Cubic critical, narrow range

Percentage Accurate: 55.0% → 91.4%

Time: 27.4s

Alternatives: 15

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: 55.0% 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: 91.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \frac{-0.3333333333333333}{\frac{0.6666666666666666}{\frac{c}{b}} + a \cdot \left(a \cdot \left(\frac{\frac{c}{\frac{b \cdot b}{-0.375}}}{b} + a \cdot \left(\left(\frac{\frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\frac{{b}^{6}}{b}}}{\frac{c \cdot c}{-1.40625}} + \frac{-0.75}{\frac{b}{\frac{\frac{c}{\frac{\frac{t\_0}{c}}{-0.375}}}{b}}}\right) + \frac{c \cdot c}{\frac{b \cdot \left(b \cdot t\_0\right)}{0.5625}}\right)\right) + \frac{-0.5}{b}\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (/
    -0.3333333333333333
    (+
     (/ 0.6666666666666666 (/ c b))
     (*
      a
      (+
       (*
        a
        (+
         (/ (/ c (/ (* b b) -0.375)) b)
         (*
          a
          (+
           (+
            (/ (/ (* c (* c (* c c))) (/ (pow b 6.0) b)) (/ (* c c) -1.40625))
            (/ -0.75 (/ b (/ (/ c (/ (/ t_0 c) -0.375)) b))))
           (/ (* c c) (/ (* b (* b t_0)) 0.5625))))))
       (/ -0.5 b)))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return -0.3333333333333333 / ((0.6666666666666666 / (c / b)) + (a * ((a * (((c / ((b * b) / -0.375)) / b) + (a * (((((c * (c * (c * c))) / (pow(b, 6.0) / b)) / ((c * c) / -1.40625)) + (-0.75 / (b / ((c / ((t_0 / c) / -0.375)) / b)))) + ((c * c) / ((b * (b * t_0)) / 0.5625)))))) + (-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 = (-0.3333333333333333d0) / ((0.6666666666666666d0 / (c / b)) + (a * ((a * (((c / ((b * b) / (-0.375d0))) / b) + (a * (((((c * (c * (c * c))) / ((b ** 6.0d0) / b)) / ((c * c) / (-1.40625d0))) + ((-0.75d0) / (b / ((c / ((t_0 / c) / (-0.375d0))) / b)))) + ((c * c) / ((b * (b * t_0)) / 0.5625d0)))))) + ((-0.5d0) / b))))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return -0.3333333333333333 / ((0.6666666666666666 / (c / b)) + (a * ((a * (((c / ((b * b) / -0.375)) / b) + (a * (((((c * (c * (c * c))) / (Math.pow(b, 6.0) / b)) / ((c * c) / -1.40625)) + (-0.75 / (b / ((c / ((t_0 / c) / -0.375)) / b)))) + ((c * c) / ((b * (b * t_0)) / 0.5625)))))) + (-0.5 / b))));
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	return -0.3333333333333333 / ((0.6666666666666666 / (c / b)) + (a * ((a * (((c / ((b * b) / -0.375)) / b) + (a * (((((c * (c * (c * c))) / (math.pow(b, 6.0) / b)) / ((c * c) / -1.40625)) + (-0.75 / (b / ((c / ((t_0 / c) / -0.375)) / b)))) + ((c * c) / ((b * (b * t_0)) / 0.5625)))))) + (-0.5 / b))))

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(-0.3333333333333333 / Float64(Float64(0.6666666666666666 / Float64(c / b)) + Float64(a * Float64(Float64(a * Float64(Float64(Float64(c / Float64(Float64(b * b) / -0.375)) / b) + Float64(a * Float64(Float64(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) / Float64((b ^ 6.0) / b)) / Float64(Float64(c * c) / -1.40625)) + Float64(-0.75 / Float64(b / Float64(Float64(c / Float64(Float64(t_0 / c) / -0.375)) / b)))) + Float64(Float64(c * c) / Float64(Float64(b * Float64(b * t_0)) / 0.5625)))))) + Float64(-0.5 / b)))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	tmp = -0.3333333333333333 / ((0.6666666666666666 / (c / b)) + (a * ((a * (((c / ((b * b) / -0.375)) / b) + (a * (((((c * (c * (c * c))) / ((b ^ 6.0) / b)) / ((c * c) / -1.40625)) + (-0.75 / (b / ((c / ((t_0 / c) / -0.375)) / b)))) + ((c * c) / ((b * (b * t_0)) / 0.5625)))))) + (-0.5 / b))));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(-0.3333333333333333 / N[(N[(0.6666666666666666 / N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(a * N[(N[(N[(c / N[(N[(b * b), $MachinePrecision] / -0.375), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 6.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] / -1.40625), $MachinePrecision]), $MachinePrecision] + N[(-0.75 / N[(b / N[(N[(c / N[(N[(t$95$0 / c), $MachinePrecision] / -0.375), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] / N[(N[(b * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] / 0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 51.6%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 51.6%

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

3. Simplified 51.6%

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

   1. remove-double-neg N/A

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

   2. sub-div N/A

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

   3. remove-double-neg N/A

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

   4. div-sub N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r* N/A

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

   8. associate-/r* N/A

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

   9. frac-2neg N/A

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

   10. associate-/l/ N/A

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

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

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

6. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(0 - a\right)}} \]

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{2}{3} \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \left(\frac{3}{8} \cdot \frac{c}{{b}^{3}} + a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{9} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{9}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right) \]

9. Simplified 93.4%

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

11. Applied egg-rr 93.6%

    \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\mathsf{fma}\left(0.6666666666666666, \frac{b}{c}, a \cdot \left(a \cdot \left(\frac{c \cdot -0.375}{b \cdot \left(b \cdot b\right)} + a \cdot \left(\frac{\frac{b}{\frac{{b}^{6}}{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}} \cdot -1.40625}{c \cdot c} + \left(\frac{\left(c \cdot c\right) \cdot 0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + -0.75 \cdot \frac{\frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}}{b \cdot b}\right)\right)\right) + \frac{-0.5}{b}\right)\right)}} \]

13. Applied egg-rr 93.6%

    \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\frac{0.6666666666666666}{\frac{c}{b}} + a \cdot \left(a \cdot \left(\frac{\frac{c}{\frac{b \cdot b}{-0.375}}}{b} + a \cdot \left(\left(\frac{\frac{c \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\frac{{b}^{6}}{b}}}{\frac{c \cdot c}{-1.40625}} + \frac{-0.75}{\frac{b}{\frac{\frac{c}{\frac{\frac{b \cdot \left(b \cdot b\right)}{c}}{-0.375}}}{b}}}\right) + \frac{c \cdot c}{\frac{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{0.5625}}\right)\right) + \frac{-0.5}{b}\right)}} \]
14. Add Preprocessing

Alternative 2: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -110:\\ \;\;\;\;\frac{\frac{\frac{a}{0.3333333333333333} \cdot \left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b\right)}{a \cdot 9}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\mathsf{fma}\left(0.6666666666666666, \frac{b}{c}, a \cdot \frac{-0.5 + \frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b}}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -110.0)
   (/
    (/
     (* (/ a 0.3333333333333333) (- (sqrt (+ (* b b) (* c (* a -3.0)))) b))
     (* a 9.0))
    a)
   (/
    -0.3333333333333333
    (fma
     0.6666666666666666
     (/ b c)
     (* a (/ (+ -0.5 (/ (* -0.375 (* c a)) (* b b))) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -110.0) {
		tmp = (((a / 0.3333333333333333) * (sqrt(((b * b) + (c * (a * -3.0)))) - b)) / (a * 9.0)) / a;
	} else {
		tmp = -0.3333333333333333 / fma(0.6666666666666666, (b / c), (a * ((-0.5 + ((-0.375 * (c * a)) / (b * b))) / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -110.0)
		tmp = Float64(Float64(Float64(Float64(a / 0.3333333333333333) * Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))) - b)) / Float64(a * 9.0)) / a);
	else
		tmp = Float64(-0.3333333333333333 / fma(0.6666666666666666, Float64(b / c), Float64(a * Float64(Float64(-0.5 + Float64(Float64(-0.375 * Float64(c * a)) / Float64(b * b))) / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -110.0], N[(N[(N[(N[(a / 0.3333333333333333), $MachinePrecision] * N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / N[(a * 9.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-0.3333333333333333 / N[(0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(a * N[(N[(-0.5 + N[(N[(-0.375 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $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)) < -110

   1. Initial program 87.2%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 87.2%

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

   3. Simplified 87.2%

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

      1. remove-double-neg N/A

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

      2. sub-div N/A

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

      3. remove-double-neg N/A

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

      4. frac-sub N/A

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

      5. associate-*r* N/A

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

      6. associate-/r* N/A

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

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

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

   6. Applied egg-rr 87.3%

      \[\leadsto \color{blue}{\frac{\frac{\frac{a}{0.3333333333333333} \cdot \left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b\right)}{a \cdot 9}}{a}} \]

   if -110 < (/.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 49.0%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 49.0%

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

   3. Simplified 49.0%

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

      1. remove-double-neg N/A

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

      2. sub-div N/A

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

      3. remove-double-neg N/A

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

      4. div-sub N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. associate-/r* N/A

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

      9. frac-2neg N/A

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

      10. associate-/l/ N/A

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

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

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

   6. Applied egg-rr 49.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(0 - a\right)}} \]

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{2}{3} \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \left(\frac{3}{8} \cdot \frac{c}{{b}^{3}} + a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{9} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{9}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right) \]

   9. Simplified 95.1%

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

   11. Applied egg-rr 95.3%

       \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\mathsf{fma}\left(0.6666666666666666, \frac{b}{c}, a \cdot \left(a \cdot \left(\frac{c \cdot -0.375}{b \cdot \left(b \cdot b\right)} + a \cdot \left(\frac{\frac{b}{\frac{{b}^{6}}{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}} \cdot -1.40625}{c \cdot c} + \left(\frac{\left(c \cdot c\right) \cdot 0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + -0.75 \cdot \frac{\frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}}{b \cdot b}\right)\right)\right) + \frac{-0.5}{b}\right)\right)}} \]

   12. Taylor expanded in b around inf

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

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

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

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

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

       5. associate-*r/ N/A

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

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

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

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

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 92.7%

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

   14. Simplified 92.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -110:\\ \;\;\;\;\frac{\frac{\frac{a}{0.3333333333333333} \cdot \left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b\right)}{a \cdot 9}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\mathsf{fma}\left(0.6666666666666666, \frac{b}{c}, a \cdot \frac{-0.5 + \frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b}}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -110:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-3}{b - \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\mathsf{fma}\left(0.6666666666666666, \frac{b}{c}, a \cdot \frac{-0.5 + \frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b}}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -110.0)
   (/ (/ 1.0 a) (/ -3.0 (- b (sqrt (+ (* b b) (* c (* a -3.0)))))))
   (/
    -0.3333333333333333
    (fma
     0.6666666666666666
     (/ b c)
     (* a (/ (+ -0.5 (/ (* -0.375 (* c a)) (* b b))) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -110.0) {
		tmp = (1.0 / a) / (-3.0 / (b - sqrt(((b * b) + (c * (a * -3.0))))));
	} else {
		tmp = -0.3333333333333333 / fma(0.6666666666666666, (b / c), (a * ((-0.5 + ((-0.375 * (c * a)) / (b * b))) / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -110.0)
		tmp = Float64(Float64(1.0 / a) / Float64(-3.0 / Float64(b - sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))))));
	else
		tmp = Float64(-0.3333333333333333 / fma(0.6666666666666666, Float64(b / c), Float64(a * Float64(Float64(-0.5 + Float64(Float64(-0.375 * Float64(c * a)) / Float64(b * b))) / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -110.0], N[(N[(1.0 / a), $MachinePrecision] / N[(-3.0 / N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(0.6666666666666666 * N[(b / c), $MachinePrecision] + N[(a * N[(N[(-0.5 + N[(N[(-0.375 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $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)) < -110

   1. Initial program 87.2%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 87.2%

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

   3. Simplified 87.2%

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

      1. remove-double-neg N/A

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

      2. sub-div N/A

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

      3. remove-double-neg N/A

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

      4. div-sub N/A

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

      5. associate-/r* N/A

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

      6. clear-num N/A

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

      7. associate-/r* N/A

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

      8. associate-/l/ N/A

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

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

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

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

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

      11. frac-2neg N/A

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

      12. metadata-eval N/A

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

   6. Applied egg-rr 87.3%

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

   if -110 < (/.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 49.0%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 49.0%

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

   3. Simplified 49.0%

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

      1. remove-double-neg N/A

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

      2. sub-div N/A

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

      3. remove-double-neg N/A

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

      4. div-sub N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. associate-/r* N/A

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

      9. frac-2neg N/A

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

      10. associate-/l/ N/A

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

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

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

   6. Applied egg-rr 49.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(0 - a\right)}} \]

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{2}{3} \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \left(\frac{3}{8} \cdot \frac{c}{{b}^{3}} + a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{9} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{9}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right) \]

   9. Simplified 95.1%

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

   11. Applied egg-rr 95.3%

       \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\mathsf{fma}\left(0.6666666666666666, \frac{b}{c}, a \cdot \left(a \cdot \left(\frac{c \cdot -0.375}{b \cdot \left(b \cdot b\right)} + a \cdot \left(\frac{\frac{b}{\frac{{b}^{6}}{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}} \cdot -1.40625}{c \cdot c} + \left(\frac{\left(c \cdot c\right) \cdot 0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + -0.75 \cdot \frac{\frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}}{b \cdot b}\right)\right)\right) + \frac{-0.5}{b}\right)\right)}} \]

   12. Taylor expanded in b around inf

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

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

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

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

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

       5. associate-*r/ N/A

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

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

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

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

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 92.7%

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

   14. Simplified 92.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -110:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-3}{b - \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\mathsf{fma}\left(0.6666666666666666, \frac{b}{c}, a \cdot \frac{-0.5 + \frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b}}{b}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -110:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-3}{b - \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{0.6666666666666666 \cdot b + c \cdot \left(\frac{a \cdot -0.5}{b} + c \cdot \left(-0.375 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}\right)\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -110.0)
   (/ (/ 1.0 a) (/ -3.0 (- b (sqrt (+ (* b b) (* c (* a -3.0)))))))
   (/
    -0.3333333333333333
    (/
     (+
      (* 0.6666666666666666 b)
      (* c (+ (/ (* a -0.5) b) (* c (* -0.375 (/ (* a a) (* b (* b b))))))))
     c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -110.0) {
		tmp = (1.0 / a) / (-3.0 / (b - sqrt(((b * b) + (c * (a * -3.0))))));
	} else {
		tmp = -0.3333333333333333 / (((0.6666666666666666 * b) + (c * (((a * -0.5) / b) + (c * (-0.375 * ((a * a) / (b * (b * b)))))))) / c);
	}
	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) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-110.0d0)) then
        tmp = (1.0d0 / a) / ((-3.0d0) / (b - sqrt(((b * b) + (c * (a * (-3.0d0)))))))
    else
        tmp = (-0.3333333333333333d0) / (((0.6666666666666666d0 * b) + (c * (((a * (-0.5d0)) / b) + (c * ((-0.375d0) * ((a * a) / (b * (b * b)))))))) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -110.0) {
		tmp = (1.0 / a) / (-3.0 / (b - Math.sqrt(((b * b) + (c * (a * -3.0))))));
	} else {
		tmp = -0.3333333333333333 / (((0.6666666666666666 * b) + (c * (((a * -0.5) / b) + (c * (-0.375 * ((a * a) / (b * (b * b)))))))) / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -110.0:
		tmp = (1.0 / a) / (-3.0 / (b - math.sqrt(((b * b) + (c * (a * -3.0))))))
	else:
		tmp = -0.3333333333333333 / (((0.6666666666666666 * b) + (c * (((a * -0.5) / b) + (c * (-0.375 * ((a * a) / (b * (b * b)))))))) / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -110.0)
		tmp = Float64(Float64(1.0 / a) / Float64(-3.0 / Float64(b - sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))))));
	else
		tmp = Float64(-0.3333333333333333 / Float64(Float64(Float64(0.6666666666666666 * b) + Float64(c * Float64(Float64(Float64(a * -0.5) / b) + Float64(c * Float64(-0.375 * Float64(Float64(a * a) / Float64(b * Float64(b * b)))))))) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -110.0)
		tmp = (1.0 / a) / (-3.0 / (b - sqrt(((b * b) + (c * (a * -3.0))))));
	else
		tmp = -0.3333333333333333 / (((0.6666666666666666 * b) + (c * (((a * -0.5) / b) + (c * (-0.375 * ((a * a) / (b * (b * b)))))))) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := 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], -110.0], N[(N[(1.0 / a), $MachinePrecision] / N[(-3.0 / N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(N[(N[(0.6666666666666666 * b), $MachinePrecision] + N[(c * N[(N[(N[(a * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(c * N[(-0.375 * N[(N[(a * a), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -110

   1. Initial program 87.2%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 87.2%

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

   3. Simplified 87.2%

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

      1. remove-double-neg N/A

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

      2. sub-div N/A

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

      3. remove-double-neg N/A

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

      4. div-sub N/A

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

      5. associate-/r* N/A

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

      6. clear-num N/A

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

      7. associate-/r* N/A

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

      8. associate-/l/ N/A

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

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

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

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

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

      11. frac-2neg N/A

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

      12. metadata-eval N/A

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

   6. Applied egg-rr 87.3%

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

   if -110 < (/.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 49.0%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 49.0%

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

   3. Simplified 49.0%

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

      1. remove-double-neg N/A

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

      2. sub-div N/A

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

      3. remove-double-neg N/A

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

      4. div-sub N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. associate-/r* N/A

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

      9. frac-2neg N/A

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

      10. associate-/l/ N/A

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

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

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

   6. Applied egg-rr 49.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(0 - a\right)}} \]

   7. Taylor expanded in c around 0

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{2}{3} \cdot b + c \cdot \left(\frac{-1}{2} \cdot \frac{a}{b} + c \cdot \left(\frac{-3}{4} \cdot \frac{{a}^{2}}{{b}^{3}} + \frac{3}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)}{c}\right)}\right) \]
   8. Step-by-step derivation

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

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

   9. Simplified 92.6%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -110:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{-3}{b - \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{0.6666666666666666 \cdot b + c \cdot \left(\frac{a \cdot -0.5}{b} + c \cdot \left(-0.375 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}\right)\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -110:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{0.6666666666666666 \cdot b + c \cdot \left(\frac{a \cdot -0.5}{b} + c \cdot \left(-0.375 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}\right)\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -110.0)
   (/ (- (sqrt (+ (* b b) (* -3.0 (* c a)))) b) (* a 3.0))
   (/
    -0.3333333333333333
    (/
     (+
      (* 0.6666666666666666 b)
      (* c (+ (/ (* a -0.5) b) (* c (* -0.375 (/ (* a a) (* b (* b b))))))))
     c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -110.0) {
		tmp = (sqrt(((b * b) + (-3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = -0.3333333333333333 / (((0.6666666666666666 * b) + (c * (((a * -0.5) / b) + (c * (-0.375 * ((a * a) / (b * (b * b)))))))) / c);
	}
	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) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-110.0d0)) then
        tmp = (sqrt(((b * b) + ((-3.0d0) * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = (-0.3333333333333333d0) / (((0.6666666666666666d0 * b) + (c * (((a * (-0.5d0)) / b) + (c * ((-0.375d0) * ((a * a) / (b * (b * b)))))))) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -110.0) {
		tmp = (Math.sqrt(((b * b) + (-3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = -0.3333333333333333 / (((0.6666666666666666 * b) + (c * (((a * -0.5) / b) + (c * (-0.375 * ((a * a) / (b * (b * b)))))))) / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -110.0:
		tmp = (math.sqrt(((b * b) + (-3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = -0.3333333333333333 / (((0.6666666666666666 * b) + (c * (((a * -0.5) / b) + (c * (-0.375 * ((a * a) / (b * (b * b)))))))) / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -110.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(-3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.3333333333333333 / Float64(Float64(Float64(0.6666666666666666 * b) + Float64(c * Float64(Float64(Float64(a * -0.5) / b) + Float64(c * Float64(-0.375 * Float64(Float64(a * a) / Float64(b * Float64(b * b)))))))) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -110.0)
		tmp = (sqrt(((b * b) + (-3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = -0.3333333333333333 / (((0.6666666666666666 * b) + (c * (((a * -0.5) / b) + (c * (-0.375 * ((a * a) / (b * (b * b)))))))) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := 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], -110.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 / N[(N[(N[(0.6666666666666666 * b), $MachinePrecision] + N[(c * N[(N[(N[(a * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(c * N[(-0.375 * N[(N[(a * a), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -110

   1. Initial program 87.2%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 87.2%

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

   3. Simplified 87.2%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 87.3%

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

   6. Applied egg-rr 87.3%

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

   if -110 < (/.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 49.0%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. sub-neg N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. metadata-eval N/A

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

      16. *-lowering-*.f64 49.0%

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

   3. Simplified 49.0%

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

      1. remove-double-neg N/A

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

      2. sub-div N/A

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

      3. remove-double-neg N/A

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

      4. div-sub N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. associate-/r* N/A

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

      9. frac-2neg N/A

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

      10. associate-/l/ N/A

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

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

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

   6. Applied egg-rr 49.0%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(0 - a\right)}} \]

   7. Taylor expanded in c around 0

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{2}{3} \cdot b + c \cdot \left(\frac{-1}{2} \cdot \frac{a}{b} + c \cdot \left(\frac{-3}{4} \cdot \frac{{a}^{2}}{{b}^{3}} + \frac{3}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)}{c}\right)}\right) \]
   8. Step-by-step derivation

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

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

   9. Simplified 92.6%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -110:\\ \;\;\;\;\frac{\sqrt{b \cdot b + -3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{0.6666666666666666 \cdot b + c \cdot \left(\frac{a \cdot -0.5}{b} + c \cdot \left(-0.375 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}\right)\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{\frac{c \cdot \left(\frac{a \cdot -0.5}{b} + c \cdot \left(-0.5625 \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \frac{c}{{b}^{5}}\right) + \frac{-0.375 \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right)\right) + 0.6666666666666666 \cdot b}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  -0.3333333333333333
  (/
   (+
    (*
     c
     (+
      (/ (* a -0.5) b)
      (*
       c
       (+
        (* -0.5625 (* (* a (* a a)) (/ c (pow b 5.0))))
        (/ (* -0.375 (* a a)) (* b (* b b)))))))
    (* 0.6666666666666666 b))
   c)))

C

double code(double a, double b, double c) {
	return -0.3333333333333333 / (((c * (((a * -0.5) / b) + (c * ((-0.5625 * ((a * (a * a)) * (c / pow(b, 5.0)))) + ((-0.375 * (a * a)) / (b * (b * b))))))) + (0.6666666666666666 * 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 = (-0.3333333333333333d0) / (((c * (((a * (-0.5d0)) / b) + (c * (((-0.5625d0) * ((a * (a * a)) * (c / (b ** 5.0d0)))) + (((-0.375d0) * (a * a)) / (b * (b * b))))))) + (0.6666666666666666d0 * b)) / c)
end function

Java

public static double code(double a, double b, double c) {
	return -0.3333333333333333 / (((c * (((a * -0.5) / b) + (c * ((-0.5625 * ((a * (a * a)) * (c / Math.pow(b, 5.0)))) + ((-0.375 * (a * a)) / (b * (b * b))))))) + (0.6666666666666666 * b)) / c);
}

Python

def code(a, b, c):
	return -0.3333333333333333 / (((c * (((a * -0.5) / b) + (c * ((-0.5625 * ((a * (a * a)) * (c / math.pow(b, 5.0)))) + ((-0.375 * (a * a)) / (b * (b * b))))))) + (0.6666666666666666 * b)) / c)

Julia

function code(a, b, c)
	return Float64(-0.3333333333333333 / Float64(Float64(Float64(c * Float64(Float64(Float64(a * -0.5) / b) + Float64(c * Float64(Float64(-0.5625 * Float64(Float64(a * Float64(a * a)) * Float64(c / (b ^ 5.0)))) + Float64(Float64(-0.375 * Float64(a * a)) / Float64(b * Float64(b * b))))))) + Float64(0.6666666666666666 * b)) / c))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.3333333333333333 / (((c * (((a * -0.5) / b) + (c * ((-0.5625 * ((a * (a * a)) * (c / (b ^ 5.0)))) + ((-0.375 * (a * a)) / (b * (b * b))))))) + (0.6666666666666666 * b)) / c);
end

Wolfram

code[a_, b_, c_] := N[(-0.3333333333333333 / N[(N[(N[(c * N[(N[(N[(a * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(c * N[(N[(-0.5625 * N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(c / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 51.6%

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

3. Simplified 51.6%

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

   1. remove-double-neg N/A

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

   2. sub-div N/A

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

   3. remove-double-neg N/A

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

   4. div-sub N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r* N/A

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

   8. associate-/r* N/A

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

   9. frac-2neg N/A

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

   10. associate-/l/ N/A

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

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

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

6. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(0 - a\right)}} \]

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{2}{3} \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \left(\frac{3}{8} \cdot \frac{c}{{b}^{3}} + a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{9} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{9}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right) \]

9. Simplified 93.4%

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

10. Taylor expanded in c around 0

    \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{2}{3} \cdot b + c \cdot \left(\frac{-1}{2} \cdot \frac{a}{b} + c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{3} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)}{c}\right)}\right) \]
11. Step-by-step derivation

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

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

12. Simplified 93.5%

    \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\frac{0.6666666666666666 \cdot b + c \cdot \left(\frac{-0.5 \cdot a}{b} + c \cdot \left(-0.5625 \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \frac{c}{{b}^{5}}\right) + \frac{-0.375 \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right)\right)}{c}}} \]

13. Final simplification 93.5%

    \[\leadsto \frac{-0.3333333333333333}{\frac{c \cdot \left(\frac{a \cdot -0.5}{b} + c \cdot \left(-0.5625 \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot \frac{c}{{b}^{5}}\right) + \frac{-0.375 \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right)\right) + 0.6666666666666666 \cdot b}{c}} \]
14. Add Preprocessing

Alternative 7: 91.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{\frac{0.6666666666666666 \cdot b}{c} + a \cdot \left(c \cdot \left(\frac{-0.5625 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{5}} + -0.375 \cdot \frac{a}{b \cdot \left(b \cdot b\right)}\right) - \frac{0.5}{b}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  -0.3333333333333333
  (+
   (/ (* 0.6666666666666666 b) c)
   (*
    a
    (-
     (*
      c
      (+
       (/ (* -0.5625 (* c (* a a))) (pow b 5.0))
       (* -0.375 (/ a (* b (* b b))))))
     (/ 0.5 b))))))

C

double code(double a, double b, double c) {
	return -0.3333333333333333 / (((0.6666666666666666 * b) / c) + (a * ((c * (((-0.5625 * (c * (a * a))) / pow(b, 5.0)) + (-0.375 * (a / (b * (b * b)))))) - (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 = (-0.3333333333333333d0) / (((0.6666666666666666d0 * b) / c) + (a * ((c * ((((-0.5625d0) * (c * (a * a))) / (b ** 5.0d0)) + ((-0.375d0) * (a / (b * (b * b)))))) - (0.5d0 / b))))
end function

Java

public static double code(double a, double b, double c) {
	return -0.3333333333333333 / (((0.6666666666666666 * b) / c) + (a * ((c * (((-0.5625 * (c * (a * a))) / Math.pow(b, 5.0)) + (-0.375 * (a / (b * (b * b)))))) - (0.5 / b))));
}

Python

def code(a, b, c):
	return -0.3333333333333333 / (((0.6666666666666666 * b) / c) + (a * ((c * (((-0.5625 * (c * (a * a))) / math.pow(b, 5.0)) + (-0.375 * (a / (b * (b * b)))))) - (0.5 / b))))

Julia

function code(a, b, c)
	return Float64(-0.3333333333333333 / Float64(Float64(Float64(0.6666666666666666 * b) / c) + Float64(a * Float64(Float64(c * Float64(Float64(Float64(-0.5625 * Float64(c * Float64(a * a))) / (b ^ 5.0)) + Float64(-0.375 * Float64(a / Float64(b * Float64(b * b)))))) - Float64(0.5 / b)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.3333333333333333 / (((0.6666666666666666 * b) / c) + (a * ((c * (((-0.5625 * (c * (a * a))) / (b ^ 5.0)) + (-0.375 * (a / (b * (b * b)))))) - (0.5 / b))));
end

Wolfram

code[a_, b_, c_] := N[(-0.3333333333333333 / N[(N[(N[(0.6666666666666666 * b), $MachinePrecision] / c), $MachinePrecision] + N[(a * N[(N[(c * N[(N[(N[(-0.5625 * N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 51.6%

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

3. Simplified 51.6%

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

   1. remove-double-neg N/A

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

   2. sub-div N/A

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

   3. remove-double-neg N/A

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

   4. div-sub N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r* N/A

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

   8. associate-/r* N/A

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

   9. frac-2neg N/A

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

   10. associate-/l/ N/A

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

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

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

6. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(0 - a\right)}} \]

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{2}{3} \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \left(\frac{3}{8} \cdot \frac{c}{{b}^{3}} + a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{9} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{9}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right) \]

9. Simplified 93.4%

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

10. Taylor expanded in c around 0

    \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \frac{2}{3}\right), c\right), \mathsf{*.f64}\left(a, \color{blue}{\left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)}\right)\right)\right) \]
11. Step-by-step derivation

    1. metadata-eval N/A

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

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

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

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

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

12. Simplified 93.4%

    \[\leadsto \frac{-0.3333333333333333}{\frac{b \cdot 0.6666666666666666}{c} + a \cdot \color{blue}{\left(c \cdot \left(\frac{-0.5625 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{5}} + -0.375 \cdot \frac{a}{b \cdot \left(b \cdot b\right)}\right) - \frac{0.5}{b}\right)}} \]

13. Final simplification 93.4%

    \[\leadsto \frac{-0.3333333333333333}{\frac{0.6666666666666666 \cdot b}{c} + a \cdot \left(c \cdot \left(\frac{-0.5625 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{5}} + -0.375 \cdot \frac{a}{b \cdot \left(b \cdot b\right)}\right) - \frac{0.5}{b}\right)} \]
14. Add Preprocessing

Alternative 8: 88.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{\frac{0.6666666666666666 \cdot b + c \cdot \left(\frac{a \cdot -0.5}{b} + c \cdot \left(-0.375 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}\right)\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  -0.3333333333333333
  (/
   (+
    (* 0.6666666666666666 b)
    (* c (+ (/ (* a -0.5) b) (* c (* -0.375 (/ (* a a) (* b (* b b))))))))
   c)))

C

double code(double a, double b, double c) {
	return -0.3333333333333333 / (((0.6666666666666666 * b) + (c * (((a * -0.5) / b) + (c * (-0.375 * ((a * a) / (b * (b * 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 = (-0.3333333333333333d0) / (((0.6666666666666666d0 * b) + (c * (((a * (-0.5d0)) / b) + (c * ((-0.375d0) * ((a * a) / (b * (b * b)))))))) / c)
end function

Java

public static double code(double a, double b, double c) {
	return -0.3333333333333333 / (((0.6666666666666666 * b) + (c * (((a * -0.5) / b) + (c * (-0.375 * ((a * a) / (b * (b * b)))))))) / c);
}

Python

def code(a, b, c):
	return -0.3333333333333333 / (((0.6666666666666666 * b) + (c * (((a * -0.5) / b) + (c * (-0.375 * ((a * a) / (b * (b * b)))))))) / c)

Julia

function code(a, b, c)
	return Float64(-0.3333333333333333 / Float64(Float64(Float64(0.6666666666666666 * b) + Float64(c * Float64(Float64(Float64(a * -0.5) / b) + Float64(c * Float64(-0.375 * Float64(Float64(a * a) / Float64(b * Float64(b * b)))))))) / c))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.3333333333333333 / (((0.6666666666666666 * b) + (c * (((a * -0.5) / b) + (c * (-0.375 * ((a * a) / (b * (b * b)))))))) / c);
end

Wolfram

code[a_, b_, c_] := N[(-0.3333333333333333 / N[(N[(N[(0.6666666666666666 * b), $MachinePrecision] + N[(c * N[(N[(N[(a * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(c * N[(-0.375 * N[(N[(a * a), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 51.6%

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

3. Simplified 51.6%

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

   1. remove-double-neg N/A

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

   2. sub-div N/A

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

   3. remove-double-neg N/A

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

   4. div-sub N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r* N/A

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

   8. associate-/r* N/A

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

   9. frac-2neg N/A

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

   10. associate-/l/ N/A

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

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

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

6. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(0 - a\right)}} \]

7. Taylor expanded in c around 0

   \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{\frac{2}{3} \cdot b + c \cdot \left(\frac{-1}{2} \cdot \frac{a}{b} + c \cdot \left(\frac{-3}{4} \cdot \frac{{a}^{2}}{{b}^{3}} + \frac{3}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)}{c}\right)}\right) \]
8. Step-by-step derivation

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

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

9. Simplified 90.6%

   \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\frac{b \cdot 0.6666666666666666 + c \cdot \left(\frac{-0.5 \cdot a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.375\right)\right)}{c}}} \]

10. Final simplification 90.6%

    \[\leadsto \frac{-0.3333333333333333}{\frac{0.6666666666666666 \cdot b + c \cdot \left(\frac{a \cdot -0.5}{b} + c \cdot \left(-0.375 \cdot \frac{a \cdot a}{b \cdot \left(b \cdot b\right)}\right)\right)}{c}} \]
11. Add Preprocessing

Alternative 9: 88.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\frac{b}{c} \cdot -0.6666666666666666 + a \cdot \left(\frac{0.5}{b} - a \cdot \left(-0.375 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}\right)\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  0.3333333333333333
  (+
   (* (/ b c) -0.6666666666666666)
   (* a (- (/ 0.5 b) (* a (* -0.375 (/ c (* b (* b b))))))))))

C

double code(double a, double b, double c) {
	return 0.3333333333333333 / (((b / c) * -0.6666666666666666) + (a * ((0.5 / b) - (a * (-0.375 * (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
    code = 0.3333333333333333d0 / (((b / c) * (-0.6666666666666666d0)) + (a * ((0.5d0 / b) - (a * ((-0.375d0) * (c / (b * (b * b))))))))
end function

Java

public static double code(double a, double b, double c) {
	return 0.3333333333333333 / (((b / c) * -0.6666666666666666) + (a * ((0.5 / b) - (a * (-0.375 * (c / (b * (b * b))))))));
}

Python

def code(a, b, c):
	return 0.3333333333333333 / (((b / c) * -0.6666666666666666) + (a * ((0.5 / b) - (a * (-0.375 * (c / (b * (b * b))))))))

Julia

function code(a, b, c)
	return Float64(0.3333333333333333 / Float64(Float64(Float64(b / c) * -0.6666666666666666) + Float64(a * Float64(Float64(0.5 / b) - Float64(a * Float64(-0.375 * Float64(c / Float64(b * Float64(b * b)))))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.3333333333333333 / (((b / c) * -0.6666666666666666) + (a * ((0.5 / b) - (a * (-0.375 * (c / (b * (b * b))))))));
end

Wolfram

code[a_, b_, c_] := N[(0.3333333333333333 / N[(N[(N[(b / c), $MachinePrecision] * -0.6666666666666666), $MachinePrecision] + N[(a * N[(N[(0.5 / b), $MachinePrecision] - N[(a * N[(-0.375 * N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 51.6%

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

3. Simplified 51.6%

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

   1. remove-double-neg N/A

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

   2. sub-div N/A

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

   3. remove-double-neg N/A

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

   4. div-sub N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r* N/A

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

   8. associate-/r* N/A

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

   9. frac-2neg N/A

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

   10. associate-/l/ N/A

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

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

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

6. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(0 - a\right)}} \]
7. Step-by-step derivation

   1. sub0-neg N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. frac-times N/A

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

   5. metadata-eval N/A

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

   6. frac-2neg N/A

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

   7. frac-times N/A

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

   8. metadata-eval N/A

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

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

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

8. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{a \cdot \frac{1}{\sqrt{b \cdot b + c \cdot \frac{a}{-0.3333333333333333}} - b}}} \]

9. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\frac{-2}{3} \cdot \frac{b}{c} + a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right) \]
10. Step-by-step derivation

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

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

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

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

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

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{/.f64}\left(b, c\right)\right), \left(a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)\right) \]

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

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

    5. +-commutative N/A

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

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

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

    7. associate-*r/ N/A

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

    8. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-2}{3}, \mathsf{/.f64}\left(b, c\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{b}\right), \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right)\right)\right)\right)\right)\right) \]

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

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

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

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

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

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

    12. distribute-rgt-out N/A

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

    13. metadata-eval N/A

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

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

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

11. Simplified 90.6%

    \[\leadsto \frac{0.3333333333333333}{\color{blue}{-0.6666666666666666 \cdot \frac{b}{c} + a \cdot \left(\frac{0.5}{b} + -1 \cdot \left(a \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375\right)\right)\right)}} \]

12. Final simplification 90.6%

    \[\leadsto \frac{0.3333333333333333}{\frac{b}{c} \cdot -0.6666666666666666 + a \cdot \left(\frac{0.5}{b} - a \cdot \left(-0.375 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}\right)\right)} \]
13. Add Preprocessing

Alternative 10: 88.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{\frac{0.6666666666666666 \cdot b}{c} + a \cdot \frac{-0.5 + \frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b}}{b}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  -0.3333333333333333
  (+
   (/ (* 0.6666666666666666 b) c)
   (* a (/ (+ -0.5 (/ (* -0.375 (* c a)) (* b b))) b)))))

C

double code(double a, double b, double c) {
	return -0.3333333333333333 / (((0.6666666666666666 * b) / c) + (a * ((-0.5 + ((-0.375 * (c * a)) / (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 = (-0.3333333333333333d0) / (((0.6666666666666666d0 * b) / c) + (a * (((-0.5d0) + (((-0.375d0) * (c * a)) / (b * b))) / b)))
end function

Java

public static double code(double a, double b, double c) {
	return -0.3333333333333333 / (((0.6666666666666666 * b) / c) + (a * ((-0.5 + ((-0.375 * (c * a)) / (b * b))) / b)));
}

Python

def code(a, b, c):
	return -0.3333333333333333 / (((0.6666666666666666 * b) / c) + (a * ((-0.5 + ((-0.375 * (c * a)) / (b * b))) / b)))

Julia

function code(a, b, c)
	return Float64(-0.3333333333333333 / Float64(Float64(Float64(0.6666666666666666 * b) / c) + Float64(a * Float64(Float64(-0.5 + Float64(Float64(-0.375 * Float64(c * a)) / Float64(b * b))) / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.3333333333333333 / (((0.6666666666666666 * b) / c) + (a * ((-0.5 + ((-0.375 * (c * a)) / (b * b))) / b)));
end

Wolfram

code[a_, b_, c_] := N[(-0.3333333333333333 / N[(N[(N[(0.6666666666666666 * b), $MachinePrecision] / c), $MachinePrecision] + N[(a * N[(N[(-0.5 + N[(N[(-0.375 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 51.6%

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

3. Simplified 51.6%

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

   1. remove-double-neg N/A

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

   2. sub-div N/A

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

   3. remove-double-neg N/A

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

   4. div-sub N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r* N/A

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

   8. associate-/r* N/A

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

   9. frac-2neg N/A

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

   10. associate-/l/ N/A

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

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

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

6. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(0 - a\right)}} \]

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{2}{3} \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \left(\frac{3}{8} \cdot \frac{c}{{b}^{3}} + a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{9} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{9}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right)\right) - \frac{1}{2} \cdot \frac{1}{b}\right)\right)}\right) \]

9. Simplified 93.4%

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

10. Taylor expanded in b around inf

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

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

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

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

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

    5. associate-*r/ N/A

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

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

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

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

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

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

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

    9. unpow2 N/A

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

    10. *-lowering-*.f64 90.5%

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

12. Simplified 90.5%

    \[\leadsto \frac{-0.3333333333333333}{\frac{b \cdot 0.6666666666666666}{c} + a \cdot \color{blue}{\frac{\frac{-0.375 \cdot \left(a \cdot c\right)}{b \cdot b} + -0.5}{b}}} \]

13. Final simplification 90.5%

    \[\leadsto \frac{-0.3333333333333333}{\frac{0.6666666666666666 \cdot b}{c} + a \cdot \frac{-0.5 + \frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b}}{b}} \]
14. Add Preprocessing

Alternative 11: 82.3% accurate, 7.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return -0.3333333333333333 / (b * ((-0.5 * (a / (b * b))) + (0.6666666666666666 / 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 = (-0.3333333333333333d0) / (b * (((-0.5d0) * (a / (b * b))) + (0.6666666666666666d0 / c)))
end function

Java

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

Python

def code(a, b, c):
	return -0.3333333333333333 / (b * ((-0.5 * (a / (b * b))) + (0.6666666666666666 / c)))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = -0.3333333333333333 / (b * ((-0.5 * (a / (b * b))) + (0.6666666666666666 / c)));
end

Wolfram

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

Derivation

1. Initial program 51.6%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 51.6%

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

3. Simplified 51.6%

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

   1. remove-double-neg N/A

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

   2. sub-div N/A

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

   3. remove-double-neg N/A

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

   4. div-sub N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r* N/A

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

   8. associate-/r* N/A

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

   9. frac-2neg N/A

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

   10. associate-/l/ N/A

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

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

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

6. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(0 - a\right)}} \]

7. Taylor expanded in b around inf

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

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

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

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

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

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

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

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

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

   5. unpow2 N/A

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

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

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. /-lowering-/.f64 85.0%

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

9. Simplified 85.0%

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

Alternative 12: 82.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{\frac{b}{c} \cdot -0.6666666666666666 + \frac{a \cdot 0.5}{b}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 0.3333333333333333 / (((b / c) * -0.6666666666666666) + ((a * 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 = 0.3333333333333333d0 / (((b / c) * (-0.6666666666666666d0)) + ((a * 0.5d0) / b))
end function

Java

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

Python

def code(a, b, c):
	return 0.3333333333333333 / (((b / c) * -0.6666666666666666) + ((a * 0.5) / b))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = 0.3333333333333333 / (((b / c) * -0.6666666666666666) + ((a * 0.5) / b));
end

Wolfram

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

Derivation

1. Initial program 51.6%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 51.6%

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

3. Simplified 51.6%

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

   1. remove-double-neg N/A

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

   2. sub-div N/A

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

   3. remove-double-neg N/A

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

   4. div-sub N/A

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

   5. clear-num N/A

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

   6. div-inv N/A

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

   7. associate-/r* N/A

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

   8. associate-/r* N/A

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

   9. frac-2neg N/A

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

   10. associate-/l/ N/A

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

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

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

6. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(0 - a\right)}} \]
7. Step-by-step derivation

   1. sub0-neg N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. frac-times N/A

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

   5. metadata-eval N/A

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

   6. frac-2neg N/A

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

   7. frac-times N/A

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

   8. metadata-eval N/A

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

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

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

8. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{a \cdot \frac{1}{\sqrt{b \cdot b + c \cdot \frac{a}{-0.3333333333333333}} - b}}} \]

9. Taylor expanded in a around 0

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

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

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

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

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

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

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

    4. associate-*r/ N/A

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

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

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

    6. *-lowering-*.f64 84.9%

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

11. Simplified 84.9%

    \[\leadsto \frac{0.3333333333333333}{\color{blue}{-0.6666666666666666 \cdot \frac{b}{c} + \frac{0.5 \cdot a}{b}}} \]

12. Final simplification 84.9%

    \[\leadsto \frac{0.3333333333333333}{\frac{b}{c} \cdot -0.6666666666666666 + \frac{a \cdot 0.5}{b}} \]
13. Add Preprocessing

Alternative 13: 82.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{\frac{a \cdot -0.5}{b} + \frac{0.6666666666666666 \cdot b}{c}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return -0.3333333333333333 / (((a * -0.5) / b) + ((0.6666666666666666 * 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 = (-0.3333333333333333d0) / (((a * (-0.5d0)) / b) + ((0.6666666666666666d0 * b) / c))
end function

Java

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

Python

def code(a, b, c):
	return -0.3333333333333333 / (((a * -0.5) / b) + ((0.6666666666666666 * b) / c))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = -0.3333333333333333 / (((a * -0.5) / b) + ((0.6666666666666666 * b) / c));
end

Wolfram

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

Derivation

1. Initial program 51.6%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 51.6%

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

3. Simplified 51.6%

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

   1. remove-double-neg N/A

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

   2. sub-div N/A

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

   3. remove-double-neg N/A

      \[\leadsto \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}{3 \cdot a} - \frac{b}{3 \cdot a} \]

   4. div-sub N/A

      \[\leadsto \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{\color{blue}{3 \cdot a}} \]

   5. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{3 \cdot a}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}}} \]

   6. div-inv N/A

      \[\leadsto \frac{1}{\left(3 \cdot a\right) \cdot \color{blue}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}}} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{3 \cdot a}}{\color{blue}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}}} \]

   8. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{3}}{a}}{\frac{\color{blue}{1}}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}} \]

   9. frac-2neg N/A

      \[\leadsto \frac{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{\mathsf{neg}\left(a\right)}}{\frac{\color{blue}{1}}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}} \]

   10. associate-/l/ N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{3}\right)}{\color{blue}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(\mathsf{neg}\left(a\right)\right)}} \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \color{blue}{\left(\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(\mathsf{neg}\left(a\right)\right)\right)}\right) \]

6. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{1}{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b} \cdot \left(0 - a\right)}} \]

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a}{b} + \frac{2}{3} \cdot \frac{b}{c}\right)}\right) \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \left(\frac{2}{3} \cdot \frac{b}{c} + \color{blue}{\frac{-1}{2} \cdot \frac{a}{b}}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{+.f64}\left(\left(\frac{2}{3} \cdot \frac{b}{c}\right), \color{blue}{\left(\frac{-1}{2} \cdot \frac{a}{b}\right)}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{+.f64}\left(\left(\frac{\frac{2}{3} \cdot b}{c}\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{a}{b}\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{3} \cdot b\right), c\right), \left(\color{blue}{\frac{-1}{2}} \cdot \frac{a}{b}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(b \cdot \frac{2}{3}\right), c\right), \left(\frac{-1}{2} \cdot \frac{a}{b}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \frac{2}{3}\right), c\right), \left(\frac{-1}{2} \cdot \frac{a}{b}\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \frac{2}{3}\right), c\right), \left(\frac{\frac{-1}{2} \cdot a}{\color{blue}{b}}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \frac{2}{3}\right), c\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot a\right), \color{blue}{b}\right)\right)\right) \]

   9. *-lowering-*.f64 84.9%

      \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b, \frac{2}{3}\right), c\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{2}, a\right), b\right)\right)\right) \]

9. Simplified 84.9%

   \[\leadsto \frac{-0.3333333333333333}{\color{blue}{\frac{b \cdot 0.6666666666666666}{c} + \frac{-0.5 \cdot a}{b}}} \]

10. Final simplification 84.9%

    \[\leadsto \frac{-0.3333333333333333}{\frac{a \cdot -0.5}{b} + \frac{0.6666666666666666 \cdot b}{c}} \]
11. Add Preprocessing

Alternative 14: 64.7% 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 51.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), \color{blue}{\left(3 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(c \cdot \left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   10. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(a \cdot 3\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   13. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   16. *-lowering-*.f64 51.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(3, \color{blue}{a}\right)\right) \]

3. Simplified 51.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
6. 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 67.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

7. Simplified 67.9%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
8. Add Preprocessing

Alternative 15: 64.6% 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 51.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), \color{blue}{\left(3 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{3} \cdot a\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(c \cdot \left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   10. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(c \cdot \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(3 \cdot a\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(a \cdot 3\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   13. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \left(a \cdot \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \left(3 \cdot a\right)\right) \]

   16. *-lowering-*.f64 51.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(3, \color{blue}{a}\right)\right) \]

3. Simplified 51.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
6. 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 67.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right) \]

7. Simplified 67.9%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
8. 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 67.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, b\right), c\right) \]

9. Applied egg-rr 67.8%

   \[\leadsto \color{blue}{\frac{-0.5}{b} \cdot c} \]

10. Final simplification 67.8%

    \[\leadsto c \cdot \frac{-0.5}{b} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(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)))