Quadratic roots, narrow range

Percentage Accurate: 55.2% → 92.2%

Time: 21.5s

Alternatives: 13

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

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot t\_0\\ \mathbf{if}\;b \leq 0.0035:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.5}{a}, \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\frac{a}{0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{t\_0} + \left(\frac{\frac{b}{\frac{b \cdot t\_1}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{t\_1} - \frac{-0.5 \cdot \left(a \cdot a\right)}{t\_1}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* (* b b) t_0)))
   (if (<= b 0.0035)
     (fma b (/ -0.5 a) (/ (sqrt (+ (* b b) (* a (* c -4.0)))) (/ a 0.5)))
     (/
      1.0
      (/
       1.0
       (/
        c
        (-
         (*
          c
          (+
           (/ a b)
           (*
            c
            (+
             (/ (* a a) t_0)
             (*
              (+
               (/ (* (/ b (/ (* b t_1) (* a (* a (* a a))))) -2.5) a)
               (* a (- (/ (* a a) t_1) (/ (* -0.5 (* a a)) t_1))))
              (* c -2.0))))))
         b)))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	double tmp;
	if (b <= 0.0035) {
		tmp = fma(b, (-0.5 / a), (sqrt(((b * b) + (a * (c * -4.0)))) / (a / 0.5)));
	} else {
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(b * b) * t_0)
	tmp = 0.0
	if (b <= 0.0035)
		tmp = fma(b, Float64(-0.5 / a), Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) / Float64(a / 0.5)));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(c / Float64(Float64(c * Float64(Float64(a / b) + Float64(c * Float64(Float64(Float64(a * a) / t_0) + Float64(Float64(Float64(Float64(Float64(b / Float64(Float64(b * t_1) / Float64(a * Float64(a * Float64(a * a))))) * -2.5) / a) + Float64(a * Float64(Float64(Float64(a * a) / t_1) - Float64(Float64(-0.5 * Float64(a * a)) / t_1)))) * Float64(c * -2.0)))))) - b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[b, 0.0035], N[(b * N[(-0.5 / a), $MachinePrecision] + N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(c / N[(N[(c * N[(N[(a / b), $MachinePrecision] + N[(c * N[(N[(N[(a * a), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[(N[(N[(b / N[(N[(b * t$95$1), $MachinePrecision] / N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.5), $MachinePrecision] / a), $MachinePrecision] + N[(a * N[(N[(N[(a * a), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(N[(-0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00350000000000000007

   1. Initial program 89.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

         \[\leadsto \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2} + \color{blue}{\left(\mathsf{neg}\left(\frac{b}{a \cdot 2}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{b}{a \cdot 2}\right)\right) + \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2}} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \frac{b}{\mathsf{neg}\left(a \cdot 2\right)} + \frac{\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}{a \cdot 2} \]

      5. div-inv N/A

         \[\leadsto b \cdot \frac{1}{\mathsf{neg}\left(a \cdot 2\right)} + \frac{\color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}}{a \cdot 2} \]

      6. fma-define N/A

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

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

         \[\leadsto \mathsf{fma.f64}\left(b, \color{blue}{\left(\frac{1}{\mathsf{neg}\left(a \cdot 2\right)}\right)}, \left(\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\right)\right) \]

      8. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{fma.f64}\left(b, \left(\mathsf{neg}\left(\frac{1}{a \cdot 2}\right)\right), \left(\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma.f64}\left(b, \left(\mathsf{neg}\left(\frac{1}{2 \cdot a}\right)\right), \left(\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\right)\right) \]

      10. associate-/r* N/A

          \[\leadsto \mathsf{fma.f64}\left(b, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{a}\right)\right), \left(\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma.f64}\left(b, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{a}\right)\right), \left(\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\right)\right) \]

      12. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma.f64}\left(b, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{a}}\right), \left(\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{a \cdot 2}\right)\right) \]

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

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

      14. metadata-eval N/A

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

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

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

   6. Applied egg-rr 89.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{-0.5}{a}, \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\frac{a}{0.5}}\right)} \]

   if 0.00350000000000000007 < b

   1. Initial program 54.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-sub N/A

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

      2. associate-/l/ N/A

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

      3. clear-num N/A

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

      4. frac-sub N/A

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

      5. clear-num N/A

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

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

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

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

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

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

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

   6. Applied egg-rr 53.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{\frac{b}{\frac{a}{0.5}}}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \frac{a}{b} - a}}} \]

   7. Taylor expanded in c around 0

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

   9. Simplified 93.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{c \cdot \left(\frac{a}{b} + c \cdot \left(-2 \cdot \left(c \cdot \left(\left(\frac{-0.125 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)\right)}{a} + \frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)}{b \cdot b}\right) + \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right)\right) - b}{c}}} \]

   11. Applied egg-rr 93.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} + \left(\frac{\frac{b}{\frac{\left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{\left(a \cdot a\right) \cdot -0.5}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0035:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.5}{a}, \frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}}{\frac{a}{0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} + \left(\frac{\frac{b}{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{-0.5 \cdot \left(a \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot t\_0\\ \mathbf{if}\;b \leq 0.0033:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4 + \frac{b \cdot b}{c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{t\_0} + \left(\frac{\frac{b}{\frac{b \cdot t\_1}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{t\_1} - \frac{-0.5 \cdot \left(a \cdot a\right)}{t\_1}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* (* b b) t_0)))
   (if (<= b 0.0033)
     (/ (- (sqrt (* c (+ (* a -4.0) (/ (* b b) c)))) b) (* a 2.0))
     (/
      1.0
      (/
       1.0
       (/
        c
        (-
         (*
          c
          (+
           (/ a b)
           (*
            c
            (+
             (/ (* a a) t_0)
             (*
              (+
               (/ (* (/ b (/ (* b t_1) (* a (* a (* a a))))) -2.5) a)
               (* a (- (/ (* a a) t_1) (/ (* -0.5 (* a a)) t_1))))
              (* c -2.0))))))
         b)))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	double tmp;
	if (b <= 0.0033) {
		tmp = (sqrt((c * ((a * -4.0) + ((b * b) / c)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b * (b * b)
    t_1 = (b * b) * t_0
    if (b <= 0.0033d0) then
        tmp = (sqrt((c * ((a * (-4.0d0)) + ((b * b) / c)))) - b) / (a * 2.0d0)
    else
        tmp = 1.0d0 / (1.0d0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * (-2.5d0)) / a) + (a * (((a * a) / t_1) - (((-0.5d0) * (a * a)) / t_1)))) * (c * (-2.0d0))))))) - b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	double tmp;
	if (b <= 0.0033) {
		tmp = (Math.sqrt((c * ((a * -4.0) + ((b * b) / c)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	t_1 = (b * b) * t_0
	tmp = 0
	if b <= 0.0033:
		tmp = (math.sqrt((c * ((a * -4.0) + ((b * b) / c)))) - b) / (a * 2.0)
	else:
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(b * b) * t_0)
	tmp = 0.0
	if (b <= 0.0033)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(Float64(a * -4.0) + Float64(Float64(b * b) / c)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(c / Float64(Float64(c * Float64(Float64(a / b) + Float64(c * Float64(Float64(Float64(a * a) / t_0) + Float64(Float64(Float64(Float64(Float64(b / Float64(Float64(b * t_1) / Float64(a * Float64(a * Float64(a * a))))) * -2.5) / a) + Float64(a * Float64(Float64(Float64(a * a) / t_1) - Float64(Float64(-0.5 * Float64(a * a)) / t_1)))) * Float64(c * -2.0)))))) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = (b * b) * t_0;
	tmp = 0.0;
	if (b <= 0.0033)
		tmp = (sqrt((c * ((a * -4.0) + ((b * b) / c)))) - b) / (a * 2.0);
	else
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[b, 0.0033], N[(N[(N[Sqrt[N[(c * N[(N[(a * -4.0), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(c / N[(N[(c * N[(N[(a / b), $MachinePrecision] + N[(c * N[(N[(N[(a * a), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[(N[(N[(b / N[(N[(b * t$95$1), $MachinePrecision] / N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.5), $MachinePrecision] / a), $MachinePrecision] + N[(a * N[(N[(N[(a * a), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(N[(-0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0033

   1. Initial program 89.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in c around inf

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

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

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

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

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

      3. *-commutative N/A

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, -4\right), \mathsf{/.f64}\left(\left({b}^{2}\right), c\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, -4\right), \mathsf{/.f64}\left(\left(b \cdot b\right), c\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      7. *-lowering-*.f64 89.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), c\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   7. Simplified 89.3%

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

   if 0.0033 < b

   1. Initial program 54.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-sub N/A

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

      2. associate-/l/ N/A

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

      3. clear-num N/A

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

      4. frac-sub N/A

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

      5. clear-num N/A

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

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

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

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

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

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

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

   6. Applied egg-rr 53.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{\frac{b}{\frac{a}{0.5}}}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \frac{a}{b} - a}}} \]

   7. Taylor expanded in c around 0

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

   9. Simplified 93.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{c \cdot \left(\frac{a}{b} + c \cdot \left(-2 \cdot \left(c \cdot \left(\left(\frac{-0.125 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)\right)}{a} + \frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)}{b \cdot b}\right) + \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right)\right) - b}{c}}} \]

   11. Applied egg-rr 93.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} + \left(\frac{\frac{b}{\frac{\left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{\left(a \cdot a\right) \cdot -0.5}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0033:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4 + \frac{b \cdot b}{c}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} + \left(\frac{\frac{b}{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{-0.5 \cdot \left(a \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot t\_0\\ \mathbf{if}\;b \leq 0.0032:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4 + \frac{b \cdot b}{a}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{t\_0} + \left(\frac{\frac{b}{\frac{b \cdot t\_1}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{t\_1} - \frac{-0.5 \cdot \left(a \cdot a\right)}{t\_1}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* (* b b) t_0)))
   (if (<= b 0.0032)
     (/ (- (sqrt (* a (+ (* c -4.0) (/ (* b b) a)))) b) (* a 2.0))
     (/
      1.0
      (/
       1.0
       (/
        c
        (-
         (*
          c
          (+
           (/ a b)
           (*
            c
            (+
             (/ (* a a) t_0)
             (*
              (+
               (/ (* (/ b (/ (* b t_1) (* a (* a (* a a))))) -2.5) a)
               (* a (- (/ (* a a) t_1) (/ (* -0.5 (* a a)) t_1))))
              (* c -2.0))))))
         b)))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	double tmp;
	if (b <= 0.0032) {
		tmp = (sqrt((a * ((c * -4.0) + ((b * b) / a)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b * (b * b)
    t_1 = (b * b) * t_0
    if (b <= 0.0032d0) then
        tmp = (sqrt((a * ((c * (-4.0d0)) + ((b * b) / a)))) - b) / (a * 2.0d0)
    else
        tmp = 1.0d0 / (1.0d0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * (-2.5d0)) / a) + (a * (((a * a) / t_1) - (((-0.5d0) * (a * a)) / t_1)))) * (c * (-2.0d0))))))) - b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	double tmp;
	if (b <= 0.0032) {
		tmp = (Math.sqrt((a * ((c * -4.0) + ((b * b) / a)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	t_1 = (b * b) * t_0
	tmp = 0
	if b <= 0.0032:
		tmp = (math.sqrt((a * ((c * -4.0) + ((b * b) / a)))) - b) / (a * 2.0)
	else:
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(b * b) * t_0)
	tmp = 0.0
	if (b <= 0.0032)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(Float64(c * -4.0) + Float64(Float64(b * b) / a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(c / Float64(Float64(c * Float64(Float64(a / b) + Float64(c * Float64(Float64(Float64(a * a) / t_0) + Float64(Float64(Float64(Float64(Float64(b / Float64(Float64(b * t_1) / Float64(a * Float64(a * Float64(a * a))))) * -2.5) / a) + Float64(a * Float64(Float64(Float64(a * a) / t_1) - Float64(Float64(-0.5 * Float64(a * a)) / t_1)))) * Float64(c * -2.0)))))) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = (b * b) * t_0;
	tmp = 0.0;
	if (b <= 0.0032)
		tmp = (sqrt((a * ((c * -4.0) + ((b * b) / a)))) - b) / (a * 2.0);
	else
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[b, 0.0032], N[(N[(N[Sqrt[N[(a * N[(N[(c * -4.0), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(c / N[(N[(c * N[(N[(a / b), $MachinePrecision] + N[(c * N[(N[(N[(a * a), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[(N[(N[(b / N[(N[(b * t$95$1), $MachinePrecision] / N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.5), $MachinePrecision] / a), $MachinePrecision] + N[(a * N[(N[(N[(a * a), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(N[(-0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00320000000000000015

   1. Initial program 89.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf

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

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

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

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

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

      3. *-commutative N/A

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(c, -4\right), \mathsf{/.f64}\left(\left({b}^{2}\right), a\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(c, -4\right), \mathsf{/.f64}\left(\left(b \cdot b\right), a\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

      7. *-lowering-*.f64 89.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(c, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, b\right), a\right)\right)\right)\right), b\right), \mathsf{*.f64}\left(a, 2\right)\right) \]

   7. Simplified 89.2%

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

   if 0.00320000000000000015 < b

   1. Initial program 54.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-sub N/A

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

      2. associate-/l/ N/A

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

      3. clear-num N/A

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

      4. frac-sub N/A

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

      5. clear-num N/A

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

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

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

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

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

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

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

   6. Applied egg-rr 53.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{\frac{b}{\frac{a}{0.5}}}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \frac{a}{b} - a}}} \]

   7. Taylor expanded in c around 0

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

   9. Simplified 93.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{c \cdot \left(\frac{a}{b} + c \cdot \left(-2 \cdot \left(c \cdot \left(\left(\frac{-0.125 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)\right)}{a} + \frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)}{b \cdot b}\right) + \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right)\right) - b}{c}}} \]

   11. Applied egg-rr 93.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} + \left(\frac{\frac{b}{\frac{\left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{\left(a \cdot a\right) \cdot -0.5}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0032:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4 + \frac{b \cdot b}{a}\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} + \left(\frac{\frac{b}{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{-0.5 \cdot \left(a \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot t\_0\\ \mathbf{if}\;b \leq 0.0034:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{t\_0} + \left(\frac{\frac{b}{\frac{b \cdot t\_1}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{t\_1} - \frac{-0.5 \cdot \left(a \cdot a\right)}{t\_1}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* (* b b) t_0)))
   (if (<= b 0.0034)
     (/ (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (* a 2.0))
     (/
      1.0
      (/
       1.0
       (/
        c
        (-
         (*
          c
          (+
           (/ a b)
           (*
            c
            (+
             (/ (* a a) t_0)
             (*
              (+
               (/ (* (/ b (/ (* b t_1) (* a (* a (* a a))))) -2.5) a)
               (* a (- (/ (* a a) t_1) (/ (* -0.5 (* a a)) t_1))))
              (* c -2.0))))))
         b)))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	double tmp;
	if (b <= 0.0034) {
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b * (b * b)
    t_1 = (b * b) * t_0
    if (b <= 0.0034d0) then
        tmp = (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b) / (a * 2.0d0)
    else
        tmp = 1.0d0 / (1.0d0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * (-2.5d0)) / a) + (a * (((a * a) / t_1) - (((-0.5d0) * (a * a)) / t_1)))) * (c * (-2.0d0))))))) - b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	double tmp;
	if (b <= 0.0034) {
		tmp = (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	t_1 = (b * b) * t_0
	tmp = 0
	if b <= 0.0034:
		tmp = (math.sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0)
	else:
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(b * b) * t_0)
	tmp = 0.0
	if (b <= 0.0034)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(c / Float64(Float64(c * Float64(Float64(a / b) + Float64(c * Float64(Float64(Float64(a * a) / t_0) + Float64(Float64(Float64(Float64(Float64(b / Float64(Float64(b * t_1) / Float64(a * Float64(a * Float64(a * a))))) * -2.5) / a) + Float64(a * Float64(Float64(Float64(a * a) / t_1) - Float64(Float64(-0.5 * Float64(a * a)) / t_1)))) * Float64(c * -2.0)))))) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = (b * b) * t_0;
	tmp = 0.0;
	if (b <= 0.0034)
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) / (a * 2.0);
	else
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[b, 0.0034], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(c / N[(N[(c * N[(N[(a / b), $MachinePrecision] + N[(c * N[(N[(N[(a * a), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[(N[(N[(b / N[(N[(b * t$95$1), $MachinePrecision] / N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.5), $MachinePrecision] / a), $MachinePrecision] + N[(a * N[(N[(N[(a * a), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(N[(-0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00339999999999999981

   1. Initial program 89.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing

   if 0.00339999999999999981 < b

   1. Initial program 54.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-sub N/A

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

      2. associate-/l/ N/A

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

      3. clear-num N/A

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

      4. frac-sub N/A

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

      5. clear-num N/A

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

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

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

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

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

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

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

   6. Applied egg-rr 53.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{\frac{b}{\frac{a}{0.5}}}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \frac{a}{b} - a}}} \]

   7. Taylor expanded in c around 0

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

   9. Simplified 93.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{c \cdot \left(\frac{a}{b} + c \cdot \left(-2 \cdot \left(c \cdot \left(\left(\frac{-0.125 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)\right)}{a} + \frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)}{b \cdot b}\right) + \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right)\right) - b}{c}}} \]

   11. Applied egg-rr 93.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} + \left(\frac{\frac{b}{\frac{\left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{\left(a \cdot a\right) \cdot -0.5}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0034:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} + \left(\frac{\frac{b}{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{-0.5 \cdot \left(a \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot t\_0\\ \mathbf{if}\;b \leq 0.0032:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{t\_0} + \left(\frac{\frac{b}{\frac{b \cdot t\_1}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{t\_1} - \frac{-0.5 \cdot \left(a \cdot a\right)}{t\_1}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* (* b b) t_0)))
   (if (<= b 0.0032)
     (/ 0.5 (/ a (- (sqrt (+ (* b b) (* a (* c -4.0)))) b)))
     (/
      1.0
      (/
       1.0
       (/
        c
        (-
         (*
          c
          (+
           (/ a b)
           (*
            c
            (+
             (/ (* a a) t_0)
             (*
              (+
               (/ (* (/ b (/ (* b t_1) (* a (* a (* a a))))) -2.5) a)
               (* a (- (/ (* a a) t_1) (/ (* -0.5 (* a a)) t_1))))
              (* c -2.0))))))
         b)))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	double tmp;
	if (b <= 0.0032) {
		tmp = 0.5 / (a / (sqrt(((b * b) + (a * (c * -4.0)))) - b));
	} else {
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b * (b * b)
    t_1 = (b * b) * t_0
    if (b <= 0.0032d0) then
        tmp = 0.5d0 / (a / (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b))
    else
        tmp = 1.0d0 / (1.0d0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * (-2.5d0)) / a) + (a * (((a * a) / t_1) - (((-0.5d0) * (a * a)) / t_1)))) * (c * (-2.0d0))))))) - b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	double tmp;
	if (b <= 0.0032) {
		tmp = 0.5 / (a / (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b));
	} else {
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	t_1 = (b * b) * t_0
	tmp = 0
	if b <= 0.0032:
		tmp = 0.5 / (a / (math.sqrt(((b * b) + (a * (c * -4.0)))) - b))
	else:
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(b * b) * t_0)
	tmp = 0.0
	if (b <= 0.0032)
		tmp = Float64(0.5 / Float64(a / Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b)));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(c / Float64(Float64(c * Float64(Float64(a / b) + Float64(c * Float64(Float64(Float64(a * a) / t_0) + Float64(Float64(Float64(Float64(Float64(b / Float64(Float64(b * t_1) / Float64(a * Float64(a * Float64(a * a))))) * -2.5) / a) + Float64(a * Float64(Float64(Float64(a * a) / t_1) - Float64(Float64(-0.5 * Float64(a * a)) / t_1)))) * Float64(c * -2.0)))))) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = (b * b) * t_0;
	tmp = 0.0;
	if (b <= 0.0032)
		tmp = 0.5 / (a / (sqrt(((b * b) + (a * (c * -4.0)))) - b));
	else
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[b, 0.0032], N[(0.5 / N[(a / N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(c / N[(N[(c * N[(N[(a / b), $MachinePrecision] + N[(c * N[(N[(N[(a * a), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[(N[(N[(b / N[(N[(b * t$95$1), $MachinePrecision] / N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.5), $MachinePrecision] / a), $MachinePrecision] + N[(a * N[(N[(N[(a * a), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(N[(-0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00320000000000000015

   1. Initial program 89.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 89.1%

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

   6. Applied egg-rr 89.1%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}} \]

   if 0.00320000000000000015 < b

   1. Initial program 54.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-sub N/A

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

      2. associate-/l/ N/A

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

      3. clear-num N/A

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

      4. frac-sub N/A

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

      5. clear-num N/A

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

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

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

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

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

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

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

   6. Applied egg-rr 53.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{\frac{b}{\frac{a}{0.5}}}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \frac{a}{b} - a}}} \]

   7. Taylor expanded in c around 0

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

   9. Simplified 93.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{c \cdot \left(\frac{a}{b} + c \cdot \left(-2 \cdot \left(c \cdot \left(\left(\frac{-0.125 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)\right)}{a} + \frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)}{b \cdot b}\right) + \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right)\right) - b}{c}}} \]

   11. Applied egg-rr 93.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} + \left(\frac{\frac{b}{\frac{\left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{\left(a \cdot a\right) \cdot -0.5}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0032:\\ \;\;\;\;\frac{0.5}{\frac{a}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} + \left(\frac{\frac{b}{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{-0.5 \cdot \left(a \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot t\_0\\ \mathbf{if}\;b \leq 0.0032:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{t\_0} + \left(\frac{\frac{b}{\frac{b \cdot t\_1}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{t\_1} - \frac{-0.5 \cdot \left(a \cdot a\right)}{t\_1}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* (* b b) t_0)))
   (if (<= b 0.0032)
     (* (- (sqrt (+ (* b b) (* a (* c -4.0)))) b) (/ 0.5 a))
     (/
      1.0
      (/
       1.0
       (/
        c
        (-
         (*
          c
          (+
           (/ a b)
           (*
            c
            (+
             (/ (* a a) t_0)
             (*
              (+
               (/ (* (/ b (/ (* b t_1) (* a (* a (* a a))))) -2.5) a)
               (* a (- (/ (* a a) t_1) (/ (* -0.5 (* a a)) t_1))))
              (* c -2.0))))))
         b)))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	double tmp;
	if (b <= 0.0032) {
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	} else {
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b * (b * b)
    t_1 = (b * b) * t_0
    if (b <= 0.0032d0) then
        tmp = (sqrt(((b * b) + (a * (c * (-4.0d0))))) - b) * (0.5d0 / a)
    else
        tmp = 1.0d0 / (1.0d0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * (-2.5d0)) / a) + (a * (((a * a) / t_1) - (((-0.5d0) * (a * a)) / t_1)))) * (c * (-2.0d0))))))) - b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	double tmp;
	if (b <= 0.0032) {
		tmp = (Math.sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	} else {
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	t_1 = (b * b) * t_0
	tmp = 0
	if b <= 0.0032:
		tmp = (math.sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a)
	else:
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(b * b) * t_0)
	tmp = 0.0
	if (b <= 0.0032)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -4.0)))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(c / Float64(Float64(c * Float64(Float64(a / b) + Float64(c * Float64(Float64(Float64(a * a) / t_0) + Float64(Float64(Float64(Float64(Float64(b / Float64(Float64(b * t_1) / Float64(a * Float64(a * Float64(a * a))))) * -2.5) / a) + Float64(a * Float64(Float64(Float64(a * a) / t_1) - Float64(Float64(-0.5 * Float64(a * a)) / t_1)))) * Float64(c * -2.0)))))) - b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = (b * b) * t_0;
	tmp = 0.0;
	if (b <= 0.0032)
		tmp = (sqrt(((b * b) + (a * (c * -4.0)))) - b) * (0.5 / a);
	else
		tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[b, 0.0032], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(c / N[(N[(c * N[(N[(a / b), $MachinePrecision] + N[(c * N[(N[(N[(a * a), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[(N[(N[(b / N[(N[(b * t$95$1), $MachinePrecision] / N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.5), $MachinePrecision] / a), $MachinePrecision] + N[(a * N[(N[(N[(a * a), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(N[(-0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00320000000000000015

   1. Initial program 89.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 89.0%

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

   6. Applied egg-rr 89.0%

      \[\leadsto \color{blue}{\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right)} \]

   if 0.00320000000000000015 < b

   1. Initial program 54.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

   3. Simplified 54.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-sub N/A

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

      2. associate-/l/ N/A

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

      3. clear-num N/A

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

      4. frac-sub N/A

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

      5. clear-num N/A

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

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

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

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

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. div-inv N/A

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

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

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

   6. Applied egg-rr 53.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{\frac{b}{\frac{a}{0.5}}}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \frac{a}{b} - a}}} \]

   7. Taylor expanded in c around 0

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

   9. Simplified 93.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{c \cdot \left(\frac{a}{b} + c \cdot \left(-2 \cdot \left(c \cdot \left(\left(\frac{-0.125 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)\right)}{a} + \frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)}{b \cdot b}\right) + \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right)\right) - b}{c}}} \]

   11. Applied egg-rr 93.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} + \left(\frac{\frac{b}{\frac{\left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{\left(a \cdot a\right) \cdot -0.5}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0032:\\ \;\;\;\;\left(\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} + \left(\frac{\frac{b}{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{-0.5 \cdot \left(a \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := \left(b \cdot b\right) \cdot t\_0\\ \frac{1}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{t\_0} + \left(\frac{\frac{b}{\frac{b \cdot t\_1}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{t\_1} - \frac{-0.5 \cdot \left(a \cdot a\right)}{t\_1}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* (* b b) t_0)))
   (/
    1.0
    (/
     1.0
     (/
      c
      (-
       (*
        c
        (+
         (/ a b)
         (*
          c
          (+
           (/ (* a a) t_0)
           (*
            (+
             (/ (* (/ b (/ (* b t_1) (* a (* a (* a a))))) -2.5) a)
             (* a (- (/ (* a a) t_1) (/ (* -0.5 (* a a)) t_1))))
            (* c -2.0))))))
       b))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	return 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    t_0 = b * (b * b)
    t_1 = (b * b) * t_0
    code = 1.0d0 / (1.0d0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * (-2.5d0)) / a) + (a * (((a * a) / t_1) - (((-0.5d0) * (a * a)) / t_1)))) * (c * (-2.0d0))))))) - b)))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	return 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	t_1 = (b * b) * t_0
	return 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)))

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(Float64(b * b) * t_0)
	return Float64(1.0 / Float64(1.0 / Float64(c / Float64(Float64(c * Float64(Float64(a / b) + Float64(c * Float64(Float64(Float64(a * a) / t_0) + Float64(Float64(Float64(Float64(Float64(b / Float64(Float64(b * t_1) / Float64(a * Float64(a * Float64(a * a))))) * -2.5) / a) + Float64(a * Float64(Float64(Float64(a * a) / t_1) - Float64(Float64(-0.5 * Float64(a * a)) / t_1)))) * Float64(c * -2.0)))))) - b))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = (b * b) * t_0;
	tmp = 1.0 / (1.0 / (c / ((c * ((a / b) + (c * (((a * a) / t_0) + (((((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a) + (a * (((a * a) / t_1) - ((-0.5 * (a * a)) / t_1)))) * (c * -2.0)))))) - b)));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(1.0 / N[(1.0 / N[(c / N[(N[(c * N[(N[(a / b), $MachinePrecision] + N[(c * N[(N[(N[(a * a), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[(N[(N[(b / N[(N[(b * t$95$1), $MachinePrecision] / N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.5), $MachinePrecision] / a), $MachinePrecision] + N[(a * N[(N[(N[(a * a), $MachinePrecision] / t$95$1), $MachinePrecision] - N[(N[(-0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 56.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 56.8%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-sub N/A

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

   2. associate-/l/ N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. clear-num N/A

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

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

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

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

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

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

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

6. Applied egg-rr 55.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{\frac{b}{\frac{a}{0.5}}}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \frac{a}{b} - a}}} \]

7. Taylor expanded in c around 0

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

9. Simplified 91.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{c \cdot \left(\frac{a}{b} + c \cdot \left(-2 \cdot \left(c \cdot \left(\left(\frac{-0.125 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)\right)}{a} + \frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)}{b \cdot b}\right) + \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right)\right) - b}{c}}} \]

11. Applied egg-rr 91.7%

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} + \left(\frac{\frac{b}{\frac{\left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot b}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{\left(a \cdot a\right) \cdot -0.5}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}}} \]

12. Final simplification 91.7%

    \[\leadsto \frac{1}{\frac{1}{\frac{c}{c \cdot \left(\frac{a}{b} + c \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} + \left(\frac{\frac{b}{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}{a \cdot \left(a \cdot \left(a \cdot a\right)\right)}} \cdot -2.5}{a} + a \cdot \left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)} - \frac{-0.5 \cdot \left(a \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right) \cdot \left(c \cdot -2\right)\right)\right) - b}}} \]
13. Add Preprocessing

Alternative 8: 91.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(a \cdot a\right)\\ \frac{1}{\frac{c \cdot \left(\frac{a}{b} + c \cdot \frac{a \cdot a + \frac{\left(c \cdot -2\right) \cdot \left(t\_0 \cdot -1.5 + 0.5 \cdot t\_0\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right) - b}{c}} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* a a))))
   (/
    1.0
    (/
     (-
      (*
       c
       (+
        (/ a b)
        (*
         c
         (/
          (+ (* a a) (/ (* (* c -2.0) (+ (* t_0 -1.5) (* 0.5 t_0))) (* b b)))
          (* b (* b b))))))
      b)
     c))))

C

double code(double a, double b, double c) {
	double t_0 = a * (a * a);
	return 1.0 / (((c * ((a / b) + (c * (((a * a) + (((c * -2.0) * ((t_0 * -1.5) + (0.5 * t_0))) / (b * b))) / (b * (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
    real(8) :: t_0
    t_0 = a * (a * a)
    code = 1.0d0 / (((c * ((a / b) + (c * (((a * a) + (((c * (-2.0d0)) * ((t_0 * (-1.5d0)) + (0.5d0 * t_0))) / (b * b))) / (b * (b * b)))))) - b) / c)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (a * a);
	return 1.0 / (((c * ((a / b) + (c * (((a * a) + (((c * -2.0) * ((t_0 * -1.5) + (0.5 * t_0))) / (b * b))) / (b * (b * b)))))) - b) / c);
}

Python

def code(a, b, c):
	t_0 = a * (a * a)
	return 1.0 / (((c * ((a / b) + (c * (((a * a) + (((c * -2.0) * ((t_0 * -1.5) + (0.5 * t_0))) / (b * b))) / (b * (b * b)))))) - b) / c)

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(a * a))
	return Float64(1.0 / Float64(Float64(Float64(c * Float64(Float64(a / b) + Float64(c * Float64(Float64(Float64(a * a) + Float64(Float64(Float64(c * -2.0) * Float64(Float64(t_0 * -1.5) + Float64(0.5 * t_0))) / Float64(b * b))) / Float64(b * Float64(b * b)))))) - b) / c))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = a * (a * a);
	tmp = 1.0 / (((c * ((a / b) + (c * (((a * a) + (((c * -2.0) * ((t_0 * -1.5) + (0.5 * t_0))) / (b * b))) / (b * (b * b)))))) - b) / c);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(N[(N[(c * N[(N[(a / b), $MachinePrecision] + N[(c * N[(N[(N[(a * a), $MachinePrecision] + N[(N[(N[(c * -2.0), $MachinePrecision] * N[(N[(t$95$0 * -1.5), $MachinePrecision] + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 56.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 56.8%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-sub N/A

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

   2. associate-/l/ N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. clear-num N/A

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

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

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

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

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

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

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

6. Applied egg-rr 55.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{\frac{b}{\frac{a}{0.5}}}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \frac{a}{b} - a}}} \]

7. Taylor expanded in c around 0

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

9. Simplified 91.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{c \cdot \left(\frac{a}{b} + c \cdot \left(-2 \cdot \left(c \cdot \left(\left(\frac{-0.125 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)\right)}{a} + \frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)}{b \cdot b}\right) + \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right)\right) - b}{c}}} \]

10. Taylor expanded in b around inf

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

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

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

12. Simplified 91.7%

    \[\leadsto \frac{1}{\frac{c \cdot \left(\frac{a}{b} + c \cdot \color{blue}{\frac{a \cdot a + \frac{\left(-2 \cdot c\right) \cdot \left(-1.5 \cdot \left(a \cdot \left(a \cdot a\right)\right) + 0.5 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}}\right) - b}{c}} \]

13. Final simplification 91.7%

    \[\leadsto \frac{1}{\frac{c \cdot \left(\frac{a}{b} + c \cdot \frac{a \cdot a + \frac{\left(c \cdot -2\right) \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -1.5 + 0.5 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot b}}{b \cdot \left(b \cdot b\right)}\right) - b}{c}} \]
14. Add Preprocessing

Alternative 9: 88.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 56.8%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-sub N/A

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

   2. associate-/l/ N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. clear-num N/A

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

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

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

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

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

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

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

6. Applied egg-rr 55.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{\frac{b}{\frac{a}{0.5}}}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \frac{a}{b} - a}}} \]

7. Taylor expanded in c around 0

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

9. Simplified 91.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{c \cdot \left(\frac{a}{b} + c \cdot \left(-2 \cdot \left(c \cdot \left(\left(\frac{-0.125 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)\right)}{a} + \frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)}{b \cdot b}\right) + \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right)\right) - b}{c}}} \]

10. Taylor expanded in c around 0

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

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

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

    2. unpow2 N/A

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

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

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

    4. cube-mult N/A

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

    5. unpow2 N/A

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

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

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

    7. unpow2 N/A

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

    8. *-lowering-*.f64 88.6%

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

12. Simplified 88.6%

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

Alternative 10: 88.4% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 56.8%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-sub N/A

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

   2. associate-/l/ N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. clear-num N/A

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

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

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

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

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

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

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

6. Applied egg-rr 55.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{\frac{b}{\frac{a}{0.5}}}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \frac{a}{b} - a}}} \]

7. Taylor expanded in c around 0

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

9. Simplified 91.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{c \cdot \left(\frac{a}{b} + c \cdot \left(-2 \cdot \left(c \cdot \left(\left(\frac{-0.125 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)\right)}{a} + \frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)}{b \cdot b}\right) + \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)\right)\right) - b}{c}}} \]

10. Taylor expanded in a around 0

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

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

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

    2. associate-*r/ N/A

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

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

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

    4. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

    9. associate-/l* N/A

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

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

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

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

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

    12. cube-mult N/A

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

    13. unpow2 N/A

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

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

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

    15. unpow2 N/A

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

    16. *-lowering-*.f64 88.5%

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

12. Simplified 88.5%

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

13. Final simplification 88.5%

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

Alternative 11: 82.4% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 56.8%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-sub N/A

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

   2. associate-/l/ N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. clear-num N/A

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

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

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

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

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

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

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

6. Applied egg-rr 55.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{\frac{b}{\frac{a}{0.5}}}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \frac{a}{b} - a}}} \]

7. Taylor expanded in c around 0

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

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

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

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

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

   7. *-lowering-*.f64 81.9%

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

9. Simplified 81.9%

   \[\leadsto \frac{1}{\color{blue}{\frac{\frac{a \cdot c}{b} - b}{c}}} \]
10. Add Preprocessing

Alternative 12: 82.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{b} - \frac{b}{c}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 1.0 / ((a / 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 / b) - (b / c))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 56.8%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-sub N/A

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

   2. associate-/l/ N/A

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

   3. clear-num N/A

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

   4. frac-sub N/A

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

   5. clear-num N/A

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

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

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

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

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

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

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

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

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

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

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

6. Applied egg-rr 55.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{a}{\frac{b}{\frac{a}{0.5}}}}{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} \cdot \frac{a}{b} - a}}} \]

7. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. /-lowering-/.f64 81.8%

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

9. Simplified 81.8%

   \[\leadsto \frac{1}{\color{blue}{\frac{a}{b} - \frac{b}{c}}} \]
10. Add Preprocessing

Alternative 13: 64.6% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (- 0.0 (/ c b)))

C

double code(double a, double b, double c) {
	return 0.0 - (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 - (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 - (c / b);
}

Python

def code(a, b, c):
	return 0.0 - (c / b)

Julia

function code(a, b, c)
	return Float64(0.0 - Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 - (c / b);
end

Wolfram

code[a_, b_, c_] := N[(0.0 - N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\right)\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), \color{blue}{\left(2 \cdot a\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(\mathsf{neg}\left(b\right)\right)\right), \left(\color{blue}{2} \cdot a\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right), b\right), \left(\color{blue}{2} \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b \cdot b + \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(b \cdot b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(4 \cdot a\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(\left(a \cdot 4\right) \cdot c\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\mathsf{neg}\left(a \cdot \left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(4 \cdot c\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(\mathsf{neg}\left(c \cdot 4\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \left(c \cdot \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, \left(\mathsf{neg}\left(4\right)\right)\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(2 \cdot a\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -4\right)\right)\right)\right), b\right), \left(a \cdot \color{blue}{2}\right)\right) \]

3. Simplified 56.8%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{\frac{c}{b}} \]

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

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

   4. /-lowering-/.f64 63.0%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

7. Simplified 63.0%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
8. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

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

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{c}{b}\right)\right) \]

   3. /-lowering-/.f64 63.0%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(c, b\right)\right) \]

9. Applied egg-rr 63.0%

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

10. Final simplification 63.0%

    \[\leadsto 0 - \frac{c}{b} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))