Cubic critical, medium range

Percentage Accurate: 31.7% → 99.4%

Time: 21.6s

Alternatives: 14

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: 31.7% 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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{c \cdot a}{\left(0 - b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (/ (* c a) (- (- 0.0 b) (sqrt (+ (* b b) (* c (* a -3.0)))))) a))

C

double code(double a, double b, double c) {
	return ((c * a) / ((0.0 - b) - sqrt(((b * b) + (c * (a * -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 = ((c * a) / ((0.0d0 - b) - sqrt(((b * b) + (c * (a * (-3.0d0))))))) / a
end function

Java

public static double code(double a, double b, double c) {
	return ((c * a) / ((0.0 - b) - Math.sqrt(((b * b) + (c * (a * -3.0)))))) / a;
}

Python

def code(a, b, c):
	return ((c * a) / ((0.0 - b) - math.sqrt(((b * b) + (c * (a * -3.0)))))) / a

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * a) / Float64(Float64(0.0 - b) - sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))))) / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * a) / ((0.0 - b) - sqrt(((b * b) + (c * (a * -3.0)))))) / a;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * a), $MachinePrecision] / N[(N[(0.0 - b), $MachinePrecision] - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 33.2%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 33.2%

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

3. Simplified 33.2%

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

6. Applied egg-rr 34.7%

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

   1. associate-/r* N/A

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

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

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

8. Applied egg-rr 33.9%

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

9. Taylor expanded in b around 0

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

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

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

    2. *-lowering-*.f64 99.2%

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

11. Simplified 99.2%

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

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

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

13. Applied egg-rr 99.5%

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

14. Final simplification 99.5%

    \[\leadsto \frac{\frac{c \cdot a}{\left(0 - b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)}}}{a} \]
15. Add Preprocessing

Alternative 2: 95.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := b \cdot t\_0\\ \left(\frac{a \cdot -0.375}{t\_0} + c \cdot \left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b \cdot t\_1} + \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot 6.328125\right)}{\left(b \cdot b\right) \cdot t\_1} \cdot \frac{c \cdot -0.16666666666666666}{a \cdot b}\right)\right) \cdot \left(c \cdot c\right) - \frac{c \cdot 0.5}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* b t_0)))
   (-
    (*
     (+
      (/ (* a -0.375) t_0)
      (*
       c
       (+
        (/ (* -0.5625 (* a a)) (* b t_1))
        (*
         (/ (* (* a a) (* (* a a) 6.328125)) (* (* b b) t_1))
         (/ (* c -0.16666666666666666) (* a b))))))
     (* c c))
    (/ (* c 0.5) b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = b * t_0;
	return ((((a * -0.375) / t_0) + (c * (((-0.5625 * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125)) / ((b * b) * t_1)) * ((c * -0.16666666666666666) / (a * b)))))) * (c * c)) - ((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
    real(8) :: t_0
    real(8) :: t_1
    t_0 = b * (b * b)
    t_1 = b * t_0
    code = ((((a * (-0.375d0)) / t_0) + (c * ((((-0.5625d0) * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125d0)) / ((b * b) * t_1)) * ((c * (-0.16666666666666666d0)) / (a * b)))))) * (c * c)) - ((c * 0.5d0) / b)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = b * t_0;
	return ((((a * -0.375) / t_0) + (c * (((-0.5625 * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125)) / ((b * b) * t_1)) * ((c * -0.16666666666666666) / (a * b)))))) * (c * c)) - ((c * 0.5) / b);
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	t_1 = b * t_0
	return ((((a * -0.375) / t_0) + (c * (((-0.5625 * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125)) / ((b * b) * t_1)) * ((c * -0.16666666666666666) / (a * b)))))) * (c * c)) - ((c * 0.5) / b)

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(b * t_0)
	return Float64(Float64(Float64(Float64(Float64(a * -0.375) / t_0) + Float64(c * Float64(Float64(Float64(-0.5625 * Float64(a * a)) / Float64(b * t_1)) + Float64(Float64(Float64(Float64(a * a) * Float64(Float64(a * a) * 6.328125)) / Float64(Float64(b * b) * t_1)) * Float64(Float64(c * -0.16666666666666666) / Float64(a * b)))))) * Float64(c * c)) - Float64(Float64(c * 0.5) / b))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = b * t_0;
	tmp = ((((a * -0.375) / t_0) + (c * (((-0.5625 * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125)) / ((b * b) * t_1)) * ((c * -0.16666666666666666) / (a * b)))))) * (c * c)) - ((c * 0.5) / b);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * t$95$0), $MachinePrecision]}, N[(N[(N[(N[(N[(a * -0.375), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(c * N[(N[(N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(c * -0.16666666666666666), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] - N[(N[(c * 0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 33.2%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 33.2%

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

3. Simplified 33.2%

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

5. Taylor expanded in c around 0

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

7. Simplified 94.8%

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

9. Applied egg-rr 95.1%

   \[\leadsto \color{blue}{\left(\frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)} + c \cdot \left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} + \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot 6.328125\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} \cdot \frac{c \cdot -0.16666666666666666}{a \cdot b}\right)\right) \cdot \left(c \cdot c\right) - \frac{c \cdot 0.5}{b}} \]
10. Add Preprocessing

Alternative 3: 95.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := b \cdot t\_0\\ c \cdot \left(\frac{c \cdot \left(a \cdot -0.375\right)}{t\_0} + \left(\left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b \cdot t\_1} + \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot 6.328125\right)}{\left(b \cdot b\right) \cdot t\_1} \cdot \frac{c \cdot -0.16666666666666666}{a \cdot b}\right) \cdot \left(c \cdot c\right) + \frac{-0.5}{b}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* b t_0)))
   (*
    c
    (+
     (/ (* c (* a -0.375)) t_0)
     (+
      (*
       (+
        (/ (* -0.5625 (* a a)) (* b t_1))
        (*
         (/ (* (* a a) (* (* a a) 6.328125)) (* (* b b) t_1))
         (/ (* c -0.16666666666666666) (* a b))))
       (* c c))
      (/ -0.5 b))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = b * t_0;
	return c * (((c * (a * -0.375)) / t_0) + (((((-0.5625 * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125)) / ((b * b) * t_1)) * ((c * -0.16666666666666666) / (a * b)))) * (c * 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
    real(8) :: t_0
    real(8) :: t_1
    t_0 = b * (b * b)
    t_1 = b * t_0
    code = c * (((c * (a * (-0.375d0))) / t_0) + ((((((-0.5625d0) * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125d0)) / ((b * b) * t_1)) * ((c * (-0.16666666666666666d0)) / (a * b)))) * (c * c)) + ((-0.5d0) / b)))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = b * t_0;
	return c * (((c * (a * -0.375)) / t_0) + (((((-0.5625 * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125)) / ((b * b) * t_1)) * ((c * -0.16666666666666666) / (a * b)))) * (c * c)) + (-0.5 / b)));
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	t_1 = b * t_0
	return c * (((c * (a * -0.375)) / t_0) + (((((-0.5625 * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125)) / ((b * b) * t_1)) * ((c * -0.16666666666666666) / (a * b)))) * (c * c)) + (-0.5 / b)))

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(b * t_0)
	return Float64(c * Float64(Float64(Float64(c * Float64(a * -0.375)) / t_0) + Float64(Float64(Float64(Float64(Float64(-0.5625 * Float64(a * a)) / Float64(b * t_1)) + Float64(Float64(Float64(Float64(a * a) * Float64(Float64(a * a) * 6.328125)) / Float64(Float64(b * b) * t_1)) * Float64(Float64(c * -0.16666666666666666) / Float64(a * b)))) * Float64(c * c)) + Float64(-0.5 / b))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = b * t_0;
	tmp = c * (((c * (a * -0.375)) / t_0) + (((((-0.5625 * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125)) / ((b * b) * t_1)) * ((c * -0.16666666666666666) / (a * b)))) * (c * c)) + (-0.5 / b)));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * t$95$0), $MachinePrecision]}, N[(c * N[(N[(N[(c * N[(a * -0.375), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[(N[(N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(c * -0.16666666666666666), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 33.2%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 33.2%

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

3. Simplified 33.2%

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

5. Taylor expanded in c around 0

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

7. Simplified 94.8%

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

9. Applied egg-rr 94.8%

   \[\leadsto c \cdot \color{blue}{\left(\frac{c \cdot \left(a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)} + \left(\left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} + \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot 6.328125\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} \cdot \frac{c \cdot -0.16666666666666666}{a \cdot b}\right) \cdot \left(c \cdot c\right) + \frac{-0.5}{b}\right)\right)} \]
10. Add Preprocessing

Alternative 4: 95.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := b \cdot t\_0\\ c \cdot \left(\frac{-0.5}{b} + c \cdot \left(\frac{a \cdot -0.375}{t\_0} + c \cdot \left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b \cdot t\_1} + \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot 6.328125\right)}{\left(b \cdot b\right) \cdot t\_1} \cdot \frac{c \cdot -0.16666666666666666}{a \cdot b}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* b t_0)))
   (*
    c
    (+
     (/ -0.5 b)
     (*
      c
      (+
       (/ (* a -0.375) t_0)
       (*
        c
        (+
         (/ (* -0.5625 (* a a)) (* b t_1))
         (*
          (/ (* (* a a) (* (* a a) 6.328125)) (* (* b b) t_1))
          (/ (* c -0.16666666666666666) (* a b)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = b * t_0;
	return c * ((-0.5 / b) + (c * (((a * -0.375) / t_0) + (c * (((-0.5625 * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125)) / ((b * b) * t_1)) * ((c * -0.16666666666666666) / (a * b))))))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    t_0 = b * (b * b)
    t_1 = b * t_0
    code = c * (((-0.5d0) / b) + (c * (((a * (-0.375d0)) / t_0) + (c * ((((-0.5625d0) * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125d0)) / ((b * b) * t_1)) * ((c * (-0.16666666666666666d0)) / (a * b))))))))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = b * t_0;
	return c * ((-0.5 / b) + (c * (((a * -0.375) / t_0) + (c * (((-0.5625 * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125)) / ((b * b) * t_1)) * ((c * -0.16666666666666666) / (a * b))))))));
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	t_1 = b * t_0
	return c * ((-0.5 / b) + (c * (((a * -0.375) / t_0) + (c * (((-0.5625 * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125)) / ((b * b) * t_1)) * ((c * -0.16666666666666666) / (a * b))))))))

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(b * t_0)
	return Float64(c * Float64(Float64(-0.5 / b) + Float64(c * Float64(Float64(Float64(a * -0.375) / t_0) + Float64(c * Float64(Float64(Float64(-0.5625 * Float64(a * a)) / Float64(b * t_1)) + Float64(Float64(Float64(Float64(a * a) * Float64(Float64(a * a) * 6.328125)) / Float64(Float64(b * b) * t_1)) * Float64(Float64(c * -0.16666666666666666) / Float64(a * b)))))))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	t_1 = b * t_0;
	tmp = c * ((-0.5 / b) + (c * (((a * -0.375) / t_0) + (c * (((-0.5625 * (a * a)) / (b * t_1)) + ((((a * a) * ((a * a) * 6.328125)) / ((b * b) * t_1)) * ((c * -0.16666666666666666) / (a * b))))))));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * t$95$0), $MachinePrecision]}, N[(c * N[(N[(-0.5 / b), $MachinePrecision] + N[(c * N[(N[(N[(a * -0.375), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(c * N[(N[(N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(c * -0.16666666666666666), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 33.2%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 33.2%

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

3. Simplified 33.2%

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

5. Taylor expanded in c around 0

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

7. Simplified 94.8%

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

9. Applied egg-rr 94.8%

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

10. Final simplification 94.8%

    \[\leadsto c \cdot \left(\frac{-0.5}{b} + c \cdot \left(\frac{a \cdot -0.375}{b \cdot \left(b \cdot b\right)} + c \cdot \left(\frac{-0.5625 \cdot \left(a \cdot a\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} + \frac{\left(a \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot 6.328125\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)} \cdot \frac{c \cdot -0.16666666666666666}{a \cdot b}\right)\right)\right) \]
11. Add Preprocessing

Alternative 5: 93.6% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (1.0d0 / a) / ((((-2.0d0) * (b / c)) + (a * ((1.5d0 * (1.0d0 / b)) - (a * ((c / (b * (b * b))) * (-1.125d0)))))) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 33.2%

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

3. Simplified 33.2%

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

   1. div-inv N/A

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

   2. flip3-- N/A

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

   3. clear-num N/A

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

   4. associate-/l/ N/A

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

   5. frac-times N/A

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

   6. div-inv N/A

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

6. Applied egg-rr 33.2%

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

7. Taylor expanded in a around 0

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

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

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

9. Simplified 93.3%

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

10. Final simplification 93.3%

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

Alternative 6: 93.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (0.3333333333333333d0 / ((((-0.6666666666666666d0) / (a / b)) + (c * ((0.5d0 / b) - (c * ((-0.375d0) / ((b * (b * b)) / a)))))) / c)) / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 33.2%

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

3. Simplified 33.2%

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

   1. flip3-- N/A

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

   2. clear-num N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

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

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

6. Applied egg-rr 33.2%

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

7. Taylor expanded in c around 0

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

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

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

9. Simplified 93.2%

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

    1. associate-/l/ N/A

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

    2. associate-/r* N/A

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

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

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

11. Applied egg-rr 93.3%

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

Alternative 7: 93.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 33.2%

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

3. Simplified 33.2%

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

   1. flip3-- N/A

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

   2. clear-num N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

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

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

6. Applied egg-rr 33.2%

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

7. Taylor expanded in c around 0

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

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

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

9. Simplified 93.2%

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

    1. div-inv N/A

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

    2. clear-num N/A

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

    3. associate-*l/ N/A

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

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

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

11. Applied egg-rr 93.3%

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

Alternative 8: 93.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 33.2%

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

3. Simplified 33.2%

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

   1. flip3-- N/A

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

   2. clear-num N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

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

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

6. Applied egg-rr 33.2%

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

7. Taylor expanded in c around 0

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

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

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

9. Simplified 93.2%

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

    1. associate-/l/ N/A

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

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

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

    3. associate-*l/ N/A

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

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

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

11. Applied egg-rr 93.2%

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

12. Final simplification 93.2%

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

Alternative 9: 93.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c * (0.3333333333333333d0 / (a * (((-0.6666666666666666d0) / (a / b)) + (c * ((0.5d0 / b) - (c * ((-0.375d0) / ((b * (b * b)) / a))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 33.2%

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

3. Simplified 33.2%

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

   1. flip3-- N/A

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

   2. clear-num N/A

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

   3. associate-/l/ N/A

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

   4. associate-/r* N/A

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

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

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

6. Applied egg-rr 33.2%

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

7. Taylor expanded in c around 0

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

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

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

9. Simplified 93.2%

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

    1. associate-/r/ N/A

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

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

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

11. Applied egg-rr 93.2%

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

12. Final simplification 93.2%

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

Alternative 10: 93.5% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b, c):
	return c * ((-0.5 / b) + (c * ((((-0.5625 * (c * (a * a))) / (b * b)) + (a * -0.375)) / (b * (b * b)))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 33.2%

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

3. Simplified 33.2%

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

5. Taylor expanded in c around 0

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

7. Simplified 94.8%

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

8. Taylor expanded in b around inf

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

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

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

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

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

   3. associate-*r/ N/A

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

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

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

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

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

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

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 93.1%

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

10. Simplified 93.1%

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

11. Final simplification 93.1%

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

Alternative 11: 90.6% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 33.2%

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

3. Simplified 33.2%

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

   1. div-inv N/A

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

   2. flip3-- N/A

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

   3. clear-num N/A

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

   4. associate-/l/ N/A

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

   5. frac-times N/A

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

   6. div-inv N/A

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

6. Applied egg-rr 33.2%

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

7. Taylor expanded in c around 0

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

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

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

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

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

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

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

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

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

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

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

   6. /-lowering-/.f64 89.7%

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

9. Simplified 89.7%

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

10. Taylor expanded in a around inf

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

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

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

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

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

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

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

    4. *-commutative N/A

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

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

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

    6. associate-*r/ N/A

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

    7. metadata-eval N/A

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

    8. /-lowering-/.f64 89.7%

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

12. Simplified 89.7%

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

Alternative 12: 90.4% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.2%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 33.2%

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

3. Simplified 33.2%

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

5. Taylor expanded in c around 0

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

7. Simplified 94.8%

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

8. Taylor expanded in b around inf

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. associate-*r/ N/A

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

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

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

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

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 89.4%

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

10. Simplified 89.4%

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

11. Final simplification 89.4%

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

Alternative 13: 81.1% 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 33.2%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 33.2%

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

3. Simplified 33.2%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}} \]
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 79.7%

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

7. Simplified 79.7%

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

Alternative 14: 80.9% 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 33.2%

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

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

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

   2. +-commutative N/A

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

   3. unsub-neg N/A

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

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

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

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

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

   6. sub-neg N/A

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

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

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

   12. *-commutative N/A

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

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

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

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

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

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

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f64 33.2%

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

3. Simplified 33.2%

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{3 \cdot a}} \]
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 79.7%

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

7. Simplified 79.7%

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

   1. *-commutative N/A

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

   2. associate-*l/ 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 79.5%

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

9. Applied egg-rr 79.5%

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

10. Final simplification 79.5%

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

Reproduce

herbie shell --seed 2024138 
(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)))