Cubic critical, narrow range

Percentage Accurate: 55.0% → 92.2%

Time: 25.1s

Alternatives: 18

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: 92.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot -3}{b}\\ t_1 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 0.0255:\\ \;\;\;\;\frac{a + \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}{3} \cdot t\_0}{a \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(a \cdot \left(a \cdot \left(\frac{\frac{b}{\frac{{b}^{6}}{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}} \cdot -4.21875}{c \cdot c} + \left(\frac{c \cdot \left(c \cdot 1.6875\right)}{\left(b \cdot b\right) \cdot t\_1} + \frac{-0.75}{b \cdot \frac{b}{-1.125 \cdot \frac{c}{\frac{t\_1}{c}}}}\right)\right) - \frac{c}{t\_1} \cdot 1.125\right) - \frac{1.5}{b}\right) - \frac{-2}{\frac{c}{b}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* a -3.0) b)) (t_1 (* b (* b b))))
   (if (<= b 0.0255)
     (/ (+ a (* (/ (sqrt (+ (* b b) (* c (* a -3.0)))) 3.0) t_0)) (* a t_0))
     (/
      -1.0
      (-
       (*
        a
        (-
         (*
          a
          (-
           (*
            a
            (+
             (/ (* (/ b (/ (pow b 6.0) (* (* c c) (* c c)))) -4.21875) (* c c))
             (+
              (/ (* c (* c 1.6875)) (* (* b b) t_1))
              (/ -0.75 (* b (/ b (* -1.125 (/ c (/ t_1 c)))))))))
           (* (/ c t_1) 1.125)))
         (/ 1.5 b)))
       (/ -2.0 (/ c b)))))))

C

double code(double a, double b, double c) {
	double t_0 = (a * -3.0) / b;
	double t_1 = b * (b * b);
	double tmp;
	if (b <= 0.0255) {
		tmp = (a + ((sqrt(((b * b) + (c * (a * -3.0)))) / 3.0) * t_0)) / (a * t_0);
	} else {
		tmp = -1.0 / ((a * ((a * ((a * ((((b / (pow(b, 6.0) / ((c * c) * (c * c)))) * -4.21875) / (c * c)) + (((c * (c * 1.6875)) / ((b * b) * t_1)) + (-0.75 / (b * (b / (-1.125 * (c / (t_1 / c))))))))) - ((c / t_1) * 1.125))) - (1.5 / b))) - (-2.0 / (c / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (a * (-3.0d0)) / b
    t_1 = b * (b * b)
    if (b <= 0.0255d0) then
        tmp = (a + ((sqrt(((b * b) + (c * (a * (-3.0d0))))) / 3.0d0) * t_0)) / (a * t_0)
    else
        tmp = (-1.0d0) / ((a * ((a * ((a * ((((b / ((b ** 6.0d0) / ((c * c) * (c * c)))) * (-4.21875d0)) / (c * c)) + (((c * (c * 1.6875d0)) / ((b * b) * t_1)) + ((-0.75d0) / (b * (b / ((-1.125d0) * (c / (t_1 / c))))))))) - ((c / t_1) * 1.125d0))) - (1.5d0 / b))) - ((-2.0d0) / (c / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * -3.0) / b;
	double t_1 = b * (b * b);
	double tmp;
	if (b <= 0.0255) {
		tmp = (a + ((Math.sqrt(((b * b) + (c * (a * -3.0)))) / 3.0) * t_0)) / (a * t_0);
	} else {
		tmp = -1.0 / ((a * ((a * ((a * ((((b / (Math.pow(b, 6.0) / ((c * c) * (c * c)))) * -4.21875) / (c * c)) + (((c * (c * 1.6875)) / ((b * b) * t_1)) + (-0.75 / (b * (b / (-1.125 * (c / (t_1 / c))))))))) - ((c / t_1) * 1.125))) - (1.5 / b))) - (-2.0 / (c / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (a * -3.0) / b
	t_1 = b * (b * b)
	tmp = 0
	if b <= 0.0255:
		tmp = (a + ((math.sqrt(((b * b) + (c * (a * -3.0)))) / 3.0) * t_0)) / (a * t_0)
	else:
		tmp = -1.0 / ((a * ((a * ((a * ((((b / (math.pow(b, 6.0) / ((c * c) * (c * c)))) * -4.21875) / (c * c)) + (((c * (c * 1.6875)) / ((b * b) * t_1)) + (-0.75 / (b * (b / (-1.125 * (c / (t_1 / c))))))))) - ((c / t_1) * 1.125))) - (1.5 / b))) - (-2.0 / (c / b)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * -3.0) / b)
	t_1 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (b <= 0.0255)
		tmp = Float64(Float64(a + Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))) / 3.0) * t_0)) / Float64(a * t_0));
	else
		tmp = Float64(-1.0 / Float64(Float64(a * Float64(Float64(a * Float64(Float64(a * Float64(Float64(Float64(Float64(b / Float64((b ^ 6.0) / Float64(Float64(c * c) * Float64(c * c)))) * -4.21875) / Float64(c * c)) + Float64(Float64(Float64(c * Float64(c * 1.6875)) / Float64(Float64(b * b) * t_1)) + Float64(-0.75 / Float64(b * Float64(b / Float64(-1.125 * Float64(c / Float64(t_1 / c))))))))) - Float64(Float64(c / t_1) * 1.125))) - Float64(1.5 / b))) - Float64(-2.0 / Float64(c / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (a * -3.0) / b;
	t_1 = b * (b * b);
	tmp = 0.0;
	if (b <= 0.0255)
		tmp = (a + ((sqrt(((b * b) + (c * (a * -3.0)))) / 3.0) * t_0)) / (a * t_0);
	else
		tmp = -1.0 / ((a * ((a * ((a * ((((b / ((b ^ 6.0) / ((c * c) * (c * c)))) * -4.21875) / (c * c)) + (((c * (c * 1.6875)) / ((b * b) * t_1)) + (-0.75 / (b * (b / (-1.125 * (c / (t_1 / c))))))))) - ((c / t_1) * 1.125))) - (1.5 / b))) - (-2.0 / (c / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * -3.0), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 0.0255], N[(N[(a + N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(a * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(a * N[(N[(a * N[(N[(a * N[(N[(N[(N[(b / N[(N[Power[b, 6.0], $MachinePrecision] / N[(N[(c * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.21875), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * N[(c * 1.6875), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(-0.75 / N[(b * N[(b / N[(-1.125 * N[(c / N[(t$95$1 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c / t$95$1), $MachinePrecision] * 1.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-2.0 / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0254999999999999984

   1. Initial program 89.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 89.9%

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

   3. Simplified 89.9%

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

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

      5. associate-/r* N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac2 N/A

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

      8. frac-add N/A

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

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

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

   6. Applied egg-rr 90.4%

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

   if 0.0254999999999999984 < b

   1. Initial program 53.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 53.0%

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

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
   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. 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 53.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{3} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{27}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) - \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}{a}\right)}\right) \]

   9. Simplified 92.6%

      \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(\left(-a\right) \cdot \left(-0.75 \cdot \frac{c \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)}{b \cdot b} + \left(\frac{-0.6666666666666666 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c} + \frac{1.6875 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right) - \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)\right)}{a}}} \]

   11. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{-2}{\frac{c}{b}} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(\left(0 - a\right) \cdot \left(\frac{\frac{b}{\frac{{b}^{6}}{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}} \cdot -4.21875}{c \cdot c} + \left(\frac{c \cdot \left(c \cdot 1.6875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{-0.75}{b \cdot \frac{b}{-1.125 \cdot \frac{c}{\frac{b \cdot \left(b \cdot b\right)}{c}}}}\right)\right) + \frac{c}{b \cdot \left(b \cdot b\right)} \cdot 1.125\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0255:\\ \;\;\;\;\frac{a + \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}{3} \cdot \frac{a \cdot -3}{b}}{a \cdot \frac{a \cdot -3}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{a \cdot \left(a \cdot \left(a \cdot \left(\frac{\frac{b}{\frac{{b}^{6}}{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}} \cdot -4.21875}{c \cdot c} + \left(\frac{c \cdot \left(c \cdot 1.6875\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{-0.75}{b \cdot \frac{b}{-1.125 \cdot \frac{c}{\frac{b \cdot \left(b \cdot b\right)}{c}}}}\right)\right) - \frac{c}{b \cdot \left(b \cdot b\right)} \cdot 1.125\right) - \frac{1.5}{b}\right) - \frac{-2}{\frac{c}{b}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot -3}{b}\\ \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;\frac{a + \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}{3} \cdot t\_0}{a \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{a} + c \cdot \left(\frac{1.5}{b} + c \cdot \left(1.125 \cdot \frac{a}{b \cdot \left(b \cdot b\right)} + \frac{1.6875 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{5}}\right)\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* a -3.0) b)))
   (if (<= b 0.027)
     (/ (+ a (* (/ (sqrt (+ (* b b) (* c (* a -3.0)))) 3.0) t_0)) (* a t_0))
     (/
      (/ 1.0 a)
      (/
       (+
        (/ (* b -2.0) a)
        (*
         c
         (+
          (/ 1.5 b)
          (*
           c
           (+
            (* 1.125 (/ a (* b (* b b))))
            (/ (* 1.6875 (* c (* a a))) (pow b 5.0)))))))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = (a * -3.0) / b;
	double tmp;
	if (b <= 0.027) {
		tmp = (a + ((sqrt(((b * b) + (c * (a * -3.0)))) / 3.0) * t_0)) / (a * t_0);
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / a) + (c * ((1.5 / b) + (c * ((1.125 * (a / (b * (b * b)))) + ((1.6875 * (c * (a * a))) / pow(b, 5.0))))))) / 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) :: t_0
    real(8) :: tmp
    t_0 = (a * (-3.0d0)) / b
    if (b <= 0.027d0) then
        tmp = (a + ((sqrt(((b * b) + (c * (a * (-3.0d0))))) / 3.0d0) * t_0)) / (a * t_0)
    else
        tmp = (1.0d0 / a) / ((((b * (-2.0d0)) / a) + (c * ((1.5d0 / b) + (c * ((1.125d0 * (a / (b * (b * b)))) + ((1.6875d0 * (c * (a * a))) / (b ** 5.0d0))))))) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * -3.0) / b;
	double tmp;
	if (b <= 0.027) {
		tmp = (a + ((Math.sqrt(((b * b) + (c * (a * -3.0)))) / 3.0) * t_0)) / (a * t_0);
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / a) + (c * ((1.5 / b) + (c * ((1.125 * (a / (b * (b * b)))) + ((1.6875 * (c * (a * a))) / Math.pow(b, 5.0))))))) / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (a * -3.0) / b
	tmp = 0
	if b <= 0.027:
		tmp = (a + ((math.sqrt(((b * b) + (c * (a * -3.0)))) / 3.0) * t_0)) / (a * t_0)
	else:
		tmp = (1.0 / a) / ((((b * -2.0) / a) + (c * ((1.5 / b) + (c * ((1.125 * (a / (b * (b * b)))) + ((1.6875 * (c * (a * a))) / math.pow(b, 5.0))))))) / c)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * -3.0) / b)
	tmp = 0.0
	if (b <= 0.027)
		tmp = Float64(Float64(a + Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))) / 3.0) * t_0)) / Float64(a * t_0));
	else
		tmp = Float64(Float64(1.0 / a) / Float64(Float64(Float64(Float64(b * -2.0) / a) + Float64(c * Float64(Float64(1.5 / b) + Float64(c * Float64(Float64(1.125 * Float64(a / Float64(b * Float64(b * b)))) + Float64(Float64(1.6875 * Float64(c * Float64(a * a))) / (b ^ 5.0))))))) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (a * -3.0) / b;
	tmp = 0.0;
	if (b <= 0.027)
		tmp = (a + ((sqrt(((b * b) + (c * (a * -3.0)))) / 3.0) * t_0)) / (a * t_0);
	else
		tmp = (1.0 / a) / ((((b * -2.0) / a) + (c * ((1.5 / b) + (c * ((1.125 * (a / (b * (b * b)))) + ((1.6875 * (c * (a * a))) / (b ^ 5.0))))))) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * -3.0), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[b, 0.027], N[(N[(a + N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(a * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] / N[(N[(N[(N[(b * -2.0), $MachinePrecision] / a), $MachinePrecision] + N[(c * N[(N[(1.5 / b), $MachinePrecision] + N[(c * N[(N[(1.125 * N[(a / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.6875 * N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0269999999999999997

   1. Initial program 89.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 89.9%

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

   3. Simplified 89.9%

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

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

      5. associate-/r* N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac2 N/A

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

      8. frac-add N/A

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

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

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

   6. Applied egg-rr 90.4%

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

   if 0.0269999999999999997 < b

   1. Initial program 53.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 53.0%

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

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
   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. 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 53.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{3} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{27}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) - \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}{a}\right)}\right) \]

   9. Simplified 92.6%

      \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(\left(-a\right) \cdot \left(-0.75 \cdot \frac{c \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)}{b \cdot b} + \left(\frac{-0.6666666666666666 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c} + \frac{1.6875 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right) - \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)\right)}{a}}} \]

   10. Taylor expanded in c around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-2 \cdot \frac{b}{a} + c \cdot \left(c \cdot \left(\frac{9}{8} \cdot \frac{a}{{b}^{3}} + \frac{27}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}}\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}{c}\right)}\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\left(-2 \cdot \frac{b}{a} + c \cdot \left(c \cdot \left(\frac{9}{8} \cdot \frac{a}{{b}^{3}} + \frac{27}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}}\right) + \frac{3}{2} \cdot \frac{1}{b}\right)\right), \color{blue}{c}\right)\right) \]

   12. Simplified 92.6%

       \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-2 \cdot b}{a} + c \cdot \left(\frac{1.5}{b} + c \cdot \left(1.125 \cdot \frac{a}{b \cdot \left(b \cdot b\right)} + \frac{1.6875 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{5}}\right)\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;\frac{a + \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}{3} \cdot \frac{a \cdot -3}{b}}{a \cdot \frac{a \cdot -3}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{a} + c \cdot \left(\frac{1.5}{b} + c \cdot \left(1.125 \cdot \frac{a}{b \cdot \left(b \cdot b\right)} + \frac{1.6875 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{5}}\right)\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot -3}{b}\\ \mathbf{if}\;b \leq 0.05:\\ \;\;\;\;\frac{a + \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}{3} \cdot t\_0}{a \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* a -3.0) b)))
   (if (<= b 0.05)
     (/ (+ a (* (/ (sqrt (+ (* b b) (* c (* a -3.0)))) 3.0) t_0)) (* a t_0))
     (/
      (/ 1.0 a)
      (/
       (+
        (/ (* b -2.0) c)
        (*
         a
         (+
          (/ 1.5 b)
          (*
           a
           (/
            (-
             (* c 1.125)
             (/ (* a (+ (* (* c c) -4.21875) (* (* c c) 2.53125))) (* b b)))
            (* b (* b b)))))))
       a)))))

C

double code(double a, double b, double c) {
	double t_0 = (a * -3.0) / b;
	double tmp;
	if (b <= 0.05) {
		tmp = (a + ((sqrt(((b * b) + (c * (a * -3.0)))) / 3.0) * t_0)) / (a * t_0);
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * (-3.0d0)) / b
    if (b <= 0.05d0) then
        tmp = (a + ((sqrt(((b * b) + (c * (a * (-3.0d0))))) / 3.0d0) * t_0)) / (a * t_0)
    else
        tmp = (1.0d0 / a) / ((((b * (-2.0d0)) / c) + (a * ((1.5d0 / b) + (a * (((c * 1.125d0) - ((a * (((c * c) * (-4.21875d0)) + ((c * c) * 2.53125d0))) / (b * b))) / (b * (b * b))))))) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * -3.0) / b;
	double tmp;
	if (b <= 0.05) {
		tmp = (a + ((Math.sqrt(((b * b) + (c * (a * -3.0)))) / 3.0) * t_0)) / (a * t_0);
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (a * -3.0) / b
	tmp = 0
	if b <= 0.05:
		tmp = (a + ((math.sqrt(((b * b) + (c * (a * -3.0)))) / 3.0) * t_0)) / (a * t_0)
	else:
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * -3.0) / b)
	tmp = 0.0
	if (b <= 0.05)
		tmp = Float64(Float64(a + Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))) / 3.0) * t_0)) / Float64(a * t_0));
	else
		tmp = Float64(Float64(1.0 / a) / Float64(Float64(Float64(Float64(b * -2.0) / c) + Float64(a * Float64(Float64(1.5 / b) + Float64(a * Float64(Float64(Float64(c * 1.125) - Float64(Float64(a * Float64(Float64(Float64(c * c) * -4.21875) + Float64(Float64(c * c) * 2.53125))) / Float64(b * b))) / Float64(b * Float64(b * b))))))) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (a * -3.0) / b;
	tmp = 0.0;
	if (b <= 0.05)
		tmp = (a + ((sqrt(((b * b) + (c * (a * -3.0)))) / 3.0) * t_0)) / (a * t_0);
	else
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * -3.0), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[b, 0.05], N[(N[(a + N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(a * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] / N[(N[(N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision] + N[(a * N[(N[(1.5 / b), $MachinePrecision] + N[(a * N[(N[(N[(c * 1.125), $MachinePrecision] - N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -4.21875), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * 2.53125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.050000000000000003

   1. Initial program 89.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 89.9%

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

   3. Simplified 89.9%

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

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

      5. associate-/r* N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac2 N/A

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

      8. frac-add N/A

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

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

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

   6. Applied egg-rr 90.4%

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

   if 0.050000000000000003 < b

   1. Initial program 53.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 53.0%

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

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
   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. 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 53.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{3} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{27}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) - \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}{a}\right)}\right) \]

   9. Simplified 92.6%

      \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(\left(-a\right) \cdot \left(-0.75 \cdot \frac{c \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)}{b \cdot b} + \left(\frac{-0.6666666666666666 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c} + \frac{1.6875 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right) - \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)\right)}{a}}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c}{{b}^{3}}\right)}\right)\right)\right)\right), a\right)\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c\right), \left({b}^{3}\right)\right)\right)\right)\right)\right), a\right)\right) \]

   12. Simplified 92.6%

       \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \color{blue}{\frac{\frac{\left(-1 \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b} + c \cdot 1.125}{b \cdot \left(b \cdot b\right)}}\right)}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.05:\\ \;\;\;\;\frac{a + \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}{3} \cdot \frac{a \cdot -3}{b}}{a \cdot \frac{a \cdot -3}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0255:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \frac{a}{b} - a}{a}}{\frac{\frac{a}{0.3333333333333333}}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0255)
   (/
    (/ (- (* (sqrt (+ (* b b) (* c (* a -3.0)))) (/ a b)) a) a)
    (/ (/ a 0.3333333333333333) b))
   (/
    (/ 1.0 a)
    (/
     (+
      (/ (* b -2.0) c)
      (*
       a
       (+
        (/ 1.5 b)
        (*
         a
         (/
          (-
           (* c 1.125)
           (/ (* a (+ (* (* c c) -4.21875) (* (* c c) 2.53125))) (* b b)))
          (* b (* b b)))))))
     a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0255) {
		tmp = (((sqrt(((b * b) + (c * (a * -3.0)))) * (a / b)) - a) / a) / ((a / 0.3333333333333333) / b);
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	}
	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 (b <= 0.0255d0) then
        tmp = (((sqrt(((b * b) + (c * (a * (-3.0d0))))) * (a / b)) - a) / a) / ((a / 0.3333333333333333d0) / b)
    else
        tmp = (1.0d0 / a) / ((((b * (-2.0d0)) / c) + (a * ((1.5d0 / b) + (a * (((c * 1.125d0) - ((a * (((c * c) * (-4.21875d0)) + ((c * c) * 2.53125d0))) / (b * b))) / (b * (b * b))))))) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0255) {
		tmp = (((Math.sqrt(((b * b) + (c * (a * -3.0)))) * (a / b)) - a) / a) / ((a / 0.3333333333333333) / b);
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.0255:
		tmp = (((math.sqrt(((b * b) + (c * (a * -3.0)))) * (a / b)) - a) / a) / ((a / 0.3333333333333333) / b)
	else:
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0255)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))) * Float64(a / b)) - a) / a) / Float64(Float64(a / 0.3333333333333333) / b));
	else
		tmp = Float64(Float64(1.0 / a) / Float64(Float64(Float64(Float64(b * -2.0) / c) + Float64(a * Float64(Float64(1.5 / b) + Float64(a * Float64(Float64(Float64(c * 1.125) - Float64(Float64(a * Float64(Float64(Float64(c * c) * -4.21875) + Float64(Float64(c * c) * 2.53125))) / Float64(b * b))) / Float64(b * Float64(b * b))))))) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.0255)
		tmp = (((sqrt(((b * b) + (c * (a * -3.0)))) * (a / b)) - a) / a) / ((a / 0.3333333333333333) / b);
	else
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0255], N[(N[(N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(a / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / a), $MachinePrecision] / N[(N[(a / 0.3333333333333333), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] / N[(N[(N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision] + N[(a * N[(N[(1.5 / b), $MachinePrecision] + N[(a * N[(N[(N[(c * 1.125), $MachinePrecision] - N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -4.21875), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * 2.53125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0254999999999999984

   1. Initial program 89.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 89.9%

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

   3. Simplified 89.9%

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

   6. Applied egg-rr 90.1%

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

   if 0.0254999999999999984 < b

   1. Initial program 53.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 53.0%

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

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
   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. 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 53.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{3} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{27}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) - \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}{a}\right)}\right) \]

   9. Simplified 92.6%

      \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(\left(-a\right) \cdot \left(-0.75 \cdot \frac{c \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)}{b \cdot b} + \left(\frac{-0.6666666666666666 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c} + \frac{1.6875 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right) - \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)\right)}{a}}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c}{{b}^{3}}\right)}\right)\right)\right)\right), a\right)\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c\right), \left({b}^{3}\right)\right)\right)\right)\right)\right), a\right)\right) \]

   12. Simplified 92.6%

       \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \color{blue}{\frac{\frac{\left(-1 \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b} + c \cdot 1.125}{b \cdot \left(b \cdot b\right)}}\right)}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0255:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot \frac{a}{b} - a}{a}}{\frac{\frac{a}{0.3333333333333333}}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0255:\\ \;\;\;\;\frac{\frac{b}{-3} - \frac{\sqrt{b \cdot b + \frac{c}{\frac{-0.3333333333333333}{a}}}}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0255)
   (/
    (- (/ b -3.0) (/ (sqrt (+ (* b b) (/ c (/ -0.3333333333333333 a)))) -3.0))
    a)
   (/
    (/ 1.0 a)
    (/
     (+
      (/ (* b -2.0) c)
      (*
       a
       (+
        (/ 1.5 b)
        (*
         a
         (/
          (-
           (* c 1.125)
           (/ (* a (+ (* (* c c) -4.21875) (* (* c c) 2.53125))) (* b b)))
          (* b (* b b)))))))
     a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0255) {
		tmp = ((b / -3.0) - (sqrt(((b * b) + (c / (-0.3333333333333333 / a)))) / -3.0)) / a;
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	}
	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 (b <= 0.0255d0) then
        tmp = ((b / (-3.0d0)) - (sqrt(((b * b) + (c / ((-0.3333333333333333d0) / a)))) / (-3.0d0))) / a
    else
        tmp = (1.0d0 / a) / ((((b * (-2.0d0)) / c) + (a * ((1.5d0 / b) + (a * (((c * 1.125d0) - ((a * (((c * c) * (-4.21875d0)) + ((c * c) * 2.53125d0))) / (b * b))) / (b * (b * b))))))) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0255) {
		tmp = ((b / -3.0) - (Math.sqrt(((b * b) + (c / (-0.3333333333333333 / a)))) / -3.0)) / a;
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.0255:
		tmp = ((b / -3.0) - (math.sqrt(((b * b) + (c / (-0.3333333333333333 / a)))) / -3.0)) / a
	else:
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0255)
		tmp = Float64(Float64(Float64(b / -3.0) - Float64(sqrt(Float64(Float64(b * b) + Float64(c / Float64(-0.3333333333333333 / a)))) / -3.0)) / a);
	else
		tmp = Float64(Float64(1.0 / a) / Float64(Float64(Float64(Float64(b * -2.0) / c) + Float64(a * Float64(Float64(1.5 / b) + Float64(a * Float64(Float64(Float64(c * 1.125) - Float64(Float64(a * Float64(Float64(Float64(c * c) * -4.21875) + Float64(Float64(c * c) * 2.53125))) / Float64(b * b))) / Float64(b * Float64(b * b))))))) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.0255)
		tmp = ((b / -3.0) - (sqrt(((b * b) + (c / (-0.3333333333333333 / a)))) / -3.0)) / a;
	else
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0255], N[(N[(N[(b / -3.0), $MachinePrecision] - N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c / N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] / N[(N[(N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision] + N[(a * N[(N[(1.5 / b), $MachinePrecision] + N[(a * N[(N[(N[(c * 1.125), $MachinePrecision] - N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -4.21875), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * 2.53125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0254999999999999984

   1. Initial program 89.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 89.9%

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

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
   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 89.9%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. div-sub N/A

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

      5. div-sub N/A

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

      6. associate-/r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. div-inv N/A

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

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

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

      14. metadata-eval N/A

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

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

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

   8. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.3333333333333333}} - \frac{\frac{\sqrt{b \cdot b + c \cdot \frac{a}{-0.3333333333333333}}}{-3}}{a}} \]
   9. Step-by-step derivation

      1. div-inv N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. sub-div N/A

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

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

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

   10. Applied egg-rr 90.0%

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

   if 0.0254999999999999984 < b

   1. Initial program 53.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 53.0%

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

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
   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. 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 53.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{3} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{27}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) - \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}{a}\right)}\right) \]

   9. Simplified 92.6%

      \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(\left(-a\right) \cdot \left(-0.75 \cdot \frac{c \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)}{b \cdot b} + \left(\frac{-0.6666666666666666 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c} + \frac{1.6875 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right) - \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)\right)}{a}}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c}{{b}^{3}}\right)}\right)\right)\right)\right), a\right)\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c\right), \left({b}^{3}\right)\right)\right)\right)\right)\right), a\right)\right) \]

   12. Simplified 92.6%

       \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \color{blue}{\frac{\frac{\left(-1 \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b} + c \cdot 1.125}{b \cdot \left(b \cdot b\right)}}\right)}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0255:\\ \;\;\;\;\frac{\frac{b}{-3} - \frac{\sqrt{b \cdot b + \frac{c}{\frac{-0.3333333333333333}{a}}}}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.027)
   (/ (* -0.3333333333333333 (- b (sqrt (+ (* b b) (* c (* a -3.0)))))) a)
   (/
    (/ 1.0 a)
    (/
     (+
      (/ (* b -2.0) c)
      (*
       a
       (+
        (/ 1.5 b)
        (*
         a
         (/
          (-
           (* c 1.125)
           (/ (* a (+ (* (* c c) -4.21875) (* (* c c) 2.53125))) (* b b)))
          (* b (* b b)))))))
     a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.027) {
		tmp = (-0.3333333333333333 * (b - sqrt(((b * b) + (c * (a * -3.0)))))) / a;
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	}
	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 (b <= 0.027d0) then
        tmp = ((-0.3333333333333333d0) * (b - sqrt(((b * b) + (c * (a * (-3.0d0))))))) / a
    else
        tmp = (1.0d0 / a) / ((((b * (-2.0d0)) / c) + (a * ((1.5d0 / b) + (a * (((c * 1.125d0) - ((a * (((c * c) * (-4.21875d0)) + ((c * c) * 2.53125d0))) / (b * b))) / (b * (b * b))))))) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.027) {
		tmp = (-0.3333333333333333 * (b - Math.sqrt(((b * b) + (c * (a * -3.0)))))) / a;
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.027:
		tmp = (-0.3333333333333333 * (b - math.sqrt(((b * b) + (c * (a * -3.0)))))) / a
	else:
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.027)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(b - sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))))) / a);
	else
		tmp = Float64(Float64(1.0 / a) / Float64(Float64(Float64(Float64(b * -2.0) / c) + Float64(a * Float64(Float64(1.5 / b) + Float64(a * Float64(Float64(Float64(c * 1.125) - Float64(Float64(a * Float64(Float64(Float64(c * c) * -4.21875) + Float64(Float64(c * c) * 2.53125))) / Float64(b * b))) / Float64(b * Float64(b * b))))))) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.027)
		tmp = (-0.3333333333333333 * (b - sqrt(((b * b) + (c * (a * -3.0)))))) / a;
	else
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.027], N[(N[(-0.3333333333333333 * N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] / N[(N[(N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision] + N[(a * N[(N[(1.5 / b), $MachinePrecision] + N[(a * N[(N[(N[(c * 1.125), $MachinePrecision] - N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -4.21875), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * 2.53125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0269999999999999997

   1. Initial program 89.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in a around inf

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

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

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

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

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

      3. *-commutative N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 89.9%

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

   7. Simplified 89.9%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. remove-double-neg N/A

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

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

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

   9. Applied egg-rr 90.0%

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

   if 0.0269999999999999997 < b

   1. Initial program 53.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 53.0%

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

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
   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. 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 53.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{3} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{27}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) - \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}{a}\right)}\right) \]

   9. Simplified 92.6%

      \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(\left(-a\right) \cdot \left(-0.75 \cdot \frac{c \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)}{b \cdot b} + \left(\frac{-0.6666666666666666 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c} + \frac{1.6875 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right) - \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)\right)}{a}}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c}{{b}^{3}}\right)}\right)\right)\right)\right), a\right)\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c\right), \left({b}^{3}\right)\right)\right)\right)\right)\right), a\right)\right) \]

   12. Simplified 92.6%

       \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \color{blue}{\frac{\frac{\left(-1 \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b} + c \cdot 1.125}{b \cdot \left(b \cdot b\right)}}\right)}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.026:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.026)
   (/ (- (sqrt (+ (* b b) (* c (* a -3.0)))) b) (* a 3.0))
   (/
    (/ 1.0 a)
    (/
     (+
      (/ (* b -2.0) c)
      (*
       a
       (+
        (/ 1.5 b)
        (*
         a
         (/
          (-
           (* c 1.125)
           (/ (* a (+ (* (* c c) -4.21875) (* (* c c) 2.53125))) (* b b)))
          (* b (* b b)))))))
     a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.026) {
		tmp = (sqrt(((b * b) + (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	}
	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 (b <= 0.026d0) then
        tmp = (sqrt(((b * b) + (c * (a * (-3.0d0))))) - b) / (a * 3.0d0)
    else
        tmp = (1.0d0 / a) / ((((b * (-2.0d0)) / c) + (a * ((1.5d0 / b) + (a * (((c * 1.125d0) - ((a * (((c * c) * (-4.21875d0)) + ((c * c) * 2.53125d0))) / (b * b))) / (b * (b * b))))))) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.026) {
		tmp = (Math.sqrt(((b * b) + (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.026:
		tmp = (math.sqrt(((b * b) + (c * (a * -3.0)))) - b) / (a * 3.0)
	else:
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.026)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(1.0 / a) / Float64(Float64(Float64(Float64(b * -2.0) / c) + Float64(a * Float64(Float64(1.5 / b) + Float64(a * Float64(Float64(Float64(c * 1.125) - Float64(Float64(a * Float64(Float64(Float64(c * c) * -4.21875) + Float64(Float64(c * c) * 2.53125))) / Float64(b * b))) / Float64(b * Float64(b * b))))))) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.026)
		tmp = (sqrt(((b * b) + (c * (a * -3.0)))) - b) / (a * 3.0);
	else
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.026], 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], N[(N[(1.0 / a), $MachinePrecision] / N[(N[(N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision] + N[(a * N[(N[(1.5 / b), $MachinePrecision] + N[(a * N[(N[(N[(c * 1.125), $MachinePrecision] - N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -4.21875), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * 2.53125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0259999999999999988

   1. Initial program 89.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 89.9%

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

   3. Simplified 89.9%

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

   if 0.0259999999999999988 < b

   1. Initial program 53.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 53.0%

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

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
   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. 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 53.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{3} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{27}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) - \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}{a}\right)}\right) \]

   9. Simplified 92.6%

      \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(\left(-a\right) \cdot \left(-0.75 \cdot \frac{c \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)}{b \cdot b} + \left(\frac{-0.6666666666666666 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c} + \frac{1.6875 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right) - \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)\right)}{a}}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c}{{b}^{3}}\right)}\right)\right)\right)\right), a\right)\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c\right), \left({b}^{3}\right)\right)\right)\right)\right)\right), a\right)\right) \]

   12. Simplified 92.6%

       \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \color{blue}{\frac{\frac{\left(-1 \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b} + c \cdot 1.125}{b \cdot \left(b \cdot b\right)}}\right)}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.026:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b + c \cdot \frac{a}{-0.3333333333333333}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.027)
   (*
    -0.3333333333333333
    (/ (- b (sqrt (+ (* b b) (* c (/ a -0.3333333333333333))))) a))
   (/
    (/ 1.0 a)
    (/
     (+
      (/ (* b -2.0) c)
      (*
       a
       (+
        (/ 1.5 b)
        (*
         a
         (/
          (-
           (* c 1.125)
           (/ (* a (+ (* (* c c) -4.21875) (* (* c c) 2.53125))) (* b b)))
          (* b (* b b)))))))
     a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.027) {
		tmp = -0.3333333333333333 * ((b - sqrt(((b * b) + (c * (a / -0.3333333333333333))))) / a);
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	}
	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 (b <= 0.027d0) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt(((b * b) + (c * (a / (-0.3333333333333333d0)))))) / a)
    else
        tmp = (1.0d0 / a) / ((((b * (-2.0d0)) / c) + (a * ((1.5d0 / b) + (a * (((c * 1.125d0) - ((a * (((c * c) * (-4.21875d0)) + ((c * c) * 2.53125d0))) / (b * b))) / (b * (b * b))))))) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.027) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt(((b * b) + (c * (a / -0.3333333333333333))))) / a);
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.027:
		tmp = -0.3333333333333333 * ((b - math.sqrt(((b * b) + (c * (a / -0.3333333333333333))))) / a)
	else:
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.027)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(Float64(b * b) + Float64(c * Float64(a / -0.3333333333333333))))) / a));
	else
		tmp = Float64(Float64(1.0 / a) / Float64(Float64(Float64(Float64(b * -2.0) / c) + Float64(a * Float64(Float64(1.5 / b) + Float64(a * Float64(Float64(Float64(c * 1.125) - Float64(Float64(a * Float64(Float64(Float64(c * c) * -4.21875) + Float64(Float64(c * c) * 2.53125))) / Float64(b * b))) / Float64(b * Float64(b * b))))))) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.027)
		tmp = -0.3333333333333333 * ((b - sqrt(((b * b) + (c * (a / -0.3333333333333333))))) / a);
	else
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.027], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a / -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] / N[(N[(N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision] + N[(a * N[(N[(1.5 / b), $MachinePrecision] + N[(a * N[(N[(N[(c * 1.125), $MachinePrecision] - N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -4.21875), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * 2.53125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0269999999999999997

   1. Initial program 89.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 89.9%

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

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
   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 89.9%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-rgt-identity N/A

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

   8. Applied egg-rr 89.9%

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

   if 0.0269999999999999997 < b

   1. Initial program 53.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 53.0%

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

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
   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. 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 53.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{3} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{27}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) - \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}{a}\right)}\right) \]

   9. Simplified 92.6%

      \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(\left(-a\right) \cdot \left(-0.75 \cdot \frac{c \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)}{b \cdot b} + \left(\frac{-0.6666666666666666 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c} + \frac{1.6875 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right) - \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)\right)}{a}}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c}{{b}^{3}}\right)}\right)\right)\right)\right), a\right)\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c\right), \left({b}^{3}\right)\right)\right)\right)\right)\right), a\right)\right) \]

   12. Simplified 92.6%

       \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \color{blue}{\frac{\frac{\left(-1 \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b} + c \cdot 1.125}{b \cdot \left(b \cdot b\right)}}\right)}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.027:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b + c \cdot \frac{a}{-0.3333333333333333}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.039:\\ \;\;\;\;\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.039)
   (* (- (sqrt (+ (* b b) (* c (* a -3.0)))) b) (/ 0.3333333333333333 a))
   (/
    (/ 1.0 a)
    (/
     (+
      (/ (* b -2.0) c)
      (*
       a
       (+
        (/ 1.5 b)
        (*
         a
         (/
          (-
           (* c 1.125)
           (/ (* a (+ (* (* c c) -4.21875) (* (* c c) 2.53125))) (* b b)))
          (* b (* b b)))))))
     a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.039) {
		tmp = (sqrt(((b * b) + (c * (a * -3.0)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	}
	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 (b <= 0.039d0) then
        tmp = (sqrt(((b * b) + (c * (a * (-3.0d0))))) - b) * (0.3333333333333333d0 / a)
    else
        tmp = (1.0d0 / a) / ((((b * (-2.0d0)) / c) + (a * ((1.5d0 / b) + (a * (((c * 1.125d0) - ((a * (((c * c) * (-4.21875d0)) + ((c * c) * 2.53125d0))) / (b * b))) / (b * (b * b))))))) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.039) {
		tmp = (Math.sqrt(((b * b) + (c * (a * -3.0)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.039:
		tmp = (math.sqrt(((b * b) + (c * (a * -3.0)))) - b) * (0.3333333333333333 / a)
	else:
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.039)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(1.0 / a) / Float64(Float64(Float64(Float64(b * -2.0) / c) + Float64(a * Float64(Float64(1.5 / b) + Float64(a * Float64(Float64(Float64(c * 1.125) - Float64(Float64(a * Float64(Float64(Float64(c * c) * -4.21875) + Float64(Float64(c * c) * 2.53125))) / Float64(b * b))) / Float64(b * Float64(b * b))))))) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.039)
		tmp = (sqrt(((b * b) + (c * (a * -3.0)))) - b) * (0.3333333333333333 / a);
	else
		tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.039], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] / N[(N[(N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision] + N[(a * N[(N[(1.5 / b), $MachinePrecision] + N[(a * N[(N[(N[(c * 1.125), $MachinePrecision] - N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -4.21875), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * 2.53125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0389999999999999999

   1. Initial program 89.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 89.9%

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

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
   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. associate-/r/ N/A

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

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

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

      8. associate-/r* N/A

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

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

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

      10. metadata-eval N/A

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

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

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

   6. Applied egg-rr 89.9%

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

   if 0.0389999999999999999 < b

   1. Initial program 53.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 53.0%

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

   3. Simplified 53.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
   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. 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 53.0%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{3} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{27}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) - \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}{a}\right)}\right) \]

   9. Simplified 92.6%

      \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(\left(-a\right) \cdot \left(-0.75 \cdot \frac{c \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)}{b \cdot b} + \left(\frac{-0.6666666666666666 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c} + \frac{1.6875 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right) - \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)\right)}{a}}} \]

   10. Taylor expanded in b around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c}{{b}^{3}}\right)}\right)\right)\right)\right), a\right)\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c\right), \left({b}^{3}\right)\right)\right)\right)\right)\right), a\right)\right) \]

   12. Simplified 92.6%

       \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \color{blue}{\frac{\frac{\left(-1 \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b} + c \cdot 1.125}{b \cdot \left(b \cdot b\right)}}\right)}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.039:\\ \;\;\;\;\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ 1.0 a)
  (/
   (+
    (/ (* b -2.0) c)
    (*
     a
     (+
      (/ 1.5 b)
      (*
       a
       (/
        (-
         (* c 1.125)
         (/ (* a (+ (* (* c c) -4.21875) (* (* c c) 2.53125))) (* b b)))
        (* b (* b b)))))))
   a)))

C

double code(double a, double b, double c) {
	return (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / 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 = (1.0d0 / a) / ((((b * (-2.0d0)) / c) + (a * ((1.5d0 / b) + (a * (((c * 1.125d0) - ((a * (((c * c) * (-4.21875d0)) + ((c * c) * 2.53125d0))) / (b * b))) / (b * (b * b))))))) / a)
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
}

Python

def code(a, b, c):
	return (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a)

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / a) / Float64(Float64(Float64(Float64(b * -2.0) / c) + Float64(a * Float64(Float64(1.5 / b) + Float64(a * Float64(Float64(Float64(c * 1.125) - Float64(Float64(a * Float64(Float64(Float64(c * c) * -4.21875) + Float64(Float64(c * c) * 2.53125))) / Float64(b * b))) / Float64(b * Float64(b * b))))))) / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + (a * (((c * 1.125) - ((a * (((c * c) * -4.21875) + ((c * c) * 2.53125))) / (b * b))) / (b * (b * b))))))) / a);
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / a), $MachinePrecision] / N[(N[(N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision] + N[(a * N[(N[(1.5 / b), $MachinePrecision] + N[(a * N[(N[(N[(c * 1.125), $MachinePrecision] - N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * -4.21875), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * 2.53125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 56.6%

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

3. Simplified 56.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
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. 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 56.6%

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

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{3} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{27}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) - \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}{a}\right)}\right) \]

9. Simplified 89.9%

   \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(\left(-a\right) \cdot \left(-0.75 \cdot \frac{c \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)}{b \cdot b} + \left(\frac{-0.6666666666666666 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c} + \frac{1.6875 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right) - \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)\right)}{a}}} \]

10. Taylor expanded in b around inf

    \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c}{{b}^{3}}\right)}\right)\right)\right)\right), a\right)\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(-2, b\right), c\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{3}{2}, b\right), \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\left(-1 \cdot \frac{a \cdot \left(\frac{-135}{32} \cdot {c}^{2} + \left(\frac{27}{32} \cdot {c}^{2} + \frac{27}{16} \cdot {c}^{2}\right)\right)}{{b}^{2}} - \frac{-9}{8} \cdot c\right), \left({b}^{3}\right)\right)\right)\right)\right)\right), a\right)\right) \]

12. Simplified 89.9%

    \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \color{blue}{\frac{\frac{\left(-1 \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b} + c \cdot 1.125}{b \cdot \left(b \cdot b\right)}}\right)}{a}} \]

13. Final simplification 89.9%

    \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \frac{c \cdot 1.125 - \frac{a \cdot \left(\left(c \cdot c\right) \cdot -4.21875 + \left(c \cdot c\right) \cdot 2.53125\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right)}{a}} \]
14. Add Preprocessing

Alternative 11: 88.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + \frac{1.125 \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right)}{a}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ 1.0 a)
  (/
   (+ (/ (* b -2.0) c) (* a (+ (/ 1.5 b) (/ (* 1.125 (* c a)) (* b (* b b))))))
   a)))

C

double code(double a, double b, double c) {
	return (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + ((1.125 * (c * a)) / (b * (b * b)))))) / 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 = (1.0d0 / a) / ((((b * (-2.0d0)) / c) + (a * ((1.5d0 / b) + ((1.125d0 * (c * a)) / (b * (b * b)))))) / a)
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + ((1.125 * (c * a)) / (b * (b * b)))))) / a);
}

Python

def code(a, b, c):
	return (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + ((1.125 * (c * a)) / (b * (b * b)))))) / a)

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / a) / Float64(Float64(Float64(Float64(b * -2.0) / c) + Float64(a * Float64(Float64(1.5 / b) + Float64(Float64(1.125 * Float64(c * a)) / Float64(b * Float64(b * b)))))) / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / a) / ((((b * -2.0) / c) + (a * ((1.5 / b) + ((1.125 * (c * a)) / (b * (b * b)))))) / a);
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / a), $MachinePrecision] / N[(N[(N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision] + N[(a * N[(N[(1.5 / b), $MachinePrecision] + N[(N[(1.125 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 56.6%

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

3. Simplified 56.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
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. 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 56.6%

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

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{3} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{27}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) - \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}{a}\right)}\right) \]

9. Simplified 89.9%

   \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(\left(-a\right) \cdot \left(-0.75 \cdot \frac{c \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)}{b \cdot b} + \left(\frac{-0.6666666666666666 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c} + \frac{1.6875 \cdot \left(c \cdot c\right)}{{b}^{5}}\right)\right) - \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)\right)}{a}}} \]

10. Taylor expanded in a around 0

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

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

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

12. Simplified 86.6%

    \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{-2 \cdot b}{c} + a \cdot \left(\frac{1.125 \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot b\right)} + \frac{1.5}{b}\right)}{a}}} \]

13. Final simplification 86.6%

    \[\leadsto \frac{\frac{1}{a}}{\frac{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + \frac{1.125 \cdot \left(c \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right)}{a}} \]
14. Add Preprocessing

Alternative 12: 88.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{a}}{\frac{-2 \cdot \frac{b}{a} + c \cdot \left(\frac{1.5}{b} - c \cdot \left(-1.125 \cdot \frac{a}{b \cdot \left(b \cdot b\right)}\right)\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ 1.0 a)
  (/
   (+
    (* -2.0 (/ b a))
    (* c (- (/ 1.5 b) (* c (* -1.125 (/ a (* b (* b b))))))))
   c)))

C

double code(double a, double b, double c) {
	return (1.0 / a) / (((-2.0 * (b / a)) + (c * ((1.5 / b) - (c * (-1.125 * (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 = (1.0d0 / a) / ((((-2.0d0) * (b / a)) + (c * ((1.5d0 / b) - (c * ((-1.125d0) * (a / (b * (b * b)))))))) / c)
end function

Java

public static double code(double a, double b, double c) {
	return (1.0 / a) / (((-2.0 * (b / a)) + (c * ((1.5 / b) - (c * (-1.125 * (a / (b * (b * b)))))))) / c);
}

Python

def code(a, b, c):
	return (1.0 / a) / (((-2.0 * (b / a)) + (c * ((1.5 / b) - (c * (-1.125 * (a / (b * (b * b)))))))) / c)

Julia

function code(a, b, c)
	return Float64(Float64(1.0 / a) / Float64(Float64(Float64(-2.0 * Float64(b / a)) + Float64(c * Float64(Float64(1.5 / b) - Float64(c * Float64(-1.125 * Float64(a / Float64(b * Float64(b * b)))))))) / c))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (1.0 / a) / (((-2.0 * (b / a)) + (c * ((1.5 / b) - (c * (-1.125 * (a / (b * (b * b)))))))) / c);
end

Wolfram

code[a_, b_, c_] := N[(N[(1.0 / a), $MachinePrecision] / N[(N[(N[(-2.0 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(1.5 / b), $MachinePrecision] - N[(c * N[(-1.125 * N[(a / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 56.6%

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

3. Simplified 56.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
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. 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 56.6%

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

7. Taylor expanded in c around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, a\right), \color{blue}{\left(\frac{-2 \cdot \frac{b}{a} + c \cdot \left(-1 \cdot \left(c \cdot \left(\frac{-9}{4} \cdot \frac{a}{{b}^{3}} + \frac{9}{8} \cdot \frac{a}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}{c}\right)}\right) \]
8. Step-by-step derivation

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

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

9. Simplified 86.6%

   \[\leadsto \frac{\frac{1}{a}}{\color{blue}{\frac{\frac{b}{a} \cdot -2 + c \cdot \left(\frac{1.5}{b} - c \cdot \left(\frac{a}{b \cdot \left(b \cdot b\right)} \cdot -1.125\right)\right)}{c}}} \]

10. Final simplification 86.6%

    \[\leadsto \frac{\frac{1}{a}}{\frac{-2 \cdot \frac{b}{a} + c \cdot \left(\frac{1.5}{b} - c \cdot \left(-1.125 \cdot \frac{a}{b \cdot \left(b \cdot b\right)}\right)\right)}{c}} \]
11. Add Preprocessing

Alternative 13: 82.4% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 56.6%

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

3. Simplified 56.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
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. 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 56.6%

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

7. Taylor expanded in c around 0

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

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

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

   2. metadata-eval N/A

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

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

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

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

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

   5. *-commutative N/A

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

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

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

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

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

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

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

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

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

   12. *-lowering-*.f64 80.3%

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

9. Simplified 80.3%

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

10. Final simplification 80.3%

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

Alternative 14: 82.4% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 56.6%

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

3. Simplified 56.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
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. 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 56.6%

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

7. Taylor expanded in b around inf

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

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

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

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

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

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

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

   6. unpow2 N/A

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

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

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 80.3%

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

9. Simplified 80.3%

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

Alternative 15: 82.0% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 56.6%

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

3. Simplified 56.6%

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

5. Taylor expanded in b around inf

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

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

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

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

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

   3. *-commutative N/A

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

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

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

   5. associate-*r/ N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. unpow2 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

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

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 79.9%

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

7. Simplified 79.9%

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

Alternative 16: 81.8% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 56.6%

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

3. Simplified 56.6%

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

5. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

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

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

   7. +-commutative N/A

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

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

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

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

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

   14. associate-*r/ N/A

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

   15. associate-*l/ N/A

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

   16. associate-*r* N/A

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

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

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

7. Simplified 79.7%

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

8. Final simplification 79.7%

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

Alternative 17: 64.8% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 56.6%

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

3. Simplified 56.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
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 63.0%

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

7. Simplified 63.0%

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

Alternative 18: 64.7% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 56.6%

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

3. Simplified 56.6%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{3 \cdot a}} \]
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 63.0%

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

7. Simplified 63.0%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
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 63.0%

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

9. Applied egg-rr 63.0%

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

10. Final simplification 63.0%

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

Reproduce

herbie shell --seed 2024164 
(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)))