Cubic critical, medium range

Percentage Accurate: 31.4% → 95.6%

Time: 24.3s

Alternatives: 10

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

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot 0.125\\ \frac{1}{\frac{b \cdot -2 + c \cdot \left(1.5 \cdot \frac{a}{b} + c \cdot \left(9 \cdot \left(t\_0 + c \cdot \left(\frac{-0.75 \cdot \left(a \cdot t\_0\right)}{b \cdot b} + \left(\frac{-0.1875 \cdot \left(a \cdot \left(a \cdot a\right)\right)}{{b}^{5}} + \frac{0.07407407407407407 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{a}\right)\right)\right)\right)\right)}{c}} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (/ (* a a) (* b (* b b))) 0.125)))
   (/
    1.0
    (/
     (+
      (* b -2.0)
      (*
       c
       (+
        (* 1.5 (/ a b))
        (*
         c
         (*
          9.0
          (+
           t_0
           (*
            c
            (+
             (/ (* -0.75 (* a t_0)) (* b b))
             (+
              (/ (* -0.1875 (* a (* a a))) (pow b 5.0))
              (/
               (*
                0.07407407407407407
                (* b (* (/ (pow a 4.0) (pow b 6.0)) 6.328125)))
               a))))))))))
     c))))

C

double code(double a, double b, double c) {
	double t_0 = ((a * a) / (b * (b * b))) * 0.125;
	return 1.0 / (((b * -2.0) + (c * ((1.5 * (a / b)) + (c * (9.0 * (t_0 + (c * (((-0.75 * (a * t_0)) / (b * b)) + (((-0.1875 * (a * (a * a))) / pow(b, 5.0)) + ((0.07407407407407407 * (b * ((pow(a, 4.0) / pow(b, 6.0)) * 6.328125))) / a)))))))))) / c);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = ((a * a) / (b * (b * b))) * 0.125d0
    code = 1.0d0 / (((b * (-2.0d0)) + (c * ((1.5d0 * (a / b)) + (c * (9.0d0 * (t_0 + (c * ((((-0.75d0) * (a * t_0)) / (b * b)) + ((((-0.1875d0) * (a * (a * a))) / (b ** 5.0d0)) + ((0.07407407407407407d0 * (b * (((a ** 4.0d0) / (b ** 6.0d0)) * 6.328125d0))) / a)))))))))) / c)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = ((a * a) / (b * (b * b))) * 0.125;
	return 1.0 / (((b * -2.0) + (c * ((1.5 * (a / b)) + (c * (9.0 * (t_0 + (c * (((-0.75 * (a * t_0)) / (b * b)) + (((-0.1875 * (a * (a * a))) / Math.pow(b, 5.0)) + ((0.07407407407407407 * (b * ((Math.pow(a, 4.0) / Math.pow(b, 6.0)) * 6.328125))) / a)))))))))) / c);
}

Python

def code(a, b, c):
	t_0 = ((a * a) / (b * (b * b))) * 0.125
	return 1.0 / (((b * -2.0) + (c * ((1.5 * (a / b)) + (c * (9.0 * (t_0 + (c * (((-0.75 * (a * t_0)) / (b * b)) + (((-0.1875 * (a * (a * a))) / math.pow(b, 5.0)) + ((0.07407407407407407 * (b * ((math.pow(a, 4.0) / math.pow(b, 6.0)) * 6.328125))) / a)))))))))) / c)

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(a * a) / Float64(b * Float64(b * b))) * 0.125)
	return Float64(1.0 / Float64(Float64(Float64(b * -2.0) + Float64(c * Float64(Float64(1.5 * Float64(a / b)) + Float64(c * Float64(9.0 * Float64(t_0 + Float64(c * Float64(Float64(Float64(-0.75 * Float64(a * t_0)) / Float64(b * b)) + Float64(Float64(Float64(-0.1875 * Float64(a * Float64(a * a))) / (b ^ 5.0)) + Float64(Float64(0.07407407407407407 * Float64(b * Float64(Float64((a ^ 4.0) / (b ^ 6.0)) * 6.328125))) / a)))))))))) / c))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = ((a * a) / (b * (b * b))) * 0.125;
	tmp = 1.0 / (((b * -2.0) + (c * ((1.5 * (a / b)) + (c * (9.0 * (t_0 + (c * (((-0.75 * (a * t_0)) / (b * b)) + (((-0.1875 * (a * (a * a))) / (b ^ 5.0)) + ((0.07407407407407407 * (b * (((a ^ 4.0) / (b ^ 6.0)) * 6.328125))) / a)))))))))) / c);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(a * a), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]}, N[(1.0 / N[(N[(N[(b * -2.0), $MachinePrecision] + N[(c * N[(N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(c * N[(9.0 * N[(t$95$0 + N[(c * N[(N[(N[(-0.75 * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.1875 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.07407407407407407 * N[(b * N[(N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 30.4%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 30.4%

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

3. Simplified 30.4%

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

6. Applied egg-rr 31.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -9}{\left(-\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot 3\right) + \frac{\frac{b}{\frac{0.3333333333333333}{a}}}{a}}}} \]

7. Taylor expanded in c around 0

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

9. Simplified 95.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot -2 + c \cdot \left(1.5 \cdot \frac{a}{b} + c \cdot \left(9 \cdot \left(c \cdot \left(\frac{-0.75 \cdot \left(a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot 0.125\right)\right)}{b \cdot b} + \left(\frac{-0.1875 \cdot \left(a \cdot \left(a \cdot a\right)\right)}{{b}^{5}} + \frac{0.07407407407407407 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{a}\right)\right) + \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot 0.125\right)\right)\right)}{c}}} \]

10. Final simplification 95.7%

    \[\leadsto \frac{1}{\frac{b \cdot -2 + c \cdot \left(1.5 \cdot \frac{a}{b} + c \cdot \left(9 \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot 0.125 + c \cdot \left(\frac{-0.75 \cdot \left(a \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot 0.125\right)\right)}{b \cdot b} + \left(\frac{-0.1875 \cdot \left(a \cdot \left(a \cdot a\right)\right)}{{b}^{5}} + \frac{0.07407407407407407 \cdot \left(b \cdot \left(\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{a}\right)\right)\right)\right)\right)}{c}} \]
11. Add Preprocessing

Alternative 2: 95.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.125 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}\\ \frac{1}{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(9 \cdot \left(t\_0 + a \cdot \left(\left(-0.75 \cdot \left(c \cdot \frac{t\_0}{b \cdot b}\right) + \frac{-0.1875 \cdot \left(c \cdot c\right)}{{b}^{5}}\right) + \frac{0.07407407407407407 \cdot \left(b \cdot \left(6.328125 \cdot \frac{{c}^{4}}{{b}^{6}}\right)\right)}{c \cdot c}\right)\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 0.125 (/ c (* b (* b b))))))
   (/
    1.0
    (+
     (/ (* b -2.0) c)
     (*
      a
      (+
       (/ 1.5 b)
       (*
        a
        (*
         9.0
         (+
          t_0
          (*
           a
           (+
            (+
             (* -0.75 (* c (/ t_0 (* b b))))
             (/ (* -0.1875 (* c c)) (pow b 5.0)))
            (/
             (*
              0.07407407407407407
              (* b (* 6.328125 (/ (pow c 4.0) (pow b 6.0)))))
             (* c c)))))))))))))

C

double code(double a, double b, double c) {
	double t_0 = 0.125 * (c / (b * (b * b)));
	return 1.0 / (((b * -2.0) / c) + (a * ((1.5 / b) + (a * (9.0 * (t_0 + (a * (((-0.75 * (c * (t_0 / (b * b)))) + ((-0.1875 * (c * c)) / pow(b, 5.0))) + ((0.07407407407407407 * (b * (6.328125 * (pow(c, 4.0) / pow(b, 6.0))))) / (c * c))))))))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = 0.125d0 * (c / (b * (b * b)))
    code = 1.0d0 / (((b * (-2.0d0)) / c) + (a * ((1.5d0 / b) + (a * (9.0d0 * (t_0 + (a * ((((-0.75d0) * (c * (t_0 / (b * b)))) + (((-0.1875d0) * (c * c)) / (b ** 5.0d0))) + ((0.07407407407407407d0 * (b * (6.328125d0 * ((c ** 4.0d0) / (b ** 6.0d0))))) / (c * c))))))))))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = 0.125 * (c / (b * (b * b)));
	return 1.0 / (((b * -2.0) / c) + (a * ((1.5 / b) + (a * (9.0 * (t_0 + (a * (((-0.75 * (c * (t_0 / (b * b)))) + ((-0.1875 * (c * c)) / Math.pow(b, 5.0))) + ((0.07407407407407407 * (b * (6.328125 * (Math.pow(c, 4.0) / Math.pow(b, 6.0))))) / (c * c))))))))));
}

Python

def code(a, b, c):
	t_0 = 0.125 * (c / (b * (b * b)))
	return 1.0 / (((b * -2.0) / c) + (a * ((1.5 / b) + (a * (9.0 * (t_0 + (a * (((-0.75 * (c * (t_0 / (b * b)))) + ((-0.1875 * (c * c)) / math.pow(b, 5.0))) + ((0.07407407407407407 * (b * (6.328125 * (math.pow(c, 4.0) / math.pow(b, 6.0))))) / (c * c))))))))))

Julia

function code(a, b, c)
	t_0 = Float64(0.125 * Float64(c / Float64(b * Float64(b * b))))
	return Float64(1.0 / Float64(Float64(Float64(b * -2.0) / c) + Float64(a * Float64(Float64(1.5 / b) + Float64(a * Float64(9.0 * Float64(t_0 + Float64(a * Float64(Float64(Float64(-0.75 * Float64(c * Float64(t_0 / Float64(b * b)))) + Float64(Float64(-0.1875 * Float64(c * c)) / (b ^ 5.0))) + Float64(Float64(0.07407407407407407 * Float64(b * Float64(6.328125 * Float64((c ^ 4.0) / (b ^ 6.0))))) / Float64(c * c)))))))))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = 0.125 * (c / (b * (b * b)));
	tmp = 1.0 / (((b * -2.0) / c) + (a * ((1.5 / b) + (a * (9.0 * (t_0 + (a * (((-0.75 * (c * (t_0 / (b * b)))) + ((-0.1875 * (c * c)) / (b ^ 5.0))) + ((0.07407407407407407 * (b * (6.328125 * ((c ^ 4.0) / (b ^ 6.0))))) / (c * c))))))))));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(0.125 * N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 / N[(N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision] + N[(a * N[(N[(1.5 / b), $MachinePrecision] + N[(a * N[(9.0 * N[(t$95$0 + N[(a * N[(N[(N[(-0.75 * N[(c * N[(t$95$0 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.1875 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.07407407407407407 * N[(b * N[(6.328125 * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 30.4%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 30.4%

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

3. Simplified 30.4%

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

6. Applied egg-rr 31.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -9}{\left(-\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot 3\right) + \frac{\frac{b}{\frac{0.3333333333333333}{a}}}{a}}}} \]

7. Taylor expanded in a around 0

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

9. Simplified 95.6%

   \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(9 \cdot \left(a \cdot \left(\left(-0.75 \cdot \left(c \cdot \frac{\frac{c}{b \cdot \left(b \cdot b\right)} \cdot 0.125}{b \cdot b}\right) + \frac{-0.1875 \cdot \left(c \cdot c\right)}{{b}^{5}}\right) + \frac{0.07407407407407407 \cdot \left(b \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right)\right)}{c \cdot c}\right) + \frac{c}{b \cdot \left(b \cdot b\right)} \cdot 0.125\right)\right)\right)}} \]

10. Final simplification 95.6%

    \[\leadsto \frac{1}{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + a \cdot \left(9 \cdot \left(0.125 \cdot \frac{c}{b \cdot \left(b \cdot b\right)} + a \cdot \left(\left(-0.75 \cdot \left(c \cdot \frac{0.125 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}}{b \cdot b}\right) + \frac{-0.1875 \cdot \left(c \cdot c\right)}{{b}^{5}}\right) + \frac{0.07407407407407407 \cdot \left(b \cdot \left(6.328125 \cdot \frac{{c}^{4}}{{b}^{6}}\right)\right)}{c \cdot c}\right)\right)\right)\right)} \]
11. Add Preprocessing

Alternative 3: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5}{b} + a \cdot \left(\left(c \cdot c\right) \cdot \frac{\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot -1.0546875\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot -0.5625 + \left(b \cdot b\right) \cdot -0.375\right)}{{b}^{7}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (/ (* c -0.5) b)
  (*
   a
   (*
    (* c c)
    (/
     (+
      (* (* c c) (* (* a a) -1.0546875))
      (* (* b b) (+ (* (* c a) -0.5625) (* (* b b) -0.375))))
     (pow b 7.0))))))

C

double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * ((c * c) * ((((c * c) * ((a * a) * -1.0546875)) + ((b * b) * (((c * a) * -0.5625) + ((b * b) * -0.375)))) / pow(b, 7.0))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((c * (-0.5d0)) / b) + (a * ((c * c) * ((((c * c) * ((a * a) * (-1.0546875d0))) + ((b * b) * (((c * a) * (-0.5625d0)) + ((b * b) * (-0.375d0))))) / (b ** 7.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * ((c * c) * ((((c * c) * ((a * a) * -1.0546875)) + ((b * b) * (((c * a) * -0.5625) + ((b * b) * -0.375)))) / Math.pow(b, 7.0))));
}

Python

def code(a, b, c):
	return ((c * -0.5) / b) + (a * ((c * c) * ((((c * c) * ((a * a) * -1.0546875)) + ((b * b) * (((c * a) * -0.5625) + ((b * b) * -0.375)))) / math.pow(b, 7.0))))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * -0.5) / b) + Float64(a * Float64(Float64(c * c) * Float64(Float64(Float64(Float64(c * c) * Float64(Float64(a * a) * -1.0546875)) + Float64(Float64(b * b) * Float64(Float64(Float64(c * a) * -0.5625) + Float64(Float64(b * b) * -0.375)))) / (b ^ 7.0)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * -0.5) / b) + (a * ((c * c) * ((((c * c) * ((a * a) * -1.0546875)) + ((b * b) * (((c * a) * -0.5625) + ((b * b) * -0.375)))) / (b ^ 7.0))));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision] + N[(a * N[(N[(c * c), $MachinePrecision] * N[(N[(N[(N[(c * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -1.0546875), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(N[(N[(c * a), $MachinePrecision] * -0.5625), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.4%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 30.4%

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

3. Simplified 30.4%

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

5. Taylor expanded in a around 0

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

7. Simplified 95.5%

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

8. Taylor expanded in c around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \color{blue}{\left({c}^{2} \cdot \left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)}\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left({c}^{2}\right), \color{blue}{\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)}\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\left(c \cdot c\right), \left(\color{blue}{c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right)} - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \left(\color{blue}{c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right)} - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{+.f64}\left(\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)\right)}\right)\right)\right)\right) \]

10. Simplified 95.5%

    \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot \left(\frac{-1.0546875 \cdot \left(c \cdot \left(a \cdot a\right)\right)}{{b}^{7}} + \frac{-0.5625 \cdot a}{{b}^{5}}\right) + \frac{-0.375}{b \cdot \left(b \cdot b\right)}\right)\right)} \]

11. Taylor expanded in b around 0

    \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \color{blue}{\left(\frac{\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot c\right) + \frac{-3}{8} \cdot {b}^{2}\right)}{{b}^{7}}\right)}\right)\right)\right) \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{/.f64}\left(\left(\frac{-135}{128} \cdot \left({a}^{2} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left(a \cdot c\right) + \frac{-3}{8} \cdot {b}^{2}\right)\right), \color{blue}{\left({b}^{7}\right)}\right)\right)\right)\right) \]

13. Simplified 95.5%

    \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\frac{\left(-1.0546875 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot -0.5625 + -0.375 \cdot \left(b \cdot b\right)\right)}{{b}^{7}}}\right) \]

14. Final simplification 95.5%

    \[\leadsto \frac{c \cdot -0.5}{b} + a \cdot \left(\left(c \cdot c\right) \cdot \frac{\left(c \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot -1.0546875\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot -0.5625 + \left(b \cdot b\right) \cdot -0.375\right)}{{b}^{7}}\right) \]
15. Add Preprocessing

Alternative 4: 94.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{b \cdot -2 + c \cdot \left(1.5 \cdot \frac{a}{b} + \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot 0.125\right) \cdot \left(c \cdot 9\right)\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (/
   (+
    (* b -2.0)
    (*
     c
     (+ (* 1.5 (/ a b)) (* (* (/ (* a a) (* b (* b b))) 0.125) (* c 9.0)))))
   c)))

C

double code(double a, double b, double c) {
	return 1.0 / (((b * -2.0) + (c * ((1.5 * (a / b)) + ((((a * a) / (b * (b * b))) * 0.125) * (c * 9.0))))) / c);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (((b * (-2.0d0)) + (c * ((1.5d0 * (a / b)) + ((((a * a) / (b * (b * b))) * 0.125d0) * (c * 9.0d0))))) / c)
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / (((b * -2.0) + (c * ((1.5 * (a / b)) + ((((a * a) / (b * (b * b))) * 0.125) * (c * 9.0))))) / c);
}

Python

def code(a, b, c):
	return 1.0 / (((b * -2.0) + (c * ((1.5 * (a / b)) + ((((a * a) / (b * (b * b))) * 0.125) * (c * 9.0))))) / c)

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(Float64(b * -2.0) + Float64(c * Float64(Float64(1.5 * Float64(a / b)) + Float64(Float64(Float64(Float64(a * a) / Float64(b * Float64(b * b))) * 0.125) * Float64(c * 9.0))))) / c))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / (((b * -2.0) + (c * ((1.5 * (a / b)) + ((((a * a) / (b * (b * b))) * 0.125) * (c * 9.0))))) / c);
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(N[(b * -2.0), $MachinePrecision] + N[(c * N[(N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(a * a), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(c * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.4%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 30.4%

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

3. Simplified 30.4%

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

6. Applied egg-rr 31.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -9}{\left(-\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot 3\right) + \frac{\frac{b}{\frac{0.3333333333333333}{a}}}{a}}}} \]

7. Taylor expanded in c around 0

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

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

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

9. Simplified 94.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot -2 + c \cdot \left(1.5 \cdot \frac{a}{b} + \left(9 \cdot c\right) \cdot \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot 0.125\right)\right)}{c}}} \]

10. Final simplification 94.1%

    \[\leadsto \frac{1}{\frac{b \cdot -2 + c \cdot \left(1.5 \cdot \frac{a}{b} + \left(\frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot 0.125\right) \cdot \left(c \cdot 9\right)\right)}{c}} \]
11. Add Preprocessing

Alternative 5: 94.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + \left(0.125 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}\right) \cdot \left(a \cdot 9\right)\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (+
   (/ (* b -2.0) c)
   (* a (+ (/ 1.5 b) (* (* 0.125 (/ c (* b (* b b)))) (* a 9.0)))))))

C

double code(double a, double b, double c) {
	return 1.0 / (((b * -2.0) / c) + (a * ((1.5 / b) + ((0.125 * (c / (b * (b * b)))) * (a * 9.0)))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (((b * (-2.0d0)) / c) + (a * ((1.5d0 / b) + ((0.125d0 * (c / (b * (b * b)))) * (a * 9.0d0)))))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / (((b * -2.0) / c) + (a * ((1.5 / b) + ((0.125 * (c / (b * (b * b)))) * (a * 9.0)))));
}

Python

def code(a, b, c):
	return 1.0 / (((b * -2.0) / c) + (a * ((1.5 / b) + ((0.125 * (c / (b * (b * b)))) * (a * 9.0)))))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(Float64(b * -2.0) / c) + Float64(a * Float64(Float64(1.5 / b) + Float64(Float64(0.125 * Float64(c / Float64(b * Float64(b * b)))) * Float64(a * 9.0))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / (((b * -2.0) / c) + (a * ((1.5 / b) + ((0.125 * (c / (b * (b * b)))) * (a * 9.0)))));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision] + N[(a * N[(N[(1.5 / b), $MachinePrecision] + N[(N[(0.125 * N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.4%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 30.4%

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

3. Simplified 30.4%

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

6. Applied egg-rr 31.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -9}{\left(-\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot 3\right) + \frac{\frac{b}{\frac{0.3333333333333333}{a}}}{a}}}} \]

7. Taylor expanded in a around 0

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

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

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

   2. associate-*r/ N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

   7. +-commutative N/A

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

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

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

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

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

   12. associate-*r* N/A

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

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

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

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

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

   15. distribute-rgt-out N/A

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

   16. metadata-eval N/A

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

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

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

9. Simplified 94.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + \left(9 \cdot a\right) \cdot \left(\frac{c}{b \cdot \left(b \cdot b\right)} \cdot 0.125\right)\right)}} \]

10. Final simplification 94.1%

    \[\leadsto \frac{1}{\frac{b \cdot -2}{c} + a \cdot \left(\frac{1.5}{b} + \left(0.125 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}\right) \cdot \left(a \cdot 9\right)\right)} \]
11. Add Preprocessing

Alternative 6: 91.1% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 30.4%

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

3. Simplified 30.4%

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

6. Applied egg-rr 31.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -9}{\left(-\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot 3\right) + \frac{\frac{b}{\frac{0.3333333333333333}{a}}}{a}}}} \]

7. Taylor expanded in c around 0

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

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

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

   2. +-commutative N/A

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

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

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

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

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

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

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 90.9%

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

9. Simplified 90.9%

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

10. Final simplification 90.9%

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

Alternative 7: 91.1% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 30.4%

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

3. Simplified 30.4%

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

6. Applied egg-rr 31.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -9}{\left(-\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} \cdot 3\right) + \frac{\frac{b}{\frac{0.3333333333333333}{a}}}{a}}}} \]

7. Taylor expanded in a around 0

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

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

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

   2. associate-*r/ N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

   7. /-lowering-/.f64 90.8%

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

9. Simplified 90.8%

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

10. Final simplification 90.8%

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

Alternative 8: 81.4% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 30.4%

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

3. Simplified 30.4%

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

5. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 82.0%

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

7. Simplified 82.0%

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

Alternative 9: 81.2% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 30.4%

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

3. Simplified 30.4%

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

5. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 82.0%

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

7. Simplified 82.0%

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

   1. clear-num N/A

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

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

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

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

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

   4. *-lowering-*.f64 81.8%

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

9. Applied egg-rr 81.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{b}{c \cdot -0.5}}} \]
10. Step-by-step derivation

    1. associate-/r* N/A

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

    2. clear-num N/A

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

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

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

    4. /-lowering-/.f64 81.8%

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

11. Applied egg-rr 81.8%

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

Alternative 10: 81.2% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.4%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

   12. *-commutative N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 30.4%

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

3. Simplified 30.4%

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

5. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 82.0%

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

7. Simplified 82.0%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. /-lowering-/.f64 81.8%

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

9. Applied egg-rr 81.8%

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

10. Final simplification 81.8%

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

Reproduce

herbie shell --seed 2024159 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))