Cubic critical, narrow range

Percentage Accurate: 55.1% → 99.5%

Time: 21.5s

Alternatives: 18

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.1% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ \frac{\frac{t\_0}{a \cdot 3}}{b + \sqrt{t\_0 + b \cdot b}} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a -3.0))))
   (/ (/ t_0 (* a 3.0)) (+ b (sqrt (+ t_0 (* b b)))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * -3.0);
	return (t_0 / (a * 3.0)) / (b + sqrt((t_0 + (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
    real(8) :: t_0
    t_0 = c * (a * (-3.0d0))
    code = (t_0 / (a * 3.0d0)) / (b + sqrt((t_0 + (b * b))))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * -3.0);
	return (t_0 / (a * 3.0)) / (b + Math.sqrt((t_0 + (b * b))));
}

Python

def code(a, b, c):
	t_0 = c * (a * -3.0)
	return (t_0 / (a * 3.0)) / (b + math.sqrt((t_0 + (b * b))))

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * -3.0))
	return Float64(Float64(t_0 / Float64(a * 3.0)) / Float64(b + sqrt(Float64(t_0 + Float64(b * b)))))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = c * (a * -3.0);
	tmp = (t_0 / (a * 3.0)) / (b + sqrt((t_0 + (b * b))));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(t$95$0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 56.6%

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

   1. cancel-sign-sub-inv N/A

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

   2. *-commutative N/A

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

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

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. div-inv N/A

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

   10. flip-- N/A

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

   11. frac-times N/A

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

4. Applied egg-rr 58.0%

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

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

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f64 99.3%

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

6. Applied egg-rr 99.3%

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

   1. +-inverses N/A

      \[\leadsto \frac{c \cdot \left(a \cdot -3\right) + 0}{\left(b + \color{blue}{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}}\right) \cdot \left(3 \cdot a\right)} \]

   2. +-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

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

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

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

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

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

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

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

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

   9. *-commutative N/A

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

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

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

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

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

8. Applied egg-rr 99.5%

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

Alternative 2: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.2:\\ \;\;\;\;\frac{\frac{\sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + \left(\frac{\frac{a \cdot -0.375}{b}}{\frac{b}{c \cdot c}} + \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b \cdot t\_0} + \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{t\_0 \cdot t\_0}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.2)
     (/ (/ (- (sqrt (+ (* c (* a -3.0)) (* b b))) b) 3.0) a)
     (/
      (+
       (* c -0.5)
       (+
        (/ (/ (* a -0.375) b) (/ b (* c c)))
        (+
         (/ (* (* c c) (* c (* -0.5625 (* a a)))) (* b t_0))
         (/
          (* (* a (* a a)) (* -1.0546875 (* c (* c (* c c)))))
          (* t_0 t_0)))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.2) {
		tmp = ((sqrt(((c * (a * -3.0)) + (b * b))) - b) / 3.0) / a;
	} else {
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (b * b)
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-0.2d0)) then
        tmp = ((sqrt(((c * (a * (-3.0d0))) + (b * b))) - b) / 3.0d0) / a
    else
        tmp = ((c * (-0.5d0)) + ((((a * (-0.375d0)) / b) / (b / (c * c))) + ((((c * c) * (c * ((-0.5625d0) * (a * a)))) / (b * t_0)) + (((a * (a * a)) * ((-1.0546875d0) * (c * (c * (c * c))))) / (t_0 * t_0))))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.2) {
		tmp = ((Math.sqrt(((c * (a * -3.0)) + (b * b))) - b) / 3.0) / a;
	} else {
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.2:
		tmp = ((math.sqrt(((c * (a * -3.0)) + (b * b))) - b) / 3.0) / a
	else:
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.2)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(c * Float64(a * -3.0)) + Float64(b * b))) - b) / 3.0) / a);
	else
		tmp = Float64(Float64(Float64(c * -0.5) + Float64(Float64(Float64(Float64(a * -0.375) / b) / Float64(b / Float64(c * c))) + Float64(Float64(Float64(Float64(c * c) * Float64(c * Float64(-0.5625 * Float64(a * a)))) / Float64(b * t_0)) + Float64(Float64(Float64(a * Float64(a * a)) * Float64(-1.0546875 * Float64(c * Float64(c * Float64(c * c))))) / Float64(t_0 * t_0))))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b * (b * b);
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.2)
		tmp = ((sqrt(((c * (a * -3.0)) + (b * b))) - b) / 3.0) / a;
	else
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.2], N[(N[(N[(N[Sqrt[N[(N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] + N[(N[(N[(N[(a * -0.375), $MachinePrecision] / b), $MachinePrecision] / N[(b / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * c), $MachinePrecision] * N[(c * N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(-1.0546875 * N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.20000000000000001

   1. Initial program 82.9%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

   4. Applied egg-rr 82.9%

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

      1. un-div-inv N/A

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

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

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

   6. Applied egg-rr 83.0%

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

   if -0.20000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 48.6%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 95.4%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]

   8. Simplified 95.5%

      \[\leadsto \color{blue}{\frac{\left(\left(-0.5 \cdot c + \frac{-0.375 \cdot a}{b} \cdot \frac{c \cdot c}{b}\right) + \frac{\left(-0.5625 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}\right) + \frac{\left(-1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot {c}^{4}}{{b}^{6}}}{b}} \]

   10. Applied egg-rr 95.5%

       \[\leadsto \frac{\color{blue}{-0.5 \cdot c + \left(\frac{\frac{-0.375 \cdot a}{b}}{\frac{b}{c \cdot c}} + \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.2:\\ \;\;\;\;\frac{\frac{\sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + \left(\frac{\frac{a \cdot -0.375}{b}}{\frac{b}{c \cdot c}} + \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.6%

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

   1. cancel-sign-sub-inv N/A

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

   2. *-commutative N/A

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

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

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. div-inv N/A

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

   10. flip-- N/A

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

   11. frac-times N/A

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

4. Applied egg-rr 58.0%

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

5. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot -3\right)\\ \frac{t\_0}{\left(b + \sqrt{b \cdot b + t\_0}\right) \cdot \left(a \cdot 3\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c -3.0))))
   (/ t_0 (* (+ b (sqrt (+ (* b b) t_0))) (* a 3.0)))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * -3.0);
	return t_0 / ((b + sqrt(((b * b) + t_0))) * (a * 3.0));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = a * (c * (-3.0d0))
    code = t_0 / ((b + sqrt(((b * b) + t_0))) * (a * 3.0d0))
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (c * -3.0);
	return t_0 / ((b + Math.sqrt(((b * b) + t_0))) * (a * 3.0));
}

Python

def code(a, b, c):
	t_0 = a * (c * -3.0)
	return t_0 / ((b + math.sqrt(((b * b) + t_0))) * (a * 3.0))

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * -3.0))
	return Float64(t_0 / Float64(Float64(b + sqrt(Float64(Float64(b * b) + t_0))) * Float64(a * 3.0)))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = a * (c * -3.0);
	tmp = t_0 / ((b + sqrt(((b * b) + t_0))) * (a * 3.0));
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 / N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 56.6%

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

   1. cancel-sign-sub-inv N/A

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

   2. *-commutative N/A

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

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

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. div-inv N/A

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

   10. flip-- N/A

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

   11. frac-times N/A

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

4. Applied egg-rr 58.0%

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

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

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f64 99.3%

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

6. Applied egg-rr 99.3%

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

   1. +-inverses N/A

      \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \left(a \cdot -3\right) + 0\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, \color{blue}{\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right)\right)}\right), \mathsf{*.f64}\left(3, a\right)\right)\right) \]

   2. +-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

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

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

   6. *-lowering-*.f64 99.1%

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

8. Applied egg-rr 99.1%

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

9. Final simplification 99.1%

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

Alternative 5: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 7.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + \left(\frac{\frac{a \cdot -0.375}{b}}{\frac{b}{c \cdot c}} + \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b \cdot t\_0} + \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{t\_0 \cdot t\_0}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= b 7.5)
     (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
     (/
      (+
       (* c -0.5)
       (+
        (/ (/ (* a -0.375) b) (/ b (* c c)))
        (+
         (/ (* (* c c) (* c (* -0.5625 (* a a)))) (* b t_0))
         (/
          (* (* a (* a a)) (* -1.0546875 (* c (* c (* c c)))))
          (* t_0 t_0)))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= 7.5) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (b * b)
    if (b <= 7.5d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = ((c * (-0.5d0)) + ((((a * (-0.375d0)) / b) / (b / (c * c))) + ((((c * c) * (c * ((-0.5625d0) * (a * a)))) / (b * t_0)) + (((a * (a * a)) * ((-1.0546875d0) * (c * (c * (c * c))))) / (t_0 * t_0))))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= 7.5) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	tmp = 0
	if b <= 7.5:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (b <= 7.5)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(c * -0.5) + Float64(Float64(Float64(Float64(a * -0.375) / b) / Float64(b / Float64(c * c))) + Float64(Float64(Float64(Float64(c * c) * Float64(c * Float64(-0.5625 * Float64(a * a)))) / Float64(b * t_0)) + Float64(Float64(Float64(a * Float64(a * a)) * Float64(-1.0546875 * Float64(c * Float64(c * Float64(c * c))))) / Float64(t_0 * t_0))))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b * (b * b);
	tmp = 0.0;
	if (b <= 7.5)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.5], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] + N[(N[(N[(N[(a * -0.375), $MachinePrecision] / b), $MachinePrecision] / N[(b / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * c), $MachinePrecision] * N[(c * N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(-1.0546875 * N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.5

   1. Initial program 80.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 80.6%

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

   3. Simplified 80.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

      6. cancel-sign-sub-inv N/A

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

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

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

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

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 80.7%

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

   6. Applied egg-rr 80.7%

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

   if 7.5 < b

   1. Initial program 49.6%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 94.8%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]

   8. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{\left(\left(-0.5 \cdot c + \frac{-0.375 \cdot a}{b} \cdot \frac{c \cdot c}{b}\right) + \frac{\left(-0.5625 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}\right) + \frac{\left(-1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot {c}^{4}}{{b}^{6}}}{b}} \]

   10. Applied egg-rr 94.9%

       \[\leadsto \frac{\color{blue}{-0.5 \cdot c + \left(\frac{\frac{-0.375 \cdot a}{b}}{\frac{b}{c \cdot c}} + \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + \left(\frac{\frac{a \cdot -0.375}{b}}{\frac{b}{c \cdot c}} + \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 7.6:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + \left(\frac{\frac{a \cdot -0.375}{b}}{\frac{b}{c \cdot c}} + \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b \cdot t\_0} + \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{t\_0 \cdot t\_0}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= b 7.6)
     (* (/ (- (sqrt (+ (* c (* a -3.0)) (* b b))) b) a) 0.3333333333333333)
     (/
      (+
       (* c -0.5)
       (+
        (/ (/ (* a -0.375) b) (/ b (* c c)))
        (+
         (/ (* (* c c) (* c (* -0.5625 (* a a)))) (* b t_0))
         (/
          (* (* a (* a a)) (* -1.0546875 (* c (* c (* c c)))))
          (* t_0 t_0)))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= 7.6) {
		tmp = ((sqrt(((c * (a * -3.0)) + (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (b * b)
    if (b <= 7.6d0) then
        tmp = ((sqrt(((c * (a * (-3.0d0))) + (b * b))) - b) / a) * 0.3333333333333333d0
    else
        tmp = ((c * (-0.5d0)) + ((((a * (-0.375d0)) / b) / (b / (c * c))) + ((((c * c) * (c * ((-0.5625d0) * (a * a)))) / (b * t_0)) + (((a * (a * a)) * ((-1.0546875d0) * (c * (c * (c * c))))) / (t_0 * t_0))))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= 7.6) {
		tmp = ((Math.sqrt(((c * (a * -3.0)) + (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	tmp = 0
	if b <= 7.6:
		tmp = ((math.sqrt(((c * (a * -3.0)) + (b * b))) - b) / a) * 0.3333333333333333
	else:
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (b <= 7.6)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(c * Float64(a * -3.0)) + Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(Float64(Float64(c * -0.5) + Float64(Float64(Float64(Float64(a * -0.375) / b) / Float64(b / Float64(c * c))) + Float64(Float64(Float64(Float64(c * c) * Float64(c * Float64(-0.5625 * Float64(a * a)))) / Float64(b * t_0)) + Float64(Float64(Float64(a * Float64(a * a)) * Float64(-1.0546875 * Float64(c * Float64(c * Float64(c * c))))) / Float64(t_0 * t_0))))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b * (b * b);
	tmp = 0.0;
	if (b <= 7.6)
		tmp = ((sqrt(((c * (a * -3.0)) + (b * b))) - b) / a) * 0.3333333333333333;
	else
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.6], N[(N[(N[(N[Sqrt[N[(N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] + N[(N[(N[(N[(a * -0.375), $MachinePrecision] / b), $MachinePrecision] / N[(b / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * c), $MachinePrecision] * N[(c * N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(-1.0546875 * N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.5999999999999996

   1. Initial program 80.7%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

   4. Applied egg-rr 80.7%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

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

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

   6. Applied egg-rr 80.7%

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

   if 7.5999999999999996 < b

   1. Initial program 49.6%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 94.8%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]

   8. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{\left(\left(-0.5 \cdot c + \frac{-0.375 \cdot a}{b} \cdot \frac{c \cdot c}{b}\right) + \frac{\left(-0.5625 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}\right) + \frac{\left(-1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot {c}^{4}}{{b}^{6}}}{b}} \]

   10. Applied egg-rr 94.9%

       \[\leadsto \frac{\color{blue}{-0.5 \cdot c + \left(\frac{\frac{-0.375 \cdot a}{b}}{\frac{b}{c \cdot c}} + \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.6:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + \left(\frac{\frac{a \cdot -0.375}{b}}{\frac{b}{c \cdot c}} + \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathbf{if}\;b \leq 7.5:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + \left(\frac{\frac{a \cdot -0.375}{b}}{\frac{b}{c \cdot c}} + \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b \cdot t\_0} + \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{t\_0 \cdot t\_0}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (if (<= b 7.5)
     (* (/ 0.3333333333333333 a) (- (sqrt (+ (* b b) (* a (* c -3.0)))) b))
     (/
      (+
       (* c -0.5)
       (+
        (/ (/ (* a -0.375) b) (/ b (* c c)))
        (+
         (/ (* (* c c) (* c (* -0.5625 (* a a)))) (* b t_0))
         (/
          (* (* a (* a a)) (* -1.0546875 (* c (* c (* c c)))))
          (* t_0 t_0)))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= 7.5) {
		tmp = (0.3333333333333333 / a) * (sqrt(((b * b) + (a * (c * -3.0)))) - b);
	} else {
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * (b * b)
    if (b <= 7.5d0) then
        tmp = (0.3333333333333333d0 / a) * (sqrt(((b * b) + (a * (c * (-3.0d0))))) - b)
    else
        tmp = ((c * (-0.5d0)) + ((((a * (-0.375d0)) / b) / (b / (c * c))) + ((((c * c) * (c * ((-0.5625d0) * (a * a)))) / (b * t_0)) + (((a * (a * a)) * ((-1.0546875d0) * (c * (c * (c * c))))) / (t_0 * t_0))))) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double tmp;
	if (b <= 7.5) {
		tmp = (0.3333333333333333 / a) * (Math.sqrt(((b * b) + (a * (c * -3.0)))) - b);
	} else {
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	tmp = 0
	if b <= 7.5:
		tmp = (0.3333333333333333 / a) * (math.sqrt(((b * b) + (a * (c * -3.0)))) - b)
	else:
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	tmp = 0.0
	if (b <= 7.5)
		tmp = Float64(Float64(0.3333333333333333 / a) * Float64(sqrt(Float64(Float64(b * b) + Float64(a * Float64(c * -3.0)))) - b));
	else
		tmp = Float64(Float64(Float64(c * -0.5) + Float64(Float64(Float64(Float64(a * -0.375) / b) / Float64(b / Float64(c * c))) + Float64(Float64(Float64(Float64(c * c) * Float64(c * Float64(-0.5625 * Float64(a * a)))) / Float64(b * t_0)) + Float64(Float64(Float64(a * Float64(a * a)) * Float64(-1.0546875 * Float64(c * Float64(c * Float64(c * c))))) / Float64(t_0 * t_0))))) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = b * (b * b);
	tmp = 0.0;
	if (b <= 7.5)
		tmp = (0.3333333333333333 / a) * (sqrt(((b * b) + (a * (c * -3.0)))) - b);
	else
		tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.5], N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * -0.5), $MachinePrecision] + N[(N[(N[(N[(a * -0.375), $MachinePrecision] / b), $MachinePrecision] / N[(b / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * c), $MachinePrecision] * N[(c * N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(-1.0546875 * N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.5

   1. Initial program 80.7%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. unsub-neg N/A

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

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

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

   4. Applied egg-rr 80.7%

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

   if 7.5 < b

   1. Initial program 49.6%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 94.8%

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

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]

   8. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{\left(\left(-0.5 \cdot c + \frac{-0.375 \cdot a}{b} \cdot \frac{c \cdot c}{b}\right) + \frac{\left(-0.5625 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}\right) + \frac{\left(-1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot {c}^{4}}{{b}^{6}}}{b}} \]

   10. Applied egg-rr 94.9%

       \[\leadsto \frac{\color{blue}{-0.5 \cdot c + \left(\frac{\frac{-0.375 \cdot a}{b}}{\frac{b}{c \cdot c}} + \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5 + \left(\frac{\frac{a \cdot -0.375}{b}}{\frac{b}{c \cdot c}} + \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)} + \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \frac{c \cdot -0.5 + \left(\frac{\frac{a \cdot -0.375}{b}}{\frac{b}{c \cdot c}} + \left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)\right)}{b \cdot t\_0} + \frac{\left(a \cdot \left(a \cdot a\right)\right) \cdot \left(-1.0546875 \cdot \left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)\right)}{t\_0 \cdot t\_0}\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (/
    (+
     (* c -0.5)
     (+
      (/ (/ (* a -0.375) b) (/ b (* c c)))
      (+
       (/ (* (* c c) (* c (* -0.5625 (* a a)))) (* b t_0))
       (/ (* (* a (* a a)) (* -1.0546875 (* c (* c (* c c))))) (* t_0 t_0)))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    t_0 = b * (b * b)
    code = ((c * (-0.5d0)) + ((((a * (-0.375d0)) / b) / (b / (c * c))) + ((((c * c) * (c * ((-0.5625d0) * (a * a)))) / (b * t_0)) + (((a * (a * a)) * ((-1.0546875d0) * (c * (c * (c * c))))) / (t_0 * t_0))))) / b
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
}

Python

def code(a, b, c):
	t_0 = b * (b * b)
	return ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return Float64(Float64(Float64(c * -0.5) + Float64(Float64(Float64(Float64(a * -0.375) / b) / Float64(b / Float64(c * c))) + Float64(Float64(Float64(Float64(c * c) * Float64(c * Float64(-0.5625 * Float64(a * a)))) / Float64(b * t_0)) + Float64(Float64(Float64(a * Float64(a * a)) * Float64(-1.0546875 * Float64(c * Float64(c * Float64(c * c))))) / Float64(t_0 * t_0))))) / b)
end

MATLAB

function tmp = code(a, b, c)
	t_0 = b * (b * b);
	tmp = ((c * -0.5) + ((((a * -0.375) / b) / (b / (c * c))) + ((((c * c) * (c * (-0.5625 * (a * a)))) / (b * t_0)) + (((a * (a * a)) * (-1.0546875 * (c * (c * (c * c))))) / (t_0 * t_0))))) / b;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(c * -0.5), $MachinePrecision] + N[(N[(N[(N[(a * -0.375), $MachinePrecision] / b), $MachinePrecision] / N[(b / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * c), $MachinePrecision] * N[(c * N[(-0.5625 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(-1.0546875 * N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in a around 0

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

5. Simplified 89.9%

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

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]

8. Simplified 89.9%

   \[\leadsto \color{blue}{\frac{\left(\left(-0.5 \cdot c + \frac{-0.375 \cdot a}{b} \cdot \frac{c \cdot c}{b}\right) + \frac{\left(-0.5625 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}\right) + \frac{\left(-1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot {c}^{4}}{{b}^{6}}}{b}} \]

10. Applied egg-rr 90.0%

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

11. Final simplification 90.0%

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

Alternative 9: 88.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(a \cdot -3\right)}{\left(a \cdot b\right) \cdot 6 + c \cdot \left(-4.5 \cdot \frac{a \cdot a}{b} + \frac{c \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -3.375\right)}{b \cdot \left(b \cdot b\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* c (* a -3.0))
  (+
   (* (* a b) 6.0)
   (*
    c
    (+
     (* -4.5 (/ (* a a) b))
     (/ (* c (* (* a (* a a)) -3.375)) (* b (* b b))))))))

C

double code(double a, double b, double c) {
	return (c * (a * -3.0)) / (((a * b) * 6.0) + (c * ((-4.5 * ((a * a) / b)) + ((c * ((a * (a * a)) * -3.375)) / (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 * (a * (-3.0d0))) / (((a * b) * 6.0d0) + (c * (((-4.5d0) * ((a * a) / b)) + ((c * ((a * (a * a)) * (-3.375d0))) / (b * (b * b))))))
end function

Java

public static double code(double a, double b, double c) {
	return (c * (a * -3.0)) / (((a * b) * 6.0) + (c * ((-4.5 * ((a * a) / b)) + ((c * ((a * (a * a)) * -3.375)) / (b * (b * b))))));
}

Python

def code(a, b, c):
	return (c * (a * -3.0)) / (((a * b) * 6.0) + (c * ((-4.5 * ((a * a) / b)) + ((c * ((a * (a * a)) * -3.375)) / (b * (b * b))))))

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(a * -3.0)) / Float64(Float64(Float64(a * b) * 6.0) + Float64(c * Float64(Float64(-4.5 * Float64(Float64(a * a) / b)) + Float64(Float64(c * Float64(Float64(a * Float64(a * a)) * -3.375)) / Float64(b * Float64(b * b)))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * (a * -3.0)) / (((a * b) * 6.0) + (c * ((-4.5 * ((a * a) / b)) + ((c * ((a * (a * a)) * -3.375)) / (b * (b * b))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a * b), $MachinePrecision] * 6.0), $MachinePrecision] + N[(c * N[(N[(-4.5 * N[(N[(a * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(N[(c * N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * -3.375), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.6%

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

   1. cancel-sign-sub-inv N/A

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

   2. *-commutative N/A

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

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

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. div-inv N/A

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

   10. flip-- N/A

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

   11. frac-times N/A

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

4. Applied egg-rr 58.0%

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

5. Taylor expanded in c around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \color{blue}{\left(6 \cdot \left(a \cdot b\right) + c \cdot \left(\frac{-9}{2} \cdot \frac{{a}^{2}}{b} + \frac{-27}{8} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right)\right)}\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(6 \cdot \left(a \cdot b\right)\right), \color{blue}{\left(c \cdot \left(\frac{-9}{2} \cdot \frac{{a}^{2}}{b} + \frac{-27}{8} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right)\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\left(\left(a \cdot b\right) \cdot 6\right), \left(\color{blue}{c} \cdot \left(\frac{-9}{2} \cdot \frac{{a}^{2}}{b} + \frac{-27}{8} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(a \cdot b\right), 6\right), \left(\color{blue}{c} \cdot \left(\frac{-9}{2} \cdot \frac{{a}^{2}}{b} + \frac{-27}{8} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(b \cdot a\right), 6\right), \left(c \cdot \left(\frac{-9}{2} \cdot \frac{{a}^{2}}{b} + \frac{-27}{8} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, a\right), 6\right), \left(c \cdot \left(\frac{-9}{2} \cdot \frac{{a}^{2}}{b} + \frac{-27}{8} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, a\right), 6\right), \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{-9}{2} \cdot \frac{{a}^{2}}{b} + \frac{-27}{8} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, a\right), 6\right), \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{-9}{2} \cdot \frac{{a}^{2}}{b}\right), \color{blue}{\left(\frac{-27}{8} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right)}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, a\right), 6\right), \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-9}{2}, \left(\frac{{a}^{2}}{b}\right)\right), \left(\color{blue}{\frac{-27}{8}} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, a\right), 6\right), \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\left({a}^{2}\right), b\right)\right), \left(\frac{-27}{8} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, a\right), 6\right), \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\left(a \cdot a\right), b\right)\right), \left(\frac{-27}{8} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, a\right), 6\right), \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right)\right), \left(\frac{-27}{8} \cdot \frac{{a}^{3} \cdot c}{{b}^{3}}\right)\right)\right)\right)\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, a\right), 6\right), \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right)\right), \left(\frac{\frac{-27}{8} \cdot \left({a}^{3} \cdot c\right)}{\color{blue}{{b}^{3}}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(c, -3\right)\right)\right), \mathsf{*.f64}\left(b, b\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, a\right), 6\right), \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, a\right), b\right)\right), \mathsf{/.f64}\left(\left(\frac{-27}{8} \cdot \left({a}^{3} \cdot c\right)\right), \color{blue}{\left({b}^{3}\right)}\right)\right)\right)\right)\right) \]

7. Simplified 47.9%

   \[\leadsto \frac{\left(b \cdot b + a \cdot \left(c \cdot -3\right)\right) - b \cdot b}{\color{blue}{\left(b \cdot a\right) \cdot 6 + c \cdot \left(-4.5 \cdot \frac{a \cdot a}{b} + \frac{-3.375 \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot b\right)}\right)}} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\left(a \cdot \left(c \cdot -3\right) + b \cdot b\right) - b \cdot b}{\color{blue}{\left(b \cdot a\right)} \cdot 6 + c \cdot \left(\frac{-9}{2} \cdot \frac{a \cdot a}{b} + \frac{\frac{-27}{8} \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot b\right)}\right)} \]

   2. associate--l+ N/A

      \[\leadsto \frac{a \cdot \left(c \cdot -3\right) + \left(b \cdot b - b \cdot b\right)}{\color{blue}{\left(b \cdot a\right) \cdot 6} + c \cdot \left(\frac{-9}{2} \cdot \frac{a \cdot a}{b} + \frac{\frac{-27}{8} \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot b\right)}\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{a \cdot \left(-3 \cdot c\right) + \left(b \cdot b - b \cdot b\right)}{\left(b \cdot \color{blue}{a}\right) \cdot 6 + c \cdot \left(\frac{-9}{2} \cdot \frac{a \cdot a}{b} + \frac{\frac{-27}{8} \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot b\right)}\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\left(a \cdot -3\right) \cdot c + \left(b \cdot b - b \cdot b\right)}{\color{blue}{\left(b \cdot a\right)} \cdot 6 + c \cdot \left(\frac{-9}{2} \cdot \frac{a \cdot a}{b} + \frac{\frac{-27}{8} \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot b\right)}\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{c \cdot \left(a \cdot -3\right) + \left(b \cdot b - b \cdot b\right)}{\color{blue}{\left(b \cdot a\right)} \cdot 6 + c \cdot \left(\frac{-9}{2} \cdot \frac{a \cdot a}{b} + \frac{\frac{-27}{8} \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot b\right)}\right)} \]

   6. +-inverses N/A

      \[\leadsto \frac{c \cdot \left(a \cdot -3\right) + 0}{\left(b \cdot a\right) \cdot \color{blue}{6} + c \cdot \left(\frac{-9}{2} \cdot \frac{a \cdot a}{b} + \frac{\frac{-27}{8} \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot b\right)}\right)} \]

   7. +-rgt-identity N/A

      \[\leadsto \frac{c \cdot \left(a \cdot -3\right)}{\color{blue}{\left(b \cdot a\right) \cdot 6} + c \cdot \left(\frac{-9}{2} \cdot \frac{a \cdot a}{b} + \frac{\frac{-27}{8} \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot b\right)}\right)} \]

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

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

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \left(a \cdot -3\right)\right), \left(\color{blue}{\left(b \cdot a\right) \cdot 6} + c \cdot \left(\frac{-9}{2} \cdot \frac{a \cdot a}{b} + \frac{\frac{-27}{8} \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot b\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \left(\left(b \cdot a\right) \cdot \color{blue}{6} + c \cdot \left(\frac{-9}{2} \cdot \frac{a \cdot a}{b} + \frac{\frac{-27}{8} \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot b\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \mathsf{+.f64}\left(\left(\left(b \cdot a\right) \cdot 6\right), \color{blue}{\left(c \cdot \left(\frac{-9}{2} \cdot \frac{a \cdot a}{b} + \frac{\frac{-27}{8} \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{b \cdot \left(b \cdot b\right)}\right)\right)}\right)\right) \]

9. Applied egg-rr 87.3%

   \[\leadsto \color{blue}{\frac{c \cdot \left(a \cdot -3\right)}{\left(a \cdot b\right) \cdot 6 + c \cdot \left(-4.5 \cdot \frac{a \cdot a}{b} + \frac{c \cdot \left(\left(a \cdot \left(a \cdot a\right)\right) \cdot -3.375\right)}{b \cdot \left(b \cdot b\right)}\right)}} \]
10. Add Preprocessing

Alternative 10: 87.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return ((c * -0.5) / b) + (a * (((((c * (c * c)) * (a * -0.5625)) / (b * b)) + (-0.375 * (c * c))) / (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) + (a * (((((c * (c * c)) * (a * (-0.5625d0))) / (b * b)) + ((-0.375d0) * (c * c))) / (b * (b * b))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in a around 0

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

5. Simplified 89.9%

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

6. Taylor expanded in b around inf

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

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

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

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

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r* N/A

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

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

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

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

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

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

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

   11. unpow2 N/A

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

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

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

   13. unpow2 N/A

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

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

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

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

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

   16. unpow2 N/A

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

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

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

   18. cube-mult N/A

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

   19. unpow2 N/A

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

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

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

   21. unpow2 N/A

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

   22. *-lowering-*.f64 87.1%

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

8. Simplified 87.1%

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

9. Final simplification 87.1%

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

Alternative 11: 87.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in c around 0

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

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

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

   2. sub-neg N/A

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

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

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

5. Simplified 87.0%

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

6. Taylor expanded in b around inf

   \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \color{blue}{\left(\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{2}} + \frac{-3}{8} \cdot a}{{b}^{3}}\right)}\right), \mathsf{/.f64}\left(\frac{-1}{2}, b\right)\right)\right) \]
7. 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 87.0%

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

8. Simplified 87.0%

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

9. Final simplification 87.0%

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

Alternative 12: 81.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \left(a \cdot -3\right) + \left(b \cdot b - b \cdot b\right)}{a \cdot \left(\frac{\left(c \cdot a\right) \cdot -4.5}{b} + b \cdot 6\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+ (* c (* a -3.0)) (- (* b b) (* b b)))
  (* a (+ (/ (* (* c a) -4.5) b) (* b 6.0)))))

C

double code(double a, double b, double c) {
	return ((c * (a * -3.0)) + ((b * b) - (b * b))) / (a * ((((c * a) * -4.5) / b) + (b * 6.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((c * (a * (-3.0d0))) + ((b * b) - (b * b))) / (a * ((((c * a) * (-4.5d0)) / b) + (b * 6.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return ((c * (a * -3.0)) + ((b * b) - (b * b))) / (a * ((((c * a) * -4.5) / b) + (b * 6.0)));
}

Python

def code(a, b, c):
	return ((c * (a * -3.0)) + ((b * b) - (b * b))) / (a * ((((c * a) * -4.5) / b) + (b * 6.0)))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(c * Float64(a * -3.0)) + Float64(Float64(b * b) - Float64(b * b))) / Float64(a * Float64(Float64(Float64(Float64(c * a) * -4.5) / b) + Float64(b * 6.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((c * (a * -3.0)) + ((b * b) - (b * b))) / (a * ((((c * a) * -4.5) / b) + (b * 6.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(N[(N[(N[(c * a), $MachinePrecision] * -4.5), $MachinePrecision] / b), $MachinePrecision] + N[(b * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.6%

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

   1. cancel-sign-sub-inv N/A

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

   2. *-commutative N/A

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

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

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. sub-neg N/A

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

   9. div-inv N/A

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

   10. flip-- N/A

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

   11. frac-times N/A

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

4. Applied egg-rr 58.0%

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

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

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f64 99.3%

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \color{blue}{\left(a \cdot \left(\frac{-9}{2} \cdot \frac{a \cdot c}{b} + 6 \cdot b\right)\right)}\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-9}{2} \cdot \frac{a \cdot c}{b} + 6 \cdot b\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-9}{2} \cdot \frac{a \cdot c}{b}\right), \color{blue}{\left(6 \cdot b\right)}\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{\frac{-9}{2} \cdot \left(a \cdot c\right)}{b}\right), \left(\color{blue}{6} \cdot b\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-9}{2} \cdot \left(a \cdot c\right)\right), b\right), \left(\color{blue}{6} \cdot b\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{2}, \left(a \cdot c\right)\right), b\right), \left(6 \cdot b\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{2}, \left(c \cdot a\right)\right), b\right), \left(6 \cdot b\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{*.f64}\left(c, a\right)\right), b\right), \left(6 \cdot b\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{*.f64}\left(c, a\right)\right), b\right), \left(b \cdot \color{blue}{6}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 81.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{*.f64}\left(a, -3\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(b, b\right)\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-9}{2}, \mathsf{*.f64}\left(c, a\right)\right), b\right), \mathsf{*.f64}\left(b, \color{blue}{6}\right)\right)\right)\right) \]

9. Simplified 81.2%

   \[\leadsto \frac{c \cdot \left(a \cdot -3\right) + \left(b \cdot b - b \cdot b\right)}{\color{blue}{a \cdot \left(\frac{-4.5 \cdot \left(c \cdot a\right)}{b} + b \cdot 6\right)}} \]

10. Final simplification 81.2%

    \[\leadsto \frac{c \cdot \left(a \cdot -3\right) + \left(b \cdot b - b \cdot b\right)}{a \cdot \left(\frac{\left(c \cdot a\right) \cdot -4.5}{b} + b \cdot 6\right)} \]
11. Add Preprocessing

Alternative 13: 81.5% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in a around 0

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

5. Simplified 89.9%

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

6. Taylor expanded in b around inf

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

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

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

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

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

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

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

   4. associate-*r/ N/A

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

   5. associate-*r* N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

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

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

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

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

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

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 80.9%

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

8. Simplified 80.9%

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

9. Final simplification 80.9%

   \[\leadsto \frac{c \cdot -0.5 + \frac{a \cdot -0.375}{b} \cdot \frac{c \cdot c}{b}}{b} \]
10. Add Preprocessing

Alternative 14: 81.4% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in c around 0

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

   1. sub-neg N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

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

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

   7. +-commutative N/A

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

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

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

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

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

   14. associate-*r/ N/A

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

   15. associate-*l/ N/A

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

   16. associate-*r* N/A

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

   17. associate-*r/ N/A

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

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

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

   19. associate-/l* N/A

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

5. Simplified 80.8%

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

Alternative 15: 81.4% accurate, 7.7× speedup

Math

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

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(Float64(c * 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[(N[(c * N[(-0.5 + N[(N[(N[(c * a), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in a around 0

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

5. Simplified 89.9%

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

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]

8. Simplified 89.9%

   \[\leadsto \color{blue}{\frac{\left(\left(-0.5 \cdot c + \frac{-0.375 \cdot a}{b} \cdot \frac{c \cdot c}{b}\right) + \frac{\left(-0.5625 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot \left(c \cdot c\right)\right)}{{b}^{4}}\right) + \frac{\left(-1.0546875 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot {c}^{4}}{{b}^{6}}}{b}} \]

9. Taylor expanded in c around 0

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

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

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

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

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

    5. associate-*r/ N/A

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

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

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

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

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

    8. *-commutative N/A

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

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

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

    10. unpow2 N/A

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

    11. *-lowering-*.f64 80.7%

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

11. Simplified 80.7%

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

12. Final simplification 80.7%

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

Alternative 16: 81.4% 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 56.6%

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

3. Taylor expanded in c around 0

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

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

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

   2. sub-neg N/A

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

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

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

5. Simplified 87.0%

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

6. Taylor expanded in b around inf

   \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b}\right)}\right) \]
7. 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 80.7%

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

8. Simplified 80.7%

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

9. Final simplification 80.7%

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

Alternative 17: 64.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 63.7%

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

5. Simplified 63.7%

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

Alternative 18: 64.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in b around inf

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 63.7%

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

5. Simplified 63.7%

   \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b}} \]
6. 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 63.6%

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

7. Applied egg-rr 63.6%

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

8. Final simplification 63.6%

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

Reproduce

herbie shell --seed 2024147 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))