Quadratic roots, medium range

Percentage Accurate: 31.5% → 95.8%

Time: 22.2s

Alternatives: 9

Speedup: 23.2×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.5% 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: 95.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	return (0.5 * c) / ((b * -0.5) + (c * ((c * (((0.5 * (a * a)) / t_0) + (c * ((a * ((((-0.5 * (a * a)) / t_0) / (b * b)) - ((a * a) / t_1))) - (((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a))))) + ((0.5 * a) / 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 = (0.5d0 * c) / ((b * (-0.5d0)) + (c * ((c * (((0.5d0 * (a * a)) / t_0) + (c * ((a * (((((-0.5d0) * (a * a)) / t_0) / (b * b)) - ((a * a) / t_1))) - (((b / ((b * t_1) / (a * (a * (a * a))))) * (-2.5d0)) / a))))) + ((0.5d0 * a) / 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 (0.5 * c) / ((b * -0.5) + (c * ((c * (((0.5 * (a * a)) / t_0) + (c * ((a * ((((-0.5 * (a * a)) / t_0) / (b * b)) - ((a * a) / t_1))) - (((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a))))) + ((0.5 * a) / b))));
}

Python

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

Julia

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

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = (b * b) * t_0;
	tmp = (0.5 * c) / ((b * -0.5) + (c * ((c * (((0.5 * (a * a)) / t_0) + (c * ((a * ((((-0.5 * (a * a)) / t_0) / (b * b)) - ((a * a) / t_1))) - (((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a))))) + ((0.5 * a) / 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[(N[(0.5 * c), $MachinePrecision] / N[(N[(b * -0.5), $MachinePrecision] + N[(c * N[(N[(c * N[(N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(c * N[(N[(a * N[(N[(N[(N[(-0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 30.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.7%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 30.7%

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

6. Applied egg-rr 30.7%

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

7. Taylor expanded in c around 0

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

9. Simplified 95.6%

   \[\leadsto \frac{0.5}{\color{blue}{\frac{b \cdot -0.5 + c \cdot \left(c \cdot \left(\left(0 - c \cdot \left(\left(\frac{-0.125 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)\right)}{a} + \frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)}{b \cdot b}\right)\right) + \frac{0.5 \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right) + \frac{a \cdot 0.5}{b}\right)}{c}}} \]

11. Applied egg-rr 95.9%

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

12. Final simplification 95.9%

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

Alternative 2: 95.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	return 0.5 / (((b * -0.5) + (c * ((c * (((0.5 * (a * a)) / t_0) + (c * ((a * ((((-0.5 * (a * a)) / t_0) / (b * b)) - ((a * a) / t_1))) - (((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a))))) + ((0.5 * a) / 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
    real(8) :: t_1
    t_0 = b * (b * b)
    t_1 = (b * b) * t_0
    code = 0.5d0 / (((b * (-0.5d0)) + (c * ((c * (((0.5d0 * (a * a)) / t_0) + (c * ((a * (((((-0.5d0) * (a * a)) / t_0) / (b * b)) - ((a * a) / t_1))) - (((b / ((b * t_1) / (a * (a * (a * a))))) * (-2.5d0)) / a))))) + ((0.5d0 * a) / b)))) / c)
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 0.5 / (((b * -0.5) + (c * ((c * (((0.5 * (a * a)) / t_0) + (c * ((a * ((((-0.5 * (a * a)) / t_0) / (b * b)) - ((a * a) / t_1))) - (((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a))))) + ((0.5 * a) / b)))) / c);
}

Python

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

Julia

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

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = (b * b) * t_0;
	tmp = 0.5 / (((b * -0.5) + (c * ((c * (((0.5 * (a * a)) / t_0) + (c * ((a * ((((-0.5 * (a * a)) / t_0) / (b * b)) - ((a * a) / t_1))) - (((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a))))) + ((0.5 * a) / b)))) / c);
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[(0.5 / N[(N[(N[(b * -0.5), $MachinePrecision] + N[(c * N[(N[(c * N[(N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(c * N[(N[(a * N[(N[(N[(N[(-0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 30.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.7%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 30.7%

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

6. Applied egg-rr 30.7%

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

7. Taylor expanded in c around 0

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

9. Simplified 95.6%

   \[\leadsto \frac{0.5}{\color{blue}{\frac{b \cdot -0.5 + c \cdot \left(c \cdot \left(\left(0 - c \cdot \left(\left(\frac{-0.125 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)\right)}{a} + \frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)}{b \cdot b}\right)\right) + \frac{0.5 \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right) + \frac{a \cdot 0.5}{b}\right)}{c}}} \]

11. Applied egg-rr 95.6%

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

12. Final simplification 95.6%

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

Alternative 3: 95.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = (b * b) * t_0;
	return c * (0.5 / ((b * -0.5) + (c * ((c * (((0.5 * (a * a)) / t_0) + (c * ((a * ((((-0.5 * (a * a)) / t_0) / (b * b)) - ((a * a) / t_1))) - (((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a))))) + ((0.5 * a) / 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 = c * (0.5d0 / ((b * (-0.5d0)) + (c * ((c * (((0.5d0 * (a * a)) / t_0) + (c * ((a * (((((-0.5d0) * (a * a)) / t_0) / (b * b)) - ((a * a) / t_1))) - (((b / ((b * t_1) / (a * (a * (a * a))))) * (-2.5d0)) / a))))) + ((0.5d0 * a) / 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 c * (0.5 / ((b * -0.5) + (c * ((c * (((0.5 * (a * a)) / t_0) + (c * ((a * ((((-0.5 * (a * a)) / t_0) / (b * b)) - ((a * a) / t_1))) - (((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a))))) + ((0.5 * a) / b)))));
}

Python

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

Julia

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

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = (b * b) * t_0;
	tmp = c * (0.5 / ((b * -0.5) + (c * ((c * (((0.5 * (a * a)) / t_0) + (c * ((a * ((((-0.5 * (a * a)) / t_0) / (b * b)) - ((a * a) / t_1))) - (((b / ((b * t_1) / (a * (a * (a * a))))) * -2.5) / a))))) + ((0.5 * a) / 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[(c * N[(0.5 / N[(N[(b * -0.5), $MachinePrecision] + N[(c * N[(N[(c * N[(N[(N[(0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(c * N[(N[(a * N[(N[(N[(N[(-0.5 * N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 30.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.7%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 30.7%

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

6. Applied egg-rr 30.7%

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

7. Taylor expanded in c around 0

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

9. Simplified 95.6%

   \[\leadsto \frac{0.5}{\color{blue}{\frac{b \cdot -0.5 + c \cdot \left(c \cdot \left(\left(0 - c \cdot \left(\left(\frac{-0.125 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 20\right)\right)}{a} + \frac{a \cdot \left(a \cdot a\right)}{{b}^{5}}\right) - \frac{a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot -0.5\right)}{b \cdot b}\right)\right) + \frac{0.5 \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot b\right)}\right) + \frac{a \cdot 0.5}{b}\right)}{c}}} \]

11. Applied egg-rr 95.5%

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

12. Final simplification 95.5%

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

Alternative 4: 94.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.7%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 30.7%

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

6. Applied egg-rr 30.7%

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

7. Taylor expanded in c around 0

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

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

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

9. Simplified 94.1%

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

    1. associate-/r/ N/A

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

    2. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

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

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

    8. +-commutative N/A

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

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

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

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

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

    11. *-commutative N/A

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

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

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

    13. associate-*r/ N/A

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

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

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

11. Applied egg-rr 94.4%

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

Alternative 5: 94.1% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.7%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 30.7%

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

6. Applied egg-rr 30.7%

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

7. Taylor expanded in c around 0

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

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

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

9. Simplified 94.1%

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

10. Taylor expanded in b around inf

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

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

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

    2. distribute-lft-out N/A

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

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

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

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

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

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

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

    6. *-commutative N/A

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

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

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

    8. unpow2 N/A

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

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

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

    10. unpow2 N/A

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

    11. *-lowering-*.f64 94.1%

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

12. Simplified 94.1%

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

Alternative 6: 94.1% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.7%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 30.7%

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

6. Applied egg-rr 30.7%

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

7. Taylor expanded in a around 0

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

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

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

   2. associate-*r/ N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

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

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

9. Simplified 94.1%

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

Alternative 7: 91.0% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.7%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 30.7%

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

6. Applied egg-rr 30.7%

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

7. Taylor expanded in c around 0

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

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

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

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

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 91.1%

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

9. Simplified 91.1%

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

Alternative 8: 91.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.7%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 30.7%

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

6. Applied egg-rr 30.7%

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

7. Taylor expanded in a around 0

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

   1. metadata-eval N/A

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

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

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

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

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

   4. associate-*r/ N/A

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. metadata-eval N/A

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

   10. associate-*r/ N/A

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

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

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

   12. *-commutative N/A

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

   13. *-lowering-*.f64 91.0%

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

9. Simplified 91.0%

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

10. Final simplification 91.0%

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

Alternative 9: 81.3% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{c}{0 - b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c (- 0.0 b)))

C

double code(double a, double b, double c) {
	return c / (0.0 - b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / (0.0d0 - b)
end function

Java

public static double code(double a, double b, double c) {
	return c / (0.0 - b);
}

Python

def code(a, b, c):
	return c / (0.0 - b)

Julia

function code(a, b, c)
	return Float64(c / Float64(0.0 - b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / (0.0 - b);
end

Wolfram

code[a_, b_, c_] := N[(c / N[(0.0 - b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 30.7%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}} \]
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 82.0%

      \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

7. Simplified 82.0%

   \[\leadsto \color{blue}{0 - \frac{c}{b}} \]
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 82.0%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(c, b\right)\right) \]

9. Applied egg-rr 82.0%

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

10. Final simplification 82.0%

    \[\leadsto \frac{c}{0 - b} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))