -0.375) (* b b)) b)
        (*
         a
         (+
          (/ (* c (* (* c c) -0.5625)) (* (* b b) t_0))
          (/
           (/ (* (* c (* c (* c c))) (* a -1.0546875)) (* (* b b) (* b t_0)))
           b)))))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -15.0) {
		tmp = -0.3333333333333333 / (1.0 / ((b - sqrt(((b * b) + ((c * a) * -3.0)))) / a));
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((c * c) * -0.375) / (b * b)) / b) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (b * b)
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-15.0d0)) then
        tmp = (-0.3333333333333333d0) / (1.0d0 / ((b - sqrt(((b * b) + ((c * a) * (-3.0d0))))) / a))
    else
        tmp = ((c * (-0.5d0)) / b) + (a * (((((c * c) * (-0.375d0)) / (b * b)) / b) + (a * (((c * ((c * c) * (-0.5625d0))) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * (-1.0546875d0))) / ((b * b) * (b * t_0))) / b)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -15.0) {
		tmp = -0.3333333333333333 / (1.0 / ((b - Math.sqrt(((b * b) + ((c * a) * -3.0)))) / a));
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((c * c) * -0.375) / (b * b)) / b) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b)))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -15.0:
		tmp = -0.3333333333333333 / (1.0 / ((b - math.sqrt(((b * b) + ((c * a) * -3.0)))) / a))
	else:
		tmp = ((c * -0.5) / b) + (a * (((((c * c) * -0.375) / (b * b)) / b) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b)))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -15.0)
		tmp = Float64(-0.3333333333333333 / Float64(1.0 / Float64(Float64(b - sqrt(Float64(Float64(b * b) + Float64(Float64(c * a) * -3.0)))) / a)));
	else
		tmp = Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)) / b) + Float64(a * Float64(Float64(Float64(c * Float64(Float64(c * c) * -0.5625)) / Float64(Float64(b * b) * t_0)) + Float64(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * -1.0546875)) / Float64(Float64(b * b) * Float64(b * t_0))) / b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b * (b * b);
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -15.0)
		tmp = -0.3333333333333333 / (1.0 / ((b - sqrt(((b * b) + ((c * a) * -3.0)))) / a));
	else
		tmp = ((c * -0.5) / b) + (a * (((((c * c) * -0.375) / (b * b)) / b) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -15.0], N[(-0.3333333333333333 / N[(1.0 / N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(c * N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * -1.0546875), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -15

   1. Initial program 90.2%

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

   4. Applied egg-rr 90.2%

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

      1. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}{a}\right), -3\right) \]

      2. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)} \cdot \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}{b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}}{a}\right), -3\right) \]

      3. associate-/l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)} \cdot \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}{a \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)}\right), -3\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)} \cdot \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right), \left(a \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right)\right), -3\right) \]

   6. Applied egg-rr 91.6%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{-3}\right), \color{blue}{\left(\frac{a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}{b \cdot b - \left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \left(\frac{\color{blue}{a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}}{b \cdot b - \left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}\right)\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \left(\frac{1}{\color{blue}{\frac{b \cdot b - \left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}{a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{b \cdot b - \left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}{a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}\right)}\right)\right) \]

   8. Applied egg-rr 90.3%

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

   if -15 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 53.3%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 92.5%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \left(\left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(\frac{405}{64} \cdot a\right)\right) \cdot \frac{-1}{6}}{b}\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(\left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(\frac{405}{64} \cdot a\right)\right) \cdot \frac{-1}{6}}{b}\right), \color{blue}{a}\right)\right)\right)\right) \]

   7. Applied egg-rr 92.5%

      \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + \color{blue}{\left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right) \cdot a}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(c \cdot \frac{c}{b \cdot b}\right) \cdot \frac{-3}{8}\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{c \cdot c}{b \cdot b} \cdot \frac{-3}{8}\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(c \cdot c\right) \cdot \frac{-3}{8}\right), \left(b \cdot b\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c \cdot c\right), \frac{-3}{8}\right), \left(b \cdot b\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-3}{8}\right), \left(b \cdot b\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{c}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

      7. *-lowering-*.f64 92.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-3}{8}\right), \mathsf{*.f64}\left(b, b\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)}\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

   9. Applied egg-rr 92.5%

      \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\frac{\color{blue}{\frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}}}{b} + \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right) \cdot a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -15:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{1}{\frac{b - \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -3}}{a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{\frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}}{b} + a \cdot \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -15:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{\frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}}{b} + a \cdot \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot t\_0} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot t\_0\right)}}{b}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -15.0)
     (* (/ -0.3333333333333333 a) (- b (sqrt (+ (* b b) (* a (* c -3.0))))))
     (+
      (/ (* c -0.5) b)
      (*
       a
       (+
        (/ (/ (* (* c c) -0.375) (* b b)) b)
        (*
         a
         (+
          (/ (* c (* (* c c) -0.5625)) (* (* b b) t_0))
          (/
           (/ (* (* c (* c (* c c))) (* a -1.0546875)) (* (* b b) (* b t_0)))
           b)))))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -15.0) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt(((b * b) + (a * (c * -3.0)))));
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((c * c) * -0.375) / (b * b)) / b) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (b * b)
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-15.0d0)) then
        tmp = ((-0.3333333333333333d0) / a) * (b - sqrt(((b * b) + (a * (c * (-3.0d0))))))
    else
        tmp = ((c * (-0.5d0)) / b) + (a * (((((c * c) * (-0.375d0)) / (b * b)) / b) + (a * (((c * ((c * c) * (-0.5625d0))) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * (-1.0546875d0))) / ((b * b) * (b * t_0))) / b)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -15.0) {
		tmp = (-0.3333333333333333 / a) * (b - Math.sqrt(((b * b) + (a * (c * -3.0)))));
	} else {
		tmp = ((c * -0.5) / b) + (a * (((((c * c) * -0.375) / (b * b)) / b) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b)))));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -15.0:
		tmp = (-0.3333333333333333 / a) * (b - math.sqrt(((b * b) + (a * (c * -3.0)))))
	else:
		tmp = ((c * -0.5) / b) + (a * (((((c * c) * -0.375) / (b * b)) / b) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b)))))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -15.0)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -3.0))))));
	else
		tmp = Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)) / b) + Float64(a * Float64(Float64(Float64(c * Float64(Float64(c * c) * -0.5625)) / Float64(Float64(b * b) * t_0)) + Float64(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * -1.0546875)) / Float64(Float64(b * b) * Float64(b * t_0))) / b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b * (b * b);
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -15.0)
		tmp = (-0.3333333333333333 / a) * (b - sqrt(((b * b) + (a * (c * -3.0)))));
	else
		tmp = ((c * -0.5) / b) + (a * (((((c * c) * -0.375) / (b * b)) / b) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -15.0], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(c * N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * -1.0546875), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -15

   1. Initial program 90.2%

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

   4. Applied egg-rr 90.3%

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

   if -15 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 53.3%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 92.5%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \left(\left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(\frac{405}{64} \cdot a\right)\right) \cdot \frac{-1}{6}}{b}\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(\left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(\frac{405}{64} \cdot a\right)\right) \cdot \frac{-1}{6}}{b}\right), \color{blue}{a}\right)\right)\right)\right) \]

   7. Applied egg-rr 92.5%

      \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + \color{blue}{\left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right) \cdot a}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(c \cdot \frac{c}{b \cdot b}\right) \cdot \frac{-3}{8}\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{c \cdot c}{b \cdot b} \cdot \frac{-3}{8}\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(c \cdot c\right) \cdot \frac{-3}{8}\right), \left(b \cdot b\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c \cdot c\right), \frac{-3}{8}\right), \left(b \cdot b\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-3}{8}\right), \left(b \cdot b\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{c}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

      7. *-lowering-*.f64 92.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-3}{8}\right), \mathsf{*.f64}\left(b, b\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)}\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

   9. Applied egg-rr 92.5%

      \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\frac{\color{blue}{\frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}}}{b} + \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right) \cdot a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -15:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{\frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}}{b} + a \cdot \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * a) / (a * ((0.0d0 - b) - sqrt(((b * b) + ((c * a) * (-3.0d0))))))
end function

Java

public static double code(double a, double b, double c) {
	return (c * a) / (a * ((0.0 - b) - Math.sqrt(((b * b) + ((c * a) * -3.0)))));
}

Python

def code(a, b, c):
	return (c * a) / (a * ((0.0 - b) - math.sqrt(((b * b) + ((c * a) * -3.0)))))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (c * a) / (a * ((0.0 - b) - sqrt(((b * b) + ((c * a) * -3.0)))));
end

Wolfram

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

Derivation

1. Initial program 56.5%

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

4. Applied egg-rr 56.5%

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

   1. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}{a}\right), -3\right) \]

   2. flip-- N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)} \cdot \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}{b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}}{a}\right), -3\right) \]

   3. associate-/l/ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)} \cdot \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}{a \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)}\right), -3\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)} \cdot \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right), \left(a \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right)\right), -3\right) \]

6. Applied egg-rr 58.0%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{b \cdot b - \left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}{-3}\right), \color{blue}{\left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b - \left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)\right), -3\right), \left(\color{blue}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   5. associate--r+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(b \cdot b - b \cdot b\right) - c \cdot \left(a \cdot -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   6. +-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - c \cdot \left(a \cdot -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(c \cdot \left(a \cdot -3\right)\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(c \cdot a\right) \cdot -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(c \cdot a\right), -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \color{blue}{\left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}\right)\right)\right) \]

   13. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(c \cdot a\right) \cdot -3\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.3%

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

9. Taylor expanded in c around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(a \cdot c\right)\right), \mathsf{*.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right)\right)\right)\right)\right) \]

    2. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(0 - a \cdot c\right), \mathsf{*.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right)\right)\right)\right)\right) \]

    3. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot c\right)\right), \mathsf{*.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right)\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(c \cdot a\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right)\right)\right)\right)\right) \]

    5. *-lowering-*.f64 99.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, a\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right)\right)\right)\right)\right) \]

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 5: 91.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \frac{c \cdot -0.5}{b} + a \cdot \left(\frac{\frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}}{b} + a \cdot \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot t\_0} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot t\_0\right)}}{b}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (+
    (/ (* c -0.5) b)
    (*
     a
     (+
      (/ (/ (* (* c c) -0.375) (* b b)) b)
      (*
       a
       (+
        (/ (* c (* (* c c) -0.5625)) (* (* b b) t_0))
        (/
         (/ (* (* c (* c (* c c))) (* a -1.0546875)) (* (* b b) (* b t_0)))
         b))))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return ((c * -0.5) / b) + (a * (((((c * c) * -0.375) / (b * b)) / b) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b)))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = b * (b * b)
    code = ((c * (-0.5d0)) / b) + (a * (((((c * c) * (-0.375d0)) / (b * b)) / b) + (a * (((c * ((c * c) * (-0.5625d0))) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * (-1.0546875d0))) / ((b * b) * (b * t_0))) / b)))))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return ((c * -0.5) / b) + (a * (((((c * c) * -0.375) / (b * b)) / b) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b)))));
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	return ((c * -0.5) / b) + (a * (((((c * c) * -0.375) / (b * b)) / b) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b)))))

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(Float64(Float64(Float64(c * c) * -0.375) / Float64(b * b)) / b) + Float64(a * Float64(Float64(Float64(c * Float64(Float64(c * c) * -0.5625)) / Float64(Float64(b * b) * t_0)) + Float64(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * -1.0546875)) / Float64(Float64(b * b) * Float64(b * t_0))) / b))))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	tmp = ((c * -0.5) / b) + (a * (((((c * c) * -0.375) / (b * b)) / b) + (a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b)))));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(c * N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * -1.0546875), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in a around 0

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

5. Simplified 89.9%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \left(\left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(\frac{405}{64} \cdot a\right)\right) \cdot \frac{-1}{6}}{b}\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(\left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(\frac{405}{64} \cdot a\right)\right) \cdot \frac{-1}{6}}{b}\right), \color{blue}{a}\right)\right)\right)\right) \]

7. Applied egg-rr 89.9%

   \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + \color{blue}{\left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right) \cdot a}\right) \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(c \cdot \frac{c}{b \cdot b}\right) \cdot \frac{-3}{8}\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{c \cdot c}{b \cdot b} \cdot \frac{-3}{8}\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot b}\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(c \cdot c\right) \cdot \frac{-3}{8}\right), \left(b \cdot b\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(c \cdot c\right), \frac{-3}{8}\right), \left(b \cdot b\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-3}{8}\right), \left(b \cdot b\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{c}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

   7. *-lowering-*.f64 89.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-3}{8}\right), \mathsf{*.f64}\left(b, b\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)}\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

9. Applied egg-rr 89.9%

   \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\frac{\color{blue}{\frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}}}{b} + \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right) \cdot a\right) \]

10. Final simplification 89.9%

    \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\frac{\frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot b}}{b} + a \cdot \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right)\right) \]
11. Add Preprocessing

Alternative 6: 91.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \frac{c \cdot -0.5}{b} + a \cdot \left(a \cdot \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot t\_0} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot t\_0\right)}}{b}\right) + \frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (+
    (/ (* c -0.5) b)
    (*
     a
     (+
      (*
       a
       (+
        (/ (* c (* (* c c) -0.5625)) (* (* b b) t_0))
        (/
         (/ (* (* c (* c (* c c))) (* a -1.0546875)) (* (* b b) (* b t_0)))
         b)))
      (/ (* -0.375 (* c (/ c (* b b)))) b))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return ((c * -0.5) / b) + (a * ((a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b))) + ((-0.375 * (c * (c / (b * b)))) / b)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = b * (b * b)
    code = ((c * (-0.5d0)) / b) + (a * ((a * (((c * ((c * c) * (-0.5625d0))) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * (-1.0546875d0))) / ((b * b) * (b * t_0))) / b))) + (((-0.375d0) * (c * (c / (b * b)))) / b)))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return ((c * -0.5) / b) + (a * ((a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b))) + ((-0.375 * (c * (c / (b * b)))) / b)));
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	return ((c * -0.5) / b) + (a * ((a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b))) + ((-0.375 * (c * (c / (b * b)))) / b)))

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(a * Float64(Float64(Float64(c * Float64(Float64(c * c) * -0.5625)) / Float64(Float64(b * b) * t_0)) + Float64(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * -1.0546875)) / Float64(Float64(b * b) * Float64(b * t_0))) / b))) + Float64(Float64(-0.375 * Float64(c * Float64(c / Float64(b * b)))) / b))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	tmp = ((c * -0.5) / b) + (a * ((a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b))) + ((-0.375 * (c * (c / (b * b)))) / b)));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(a * N[(N[(N[(c * N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * -1.0546875), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(c * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in a around 0

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

5. Simplified 89.9%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \left(\left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(\frac{405}{64} \cdot a\right)\right) \cdot \frac{-1}{6}}{b}\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(\left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(\frac{405}{64} \cdot a\right)\right) \cdot \frac{-1}{6}}{b}\right), \color{blue}{a}\right)\right)\right)\right) \]

7. Applied egg-rr 89.9%

   \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + \color{blue}{\left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right) \cdot a}\right) \]

8. Final simplification 89.9%

   \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(a \cdot \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right) + \frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b}\right) \]
9. Add Preprocessing

Alternative 7: 91.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ a \cdot \left(a \cdot \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot t\_0} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot t\_0\right)}}{b}\right) + \frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b}\right) + c \cdot \frac{-0.5}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (+
    (*
     a
     (+
      (*
       a
       (+
        (/ (* c (* (* c c) -0.5625)) (* (* b b) t_0))
        (/
         (/ (* (* c (* c (* c c))) (* a -1.0546875)) (* (* b b) (* b t_0)))
         b)))
      (/ (* -0.375 (* c (/ c (* b b)))) b)))
    (* c (/ -0.5 b)))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (a * ((a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b))) + ((-0.375 * (c * (c / (b * b)))) / b))) + (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
    real(8) :: t_0
    t_0 = b * (b * b)
    code = (a * ((a * (((c * ((c * c) * (-0.5625d0))) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * (-1.0546875d0))) / ((b * b) * (b * t_0))) / b))) + (((-0.375d0) * (c * (c / (b * b)))) / b))) + (c * ((-0.5d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return (a * ((a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b))) + ((-0.375 * (c * (c / (b * b)))) / b))) + (c * (-0.5 / b));
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	return (a * ((a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b))) + ((-0.375 * (c * (c / (b * b)))) / b))) + (c * (-0.5 / b))

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(Float64(a * Float64(Float64(a * Float64(Float64(Float64(c * Float64(Float64(c * c) * -0.5625)) / Float64(Float64(b * b) * t_0)) + Float64(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * -1.0546875)) / Float64(Float64(b * b) * Float64(b * t_0))) / b))) + Float64(Float64(-0.375 * Float64(c * Float64(c / Float64(b * b)))) / b))) + Float64(c * Float64(-0.5 / b)))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	tmp = (a * ((a * (((c * ((c * c) * -0.5625)) / ((b * b) * t_0)) + ((((c * (c * (c * c))) * (a * -1.0546875)) / ((b * b) * (b * t_0))) / b))) + ((-0.375 * (c * (c / (b * b)))) / b))) + (c * (-0.5 / b));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(a * N[(N[(a * N[(N[(N[(c * N[(N[(c * c), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * -1.0546875), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(c * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in a around 0

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

5. Simplified 89.9%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \left(\left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(\frac{405}{64} \cdot a\right)\right) \cdot \frac{-1}{6}}{b}\right) \cdot \color{blue}{a}\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(\left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(\frac{405}{64} \cdot a\right)\right) \cdot \frac{-1}{6}}{b}\right), \color{blue}{a}\right)\right)\right)\right) \]

7. Applied egg-rr 89.9%

   \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + \color{blue}{\left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right) \cdot a}\right) \]
8. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(c \cdot \frac{\frac{-1}{2}}{b}\right), \mathsf{*.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{b} \cdot c\right), \mathsf{*.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), c\right), \mathsf{*.f64}\left(\color{blue}{a}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

   4. /-lowering-/.f64 89.8%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(c, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \frac{-9}{16}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(a, \frac{-135}{128}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, b\right)\right)\right)\right)\right), b\right)\right), a\right)\right)\right)\right) \]

9. Applied egg-rr 89.8%

   \[\leadsto \color{blue}{\frac{-0.5}{b} \cdot c} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right) \cdot a\right) \]

10. Final simplification 89.8%

    \[\leadsto a \cdot \left(a \cdot \left(\frac{c \cdot \left(\left(c \cdot c\right) \cdot -0.5625\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot -1.0546875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}\right) + \frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b}\right) + c \cdot \frac{-0.5}{b} \]
11. Add Preprocessing

Alternative 8: 88.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(c \cdot a\right) \cdot -3}{-3}}{a \cdot \left(a \cdot \left(\frac{-1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{0 - b \cdot \left(b \cdot b\right)} - \frac{c \cdot -1.5}{b}\right) - b \cdot 2\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (* (* c a) -3.0) -3.0)
  (*
   a
   (-
    (*
     a
     (- (/ (* -1.125 (* a (* c c))) (- 0.0 (* b (* b b)))) (/ (* c -1.5) b)))
    (* b 2.0)))))

C

double code(double a, double b, double c) {
	return (((c * a) * -3.0) / -3.0) / (a * ((a * (((-1.125 * (a * (c * c))) / (0.0 - (b * (b * b)))) - ((c * -1.5) / b))) - (b * 2.0)));
}

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 * a) * (-3.0d0)) / (-3.0d0)) / (a * ((a * ((((-1.125d0) * (a * (c * c))) / (0.0d0 - (b * (b * b)))) - ((c * (-1.5d0)) / b))) - (b * 2.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (((c * a) * -3.0) / -3.0) / (a * ((a * (((-1.125 * (a * (c * c))) / (0.0 - (b * (b * b)))) - ((c * -1.5) / b))) - (b * 2.0)));
}

Python

def code(a, b, c):
	return (((c * a) * -3.0) / -3.0) / (a * ((a * (((-1.125 * (a * (c * c))) / (0.0 - (b * (b * b)))) - ((c * -1.5) / b))) - (b * 2.0)))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(c * a) * -3.0) / -3.0) / Float64(a * Float64(Float64(a * Float64(Float64(Float64(-1.125 * Float64(a * Float64(c * c))) / Float64(0.0 - Float64(b * Float64(b * b)))) - Float64(Float64(c * -1.5) / b))) - Float64(b * 2.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((c * a) * -3.0) / -3.0) / (a * ((a * (((-1.125 * (a * (c * c))) / (0.0 - (b * (b * b)))) - ((c * -1.5) / b))) - (b * 2.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision] / -3.0), $MachinePrecision] / N[(a * N[(N[(a * N[(N[(N[(-1.125 * N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0 - N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * -1.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

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

4. Applied egg-rr 56.5%

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

   1. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}{a}\right), -3\right) \]

   2. flip-- N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)} \cdot \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}{b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}}{a}\right), -3\right) \]

   3. associate-/l/ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{b \cdot b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)} \cdot \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}}{a \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)}\right), -3\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)} \cdot \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right), \left(a \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)\right)\right), -3\right) \]

6. Applied egg-rr 58.0%

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

   1. associate-/l/ N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{b \cdot b - \left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}{-3}\right), \color{blue}{\left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(b \cdot b - \left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)\right), -3\right), \left(\color{blue}{a} \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   5. associate--r+ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(b \cdot b - b \cdot b\right) - c \cdot \left(a \cdot -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   6. +-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - c \cdot \left(a \cdot -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(c \cdot \left(a \cdot -3\right)\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(\left(c \cdot a\right) \cdot -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(c \cdot a\right), -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \left(a \cdot \left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \color{blue}{\left(b + \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \color{blue}{\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}\right)\right)\right) \]

   13. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(a \cdot -3\right)\right)\right)\right)\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(b, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\left(c \cdot a\right) \cdot -3\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.3%

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

9. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \color{blue}{\left(2 \cdot b + a \cdot \left(\frac{-3}{2} \cdot \frac{c}{b} + \frac{-9}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right)}\right)\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \left(a \cdot \left(\frac{-3}{2} \cdot \frac{c}{b} + \frac{-9}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + \color{blue}{2 \cdot b}\right)\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, a\right), -3\right)\right), -3\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(a \cdot \left(\frac{-3}{2} \cdot \frac{c}{b} + \frac{-9}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right), \color{blue}{\left(2 \cdot b\right)}\right)\right)\right) \]

11. Simplified 87.1%

    \[\leadsto \frac{\frac{0 - \left(c \cdot a\right) \cdot -3}{-3}}{a \cdot \color{blue}{\left(a \cdot \left(\frac{c \cdot -1.5}{b} + \frac{-1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{b \cdot \left(b \cdot b\right)}\right) + b \cdot 2\right)}} \]

12. Final simplification 87.1%

    \[\leadsto \frac{\frac{\left(c \cdot a\right) \cdot -3}{-3}}{a \cdot \left(a \cdot \left(\frac{-1.125 \cdot \left(a \cdot \left(c \cdot c\right)\right)}{0 - b \cdot \left(b \cdot b\right)} - \frac{c \cdot -1.5}{b}\right) - b \cdot 2\right)} \]
13. Add Preprocessing

Alternative 9: 88.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(c \cdot -0.5 + a \cdot \left(-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)\right)\right) + \frac{-0.5625 \cdot \left(c \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (+ (* c -0.5) (* a (* -0.375 (* c (/ c (* b b))))))
   (/ (* -0.5625 (* c (* c (* c (* a a))))) (* (* b b) (* b b))))
  b))

C

double code(double a, double b, double c) {
	return (((c * -0.5) + (a * (-0.375 * (c * (c / (b * b)))))) + ((-0.5625 * (c * (c * (c * (a * a))))) / ((b * b) * (b * b)))) / 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)) + (a * ((-0.375d0) * (c * (c / (b * b)))))) + (((-0.5625d0) * (c * (c * (c * (a * a))))) / ((b * b) * (b * b)))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (((c * -0.5) + (a * (-0.375 * (c * (c / (b * b)))))) + ((-0.5625 * (c * (c * (c * (a * a))))) / ((b * b) * (b * b)))) / b;
}

Python

def code(a, b, c):
	return (((c * -0.5) + (a * (-0.375 * (c * (c / (b * b)))))) + ((-0.5625 * (c * (c * (c * (a * a))))) / ((b * b) * (b * b)))) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(c * -0.5) + Float64(a * Float64(-0.375 * Float64(c * Float64(c / Float64(b * b)))))) + Float64(Float64(-0.5625 * Float64(c * Float64(c * Float64(c * Float64(a * a))))) / Float64(Float64(b * b) * Float64(b * b)))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((c * -0.5) + (a * (-0.375 * (c * (c / (b * b)))))) + ((-0.5625 * (c * (c * (c * (a * a))))) / ((b * b) * (b * b)))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(c * -0.5), $MachinePrecision] + N[(a * N[(-0.375 * N[(c * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5625 * N[(c * N[(c * N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in b around inf

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

5. Simplified 86.8%

   \[\leadsto \color{blue}{\frac{\left(c \cdot -0.5 + a \cdot \left(-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)\right)\right) + \frac{-0.5625 \cdot \left(c \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}}{b}} \]
6. Add Preprocessing

Alternative 10: 88.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot -0.5625\right)}{b \cdot b} + \left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (/ (* c -0.5) b)
  (*
   a
   (/
    (+ (/ (* (* c (* c c)) (* a -0.5625)) (* b b)) (* (* c c) -0.375))
    (* b (* b b))))))

C

double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * (((((c * (c * c)) * (a * -0.5625)) / (b * b)) + ((c * c) * -0.375)) / (b * (b * 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) + (a * (((((c * (c * c)) * (a * (-0.5625d0))) / (b * b)) + ((c * c) * (-0.375d0))) / (b * (b * b))))
end function

Java

public static double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * (((((c * (c * c)) * (a * -0.5625)) / (b * b)) + ((c * c) * -0.375)) / (b * (b * b))));
}

Python

def code(a, b, c):
	return ((c * -0.5) / b) + (a * (((((c * (c * c)) * (a * -0.5625)) / (b * b)) + ((c * c) * -0.375)) / (b * (b * b))))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(Float64(Float64(Float64(c * Float64(c * c)) * Float64(a * -0.5625)) / Float64(b * b)) + Float64(Float64(c * c) * -0.375)) / Float64(b * Float64(b * b)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) / b) + (a * (((((c * (c * c)) * (a * -0.5625)) / (b * b)) + ((c * c) * -0.375)) / (b * (b * b))));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(a * -0.5625), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in a around 0

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

5. Simplified 89.9%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b} + a \cdot \left(\frac{-0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right)}{b} + a \cdot \left(\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}} + \frac{\left(\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)\right) \cdot -0.16666666666666666}{b}\right)\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}\right)}\right)\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({\color{blue}{b}}^{3}\right)\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)}{{b}^{2}}\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-9}{16} \cdot \left(a \cdot {c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{-9}{16} \cdot a\right) \cdot {c}^{3}\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-9}{16} \cdot a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left({c}^{3}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \left(c \cdot {c}^{2}\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left({c}^{2}\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \left(c \cdot c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{2}\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \left(\frac{-3}{8} \cdot {c}^{2}\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left({c}^{2}\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \left(c \cdot c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left({b}^{3}\right)\right)\right)\right) \]

   18. cube-mult N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right) \]

   19. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \left(b \cdot {b}^{\color{blue}{2}}\right)\right)\right)\right) \]

   20. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left({b}^{2}\right)}\right)\right)\right)\right) \]

   21. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \left(b \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

   22. *-lowering-*.f64 86.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-9}{16}, a\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(\frac{-3}{8}, \mathsf{*.f64}\left(c, c\right)\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right)\right) \]

8. Simplified 86.7%

   \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \color{blue}{\frac{\frac{\left(-0.5625 \cdot a\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b} + -0.375 \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}} \]

9. Final simplification 86.7%

   \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \frac{\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot -0.5625\right)}{b \cdot b} + \left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)} \]
10. Add Preprocessing

Alternative 11: 82.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{b \cdot -2 + \frac{c \cdot \left(a \cdot 1.5\right)}{b}}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (/ (+ (* b -2.0) (/ (* c (* a 1.5)) b)) c)))

C

double code(double a, double b, double c) {
	return 1.0 / (((b * -2.0) + ((c * (a * 1.5)) / b)) / c);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (((b * (-2.0d0)) + ((c * (a * 1.5d0)) / b)) / c)
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / (((b * -2.0) + ((c * (a * 1.5)) / b)) / c);
}

Python

def code(a, b, c):
	return 1.0 / (((b * -2.0) + ((c * (a * 1.5)) / b)) / c)

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(Float64(b * -2.0) + Float64(Float64(c * Float64(a * 1.5)) / b)) / c))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / (((b * -2.0) + ((c * (a * 1.5)) / b)) / c);
end

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in b around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-3}{2} \cdot \left(a \cdot c\right) + \frac{-9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}}{b}\right)}, \mathsf{*.f64}\left(3, a\right)\right) \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-3}{2} \cdot \left(a \cdot c\right) + \frac{-9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right), b\right), \mathsf{*.f64}\left(\color{blue}{3}, a\right)\right) \]

5. Simplified 80.0%

   \[\leadsto \frac{\color{blue}{\frac{a \cdot \left(c \cdot -1.5\right) + \frac{-1.125 \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}}{b}}}{3 \cdot a} \]
6. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{a \cdot \left(c \cdot \frac{-3}{2}\right) + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}}{\color{blue}{\left(3 \cdot a\right) \cdot b}} \]

   2. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(3 \cdot a\right) \cdot b}{a \cdot \left(c \cdot \frac{-3}{2}\right) + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(3 \cdot a\right) \cdot b}{a \cdot \left(c \cdot \frac{-3}{2}\right) + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(3 \cdot a\right) \cdot b\right), \color{blue}{\left(a \cdot \left(c \cdot \frac{-3}{2}\right) + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(b \cdot \left(3 \cdot a\right)\right), \left(\color{blue}{a \cdot \left(c \cdot \frac{-3}{2}\right)} + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \left(3 \cdot a\right)\right), \left(\color{blue}{a \cdot \left(c \cdot \frac{-3}{2}\right)} + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \left(a \cdot 3\right)\right), \left(a \cdot \color{blue}{\left(c \cdot \frac{-3}{2}\right)} + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \left(a \cdot \color{blue}{\left(c \cdot \frac{-3}{2}\right)} + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \mathsf{+.f64}\left(\left(a \cdot \left(c \cdot \frac{-3}{2}\right)\right), \color{blue}{\left(\frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}\right)}\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \mathsf{+.f64}\left(\left(\left(c \cdot \frac{-3}{2}\right) \cdot a\right), \left(\frac{\color{blue}{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}}{b \cdot b}\right)\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \mathsf{+.f64}\left(\left(c \cdot \left(\frac{-3}{2} \cdot a\right)\right), \left(\frac{\color{blue}{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}}{b \cdot b}\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \mathsf{+.f64}\left(\left(c \cdot \left(a \cdot \frac{-3}{2}\right)\right), \left(\frac{\frac{-9}{8} \cdot \color{blue}{\left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}}{b \cdot b}\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot \frac{-3}{2}\right)\right), \left(\frac{\color{blue}{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}}{b \cdot b}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, \frac{-3}{2}\right)\right), \left(\frac{\frac{-9}{8} \cdot \color{blue}{\left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}}{b \cdot b}\right)\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, \frac{-3}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\right), \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right) \]

7. Applied egg-rr 80.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot \left(a \cdot 3\right)}{c \cdot \left(a \cdot -1.5\right) + \frac{-1.125 \cdot \left(c \cdot \left(a \cdot \left(c \cdot a\right)\right)\right)}{b \cdot b}}}} \]

8. Taylor expanded in c around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{-2 \cdot b + \frac{3}{2} \cdot \frac{a \cdot c}{b}}{c}\right)}\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(-2 \cdot b + \frac{3}{2} \cdot \frac{a \cdot c}{b}\right), \color{blue}{c}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-2 \cdot b\right), \left(\frac{3}{2} \cdot \frac{a \cdot c}{b}\right)\right), c\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(b \cdot -2\right), \left(\frac{3}{2} \cdot \frac{a \cdot c}{b}\right)\right), c\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, -2\right), \left(\frac{3}{2} \cdot \frac{a \cdot c}{b}\right)\right), c\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, -2\right), \left(\frac{\frac{3}{2} \cdot \left(a \cdot c\right)}{b}\right)\right), c\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{/.f64}\left(\left(\frac{3}{2} \cdot \left(a \cdot c\right)\right), b\right)\right), c\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{/.f64}\left(\left(\left(\frac{3}{2} \cdot a\right) \cdot c\right), b\right)\right), c\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{3}{2} \cdot a\right), c\right), b\right)\right), c\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(a \cdot \frac{3}{2}\right), c\right), b\right)\right), c\right)\right) \]

   10. *-lowering-*.f64 80.8%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, -2\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \frac{3}{2}\right), c\right), b\right)\right), c\right)\right) \]

10. Simplified 80.8%

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

11. Final simplification 80.8%

    \[\leadsto \frac{1}{\frac{b \cdot -2 + \frac{c \cdot \left(a \cdot 1.5\right)}{b}}{c}} \]
12. Add Preprocessing

Alternative 12: 82.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + \frac{a \cdot 1.5}{b}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (+ (* -2.0 (/ b c)) (/ (* a 1.5) b))))

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + ((a * 1.5) / 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 = 1.0d0 / (((-2.0d0) * (b / c)) + ((a * 1.5d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + ((a * 1.5) / b));
}

Python

def code(a, b, c):
	return 1.0 / ((-2.0 * (b / c)) + ((a * 1.5) / b))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(Float64(a * 1.5) / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((-2.0 * (b / c)) + ((a * 1.5) / b));
end

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in b around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{-3}{2} \cdot \left(a \cdot c\right) + \frac{-9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}}{b}\right)}, \mathsf{*.f64}\left(3, a\right)\right) \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-3}{2} \cdot \left(a \cdot c\right) + \frac{-9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}\right), b\right), \mathsf{*.f64}\left(\color{blue}{3}, a\right)\right) \]

5. Simplified 80.0%

   \[\leadsto \frac{\color{blue}{\frac{a \cdot \left(c \cdot -1.5\right) + \frac{-1.125 \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}}{b}}}{3 \cdot a} \]
6. Step-by-step derivation

   1. associate-/l/ N/A

      \[\leadsto \frac{a \cdot \left(c \cdot \frac{-3}{2}\right) + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}}{\color{blue}{\left(3 \cdot a\right) \cdot b}} \]

   2. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(3 \cdot a\right) \cdot b}{a \cdot \left(c \cdot \frac{-3}{2}\right) + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(3 \cdot a\right) \cdot b}{a \cdot \left(c \cdot \frac{-3}{2}\right) + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(3 \cdot a\right) \cdot b\right), \color{blue}{\left(a \cdot \left(c \cdot \frac{-3}{2}\right) + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(b \cdot \left(3 \cdot a\right)\right), \left(\color{blue}{a \cdot \left(c \cdot \frac{-3}{2}\right)} + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \left(3 \cdot a\right)\right), \left(\color{blue}{a \cdot \left(c \cdot \frac{-3}{2}\right)} + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \left(a \cdot 3\right)\right), \left(a \cdot \color{blue}{\left(c \cdot \frac{-3}{2}\right)} + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \left(a \cdot \color{blue}{\left(c \cdot \frac{-3}{2}\right)} + \frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \mathsf{+.f64}\left(\left(a \cdot \left(c \cdot \frac{-3}{2}\right)\right), \color{blue}{\left(\frac{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}\right)}\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \mathsf{+.f64}\left(\left(\left(c \cdot \frac{-3}{2}\right) \cdot a\right), \left(\frac{\color{blue}{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}}{b \cdot b}\right)\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \mathsf{+.f64}\left(\left(c \cdot \left(\frac{-3}{2} \cdot a\right)\right), \left(\frac{\color{blue}{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}}{b \cdot b}\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \mathsf{+.f64}\left(\left(c \cdot \left(a \cdot \frac{-3}{2}\right)\right), \left(\frac{\frac{-9}{8} \cdot \color{blue}{\left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}}{b \cdot b}\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot \frac{-3}{2}\right)\right), \left(\frac{\color{blue}{\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}}{b \cdot b}\right)\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, \frac{-3}{2}\right)\right), \left(\frac{\frac{-9}{8} \cdot \color{blue}{\left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}}{b \cdot b}\right)\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(a, 3\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, \frac{-3}{2}\right)\right), \mathsf{/.f64}\left(\left(\frac{-9}{8} \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\right), \color{blue}{\left(b \cdot b\right)}\right)\right)\right)\right) \]

7. Applied egg-rr 80.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{b \cdot \left(a \cdot 3\right)}{c \cdot \left(a \cdot -1.5\right) + \frac{-1.125 \cdot \left(c \cdot \left(a \cdot \left(c \cdot a\right)\right)\right)}{b \cdot b}}}} \]

8. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-2 \cdot \frac{b}{c} + \frac{3}{2} \cdot \frac{a}{b}\right)}\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(-2 \cdot \frac{b}{c}\right), \color{blue}{\left(\frac{3}{2} \cdot \frac{a}{b}\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \left(\frac{b}{c}\right)\right), \left(\color{blue}{\frac{3}{2}} \cdot \frac{a}{b}\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b, c\right)\right), \left(\frac{3}{2} \cdot \frac{a}{b}\right)\right)\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b, c\right)\right), \left(\frac{\frac{3}{2} \cdot a}{\color{blue}{b}}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b, c\right)\right), \mathsf{/.f64}\left(\left(\frac{3}{2} \cdot a\right), \color{blue}{b}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b, c\right)\right), \mathsf{/.f64}\left(\left(a \cdot \frac{3}{2}\right), b\right)\right)\right) \]

   7. *-lowering-*.f64 80.8%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b, c\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{3}{2}\right), b\right)\right)\right) \]

10. Simplified 80.8%

    \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + \frac{a \cdot 1.5}{b}}} \]
11. Add Preprocessing

Alternative 13: 64.6% accurate, 23.2× 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 56.5%

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

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

   4. *-lowering-*.f64 63.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

5. Simplified 63.6%

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

Alternative 14: 64.5% accurate, 23.2× 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 56.5%

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

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

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\right) \]

   4. *-lowering-*.f64 63.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

5. Simplified 63.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{-1}{2}}{b}\right), \color{blue}{c}\right) \]

   4. /-lowering-/.f64 63.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), c\right) \]

7. Applied egg-rr 63.5%

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

8. Final simplification 63.5%

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

Reproduce

herbie shell --seed 2024191 
(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)))